INTERNATIONAL UNION OF PURE AND APPLIED CHEMISTRY Physical and Biophysical Chemistry Division IUPAC Quantities, Units and SymD^s in Physical Chemistry Third Edition Prepared for publication by E. Richard Cohen Tomislav Cvitai^ Bertil Holmström Ian Mills Kozo Kuchitsu Franco Pavese Jürgen Stohner Herbert L. Strauss Jeremy G. Frey Roberto Marquardt Martin Quack Michio Takami Anders J Thor The first and second editions were prepared for publication by C j Ian Mills Tomislav Cvitas Klaus Homann Nikola Kallay Kozo Kuchitsu IUPAC 2007 RSC Publishing Professor E. Richard Cohen 17735, Corinthian Drive Encino, CA 91316-3704 USA email: ercohen@aol.com Professor Tom Cvitas University of Zagreb Department of Chemistry Horvatovac 102a HR-10000 Zagreb Croatia email: cvitas@chem.pmf.hr Professor Jeremy G. Frey University of Southampton Department of Chemistry Southampton, SO 17 1BJ United Kingdom email: j.g.frey@soton.ac.uk Professor Bertil Holmström Ulveliden 15 SE-41674 Göteborg Sweden email: bertilh@brikks.com Professor Kozo Kuchitsu Tokyo University of Agriculture and Technology Graduate School of BASE Naka-cho, Koganei Tokyo 184-8588 Japan email: kuchitsu@cc.tuat.ac.jp Professor Roberto Marquardt Laboratoire de Chimie Quantique Institut de Chimie Universitě Louis Pasteur 4, Rue Blaise Pascal F-67000 Strasbourg France email: roberto.marquardt@chimie.u-strasbg.fr Professor Ian Mills University of Reading Department of Chemistry Reading, RG6 6AD United Kingdom email: i.m.mills@rdg.ac.uk Professor Franco Pavese Instituto Nazionale di Ricerca Metrologica (INRIM) strada delle Cacce 73-91 1-10135 Torino Italia email: F.Pavese@imgc.cnr.it Professor Martin Quack ETH Zurich Physical Chemistry CH-8093 Zurich Switzerland email: Martin@Quack.ch or quack@ir.phys.chem.ethz.ch Professor Jürgen Stohner ZHAW Zürich University of Applied Sciences ICBC Institute of Chemistry & Biological Chemistry Campus Reidbach T, Einsiedlerstr. 31 CH-8820 Wädenswil Switzerland email: sthj@zhaw.ch or just@ir.phys.chem.ethz.ch Professor Michio Takami 3-10-8 Atago Niiza, Saitama 352-0021 Japan email: takamimy@d6.dion.ne.jp Dr. Anders J Thor Secretariat of ISO/TC 12 SIS Swedish Standards Institute Sankt Paulsgatan 6 SE-11880 Stockholm Sweden email: anders.j.thor@sis.se Professor Herbert L. Strauss University of California Berkeley, CA 94720-1460 USA email: hls@berkeley.edu Please cite this document as follows: E.R. Cohen, T. Cvitas, J.G. Frey, B. Holmström, K. Kuchitsu, R. Marquardt, I. Mills, F. Pavese, M. Quack, J. Stohner, H.L. Strauss, M. Takami, and A.J. Thor, "Quantities, Units and Symbols in Physical Chemistry", IUPAC Green Book, 3rd Edition, 2nd Printing, IUPAC & RSC Publishing, Cambridge (2008) ISBN: 978-0-85404-433-7 A catalogue record for this book is available from the British Library © International Union of Pure and Applied Chemistry 2007 Reprinted 2008 All rights reserved Apart from fair dealing for the purposes of research for non-commercial purposes or for private study, criticism or review, as permitted under the Qopyright, Designs and Patents Act 1988 and the Copyright and Related Rights Regulations 2003, this publication may not be reproduced, stored or transmitted, in any form or by any means, without the prior permission in writing of The Royal Society of Chemistry, or in the case of reproduction in accordance with the terms of licences issued by the Copyright Licensing Agency in the UK, or in accordance with the terms of the licences issued by the appropriate Reproduction Rights Organization outside the UK. 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Published by The Royal Society of Chemistry, Thomas Graham House, Science Park, Milton Road, Cambridge CB4^VF, UK Registered Charity Number 207890 Ffcr furiflll information see our web site at www.rsc.org CONTENTS v PREFACE ix HISTORICAL INTRODUCTION xi 1 PHYSICAL QUANTITIES AND UNITS 1 1.1 Physical quantities and quantity calculus.................>ls/^y. ... 3 1.2 Base quantities and derived quantities.................>^L • j..... ^ 1.3 Symbols for physical quantities and units ............../^/^^..... 5 1.3.1 General rules for symbols for quantities.................. ^ 1.3.2 General rules for symbols for units..................... 5 1.4 Use of the words "extensive", "intensive", "specific", and "molar" y^^^y......... 6 1.5 Products and quotients of physical quantities and units .... /C^'......... 7 1.6 The use of italic and Roman fonts for symbols in scientific jmbjW&Kcpis........ 7 2 TABLES OF PHYSICAL QUANTITIES 11 2.1 Space and time.................................... ^ 2.2 Classical mechanics....................A r............. 14 2.3 Electricity and magnetism.............../^^T^.............. 16 2.4 Quantum mechanics and quantum chemistry . . f^^mr................ 18 2.4.1 Ab initio Hartree-Fock self-consistent field theory (ab initio SCF)....... 20 2.4.2 Hartree-Fock-Roothaan SCF theory, using molecular orbitals expanded as linear combinations of atomic-orbital basis functions (LCAO-MO theory) . . 21 2.5 Atoms and molecules.............f^?*\J................... 22 2.6 Spectroscopy ...............#. ..................... 25 2.6.1 Symbols for angular momentum QBHerafcors and quantum numbers ...... 30 2.6.2 Symbols for symmetry operatorsaq^Mabels for symmetry species....... 31 2.6.3 Other symbols and conventions in optical spectroscopy............. 32 2.7 Electromagnetic radiation.....£-7......................... 34 2.7.1 Quantities and symbols concerned with the measurement of absorption intensity 38 2.7.2 Conventions for absorption intensities in condensed phases........... 40 2.8 Solid state....................................... 42 2.8.1 Symbols for planes ana^iuections in crystals .................. 44 2.9 Statistical thermodynamics^^^'.............................. 45 2.10 General chemistry . .............................. 47 2.10.1 Other symbols and conventions in chemistry .................. 49 2.11 Chemical thermodynamics................................. 56 2.11.1 Other symbols and conventions in chemical thermodynamics.......... 59 2.12 Chemical kinetics and photochemistry.......................... 63 2.12.1 Other symbols, terms, and conventions used in chemical kinetics ....... 68 2.13 Electrochemistry...................................... 70 2.13.1 Sign and ijotation conventions in electrochemistry................ 73 2.14 Colloid and surface chemistry............................... 77 2.14.1 S*fac«rstructure.................................. 79 2.15 Transport properties.................................... 81 2.15^~/^msport characteristic numbers: Quantities of dimension one........ 82 v 3 DEFINITIONS AND SYMBOLS FOR UNITS 83 3.1 The International System of Units (SI)........................ 85 3.2 Names and symbols for the SI base units......................L. . ) 86 3.3 Definitions of the SI base units.......................... . 87 3.4 SI derived units with special names and symbols ................^Y. T . 89 3.5 SI derived units for other quantities.....................j^^- . . 90 3.6 SI prefixes and prefixes for binary multiples...............A^^y. ... 91 3.7 Non-SI units accepted for use with the SI ...............•)..... ^2 3.8 Coherent units and checking dimensions...............^^^L....... 93 3.9 Fundamental physical constants used as units..............J'....... 94 3.9.1 Atomic units................................. 94 3.9.2 The equations of quantum chemistry expressed in terms of reduced quantities using atomic units............................. 95 3.10 Dimensionless quantities....................^^^)^/......... 97 3.10.1 Fractions (relative values, yields, and efficiencies) -.'^^............ 97 3.10.2 Deprecated usage............................... 97 3.10.3 Units for logarithmic quantities: neper, bel, and decibel............ 98 4 RECOMMENDED MATHEMATICAL SYMBOLS^ 101 4.1 Printing of numbers and mathematical symbols ( . . J.................103 4.2 Symbols, operators, and functions.......^^^r..................105 5 FUNDAMENTAL PHYSICAL CONSTANTS 109 6 PROPERTIES OF PARTICLES, ELEMENTS, AND NUCLIDES 113 6.1 Properties of selected particles...../^^^....................... 6.2 Standard atomic weights of the elements 2005 ...................... 117 6.3 Properties of nuclides.................................121 7 CONVERSION OF UNITS 129 7.1 The use of quantity calculus .............................131 7.2 Conversion tables for units . .............................135 7.3 The esu, emu, Gaussian, arj^l|ESmic unit systems in relation to the SI........143 7.4 Transformation of equaticVisvof electromagnetic theory between the ISQ(SI) and Gaussian forms . . ^. . . if................................146 ( 8 UNCERTAINTY 149 8.1 Reporting uncertainty for a single measured quantity..................151 8.2 Propagation of uncertainty for uncorrelated measurements...............153 8.3 Reporting uncertainties in terms of confidence intervals.................154 9 ABBREVIATIONS AND ACRONYMS 155 10 REFEREf^O 165 10.1 Primary sources.......................................167 10.2 Secondary sources .....................................169 vi 11 GREEK ALPHABET 12 INDEX OF SYMBOLS 13 SUBJECT INDEX NOTES PRESSURE CONVERSION FACTORS NUMERICAL ENERGY CONVERSION FACTORS IUPAC PERIODIC TABLE OF THE ELEMENTS o vii viii PREFACE The purpose of this manual is to improve the exchange of scientific information among the readers in different disciplines and across different nations. As the volume of scientific literatuWexpands, each discipline has a tendency to retreat into its own jargon. This book attempts to provide a readable compilation of widely used terms and symbols from many sources together with brief understandable definitions. This Third Edition reflects the experience of the contributors with the previous editions and we are grateful for the many thoughtful comments we have received. Most of the material in this book is "standard", but a few definitions and symbdls^n^not universally accepted. In such cases, we have attempted to list acceptable alternatives. Oj£/V[isliap Investigation Board (Phase I Report, November 10, 1999)1 found that the root q^jse'Ebr the loss of the MCO was "the failure to use metric units in the coding of the ground (based) software file". The impulse was reported in Imperial units of pounds (force)-seconds (lbf-s) r^merVhan in the metric units of Newton (force)-seconds (N-s). This caused an error of a factor of 4.45 and threw the satellite off course.2 We urge the users of this book always to define exp£f5tl|yThe terms, the units, and the symbols that they use. This edition has been compiled in machine-readable form by Martin Quack and Jürgen Stohner. The entire text of the manual will be available on the Internet some time after the publication of the book and will be accessible via the IUPAC web site^tjEp: //www. iupac. org. Suggestions and comments are welcome and may be addressed in care of the IUPAC Secretariat PO Box 13757 Research Triangle Park, NC 27709-3757, USA email: secretariat@iupac.org Corrections to the manual will be listed periodically. The book has been systematically brought up to date and new sections have been added. As in previous editions, the first chapter descHkes the use of quantity calculus for handling physical quantities and the general rules for\jÜie symbolism of quantities and units and includes an expanded description on the use of roman and italic fonts in scientific printing. The second chapter lists the symbols for quantities in a^lde range of topics used in physical chemistry. New parts of this chapter include a section on surface structure. The third chapter describes the use of the International System of units (SI) and of a few other systems such as atomic units. Chapter 4 outlines mathematical symbols and their use in print. Chapter 5 presents the 2006 revision of the fundamental physical constants, and Chapter 6 the properties of elementary particles, elements and nuclides. Conversion of units follows in Chapter 7, together with the equations of electricity and magnetism in their various forms. Chapter 8 is entirely new and outlines the treatment of uncertainty in physical measurements. Chapter 9 lists abbreviations and acronyms. Chapter 10 provides the references, and Chapter 11, the Greek alphabet. In Chapters 12 and 13, we end with indexes. The IUPAC Periodic Table of the Elements is shown on the inside back cover and the preceding pages list frequently used conversion factors for pressure and energy units. 1 The MCO report can be found at ftp://ftp.hq.nasa.gov/pub/pao/reports/1999/MCO_report.pdf. 2 Impulse (change of momentum) means here the time-integral of the force. ix Many people have contributed to this volume. The people most directly responsible are acknowledged in the Historical Introduction. Many of the members of IUPAC 1.1 have continued to make active contributions long after their terms on the Commission expired. We also wish to acknowledge the members of the other Commissions of the Physical Chemistry Division: Chemical Kinetics, Colloid and Surface Chemistry, Electrochemistry, Spectroscopy, and Thermodynamics, who have each contributed to the sections of the book that concern their various interests. We also thank all those who have contributed whom we have inadvertently my^ea out of these lists. Commission on Physicochemical Symbols, Terminology and Units Jeremy G. Frey and Herbert L. Strauss o x HISTORICAL INTRODUCTION The Manual of Symbols and Terminology for Physicochemical Quantities and Units [l^^l^Twhich this is a direct successor, was first prepared for publication on behalf of the Physical Chemistry Division of IUPAC by M. L. McGlashan in 1969, when he was chairman of the Commission on Physicochemical Symbols, Terminology and Units (1.1). He made a substantial contribution towards the objective which he described in the preface to that first edition as being "tl^ecu)e clarity and precision, and wider agreement in the use of symbols, by chemists in diffepw^countries, among physicists, chemists and engineers, and by editors of scientific journals". The second edition of that manual prepared for publication by M. A. Paul in 1973 [l.b], and the third edition prepared by D. H. Whiffen in 1976 [l.c], were revisions to take account of various developments in the Systeme International d'Unites (International System of Units, international ahlfrewation SI), and other developments in terminology. The first edition of Quantities, Units and Symbols in Physical Chemistry published in 1988 [2.a] was a substantially revised and extended version of the earlier manuals. The decision to embark on this project originally proposed by N. Kallay was taken at the IUPAC General Assembly at Leuven in 1981, when D. R. Lide was chairman of the Commission. The working party was established at the 1983 meeting in Lyngby, when K. Kuchitsu was chairman, and the project has received strong support throughout from all present and past members of the Commission 1.1 and other Physical Chemistry Commissions, particularly D. R. Lide, D. H. Whiffen, and N. Sheppard. The extensions included some of the material previously published in appendices [l.d— l.k]; all the newer resolutions and recommendations on unitg^^the Conference Generale des Poids et Mesures (CGPM); and the recommendations of the International Union of Pure and Applied Physics (IUPAP) of 1978 and of Technical Committee 12 of the International Organization for Standardization, Quantities, units, symbols, conversion factors (ISO/TC 12). The tables of physical quantities (Chapter 2) were extended to include defining equations and SI units for each quantity. The style was also slightly changed from being a book of rules towards a manual of advice and assistance for the day-to-day use of practidrryO^entists. Examples of this are the inclusion of extensive notes and explanatory text inserts in Chapter 2, the introduction to quantity calculus, and the tables of conversion factors between SI and non-SI units and equations in Chapter 7. The second edition (1993) was a revisfc^and extended version of the previous edition. The revisions were based on the recent resolutions of the CGPM [3]; the new recommendations by IUPAP [4]; the new international stafcdjFrls ISO 31 [5,6]; some recommendations published by other IUPAC commissions; and numerous comments we have received from chemists throughout the world. The revisions in the second edition were mainly carried out by Ian Mills and Tom Cvitas with substantial input from Robert Alberty, Kozo Kuchitsu, Martin Quack as well as from other members of the IUPAC Commission on Physicochemical Symbols, Terminology and Units. The manual has found wide acceptance in the chemical community, and various editions have been translated into Russian [2.c], Hungarian [2.d], Japanese [2.e], German [2.f], Romanian [2.g], Spanish [2.h], and Catalan [2.i]. Large parts of it have been reproduced in the 71st and subsequent editions of the Handbook of Chemistry and Physics published by CRC Press. The work on revisions of the second edition started immediately after its publication and between 1995 and 1997 it was discussed to change the title to "Physical-Chemical Quantities, Units and Symbols" ana? tOMjply rather complete revisions in various parts. It was emphasized that the book covers as much the field generally called "physical chemistry" as the field called "chemical physics". Indeed we consider the larger interdisciplinary field where the boundary between physics and chemistry has largely disappeared [10]. At the same time it was decided to produce the whole book as a text file in computer readable form to allow for future access directly by computer, some xi time after the printed version would be available. Support for this decision came from the IUPAC secretariat in the Research Triangle Park, NC (USA) (John W. Jost). The practical work on the revisions was carried out at the ETH Zürich, while the major input on this edition came from the group of editors listed now in full on the cover. It fits with the new structure of IUPAC that these are defined as project members and not only through membership in the commission. The basic structure of this edition was finally established at a working meeting of the project members in Engelberg, Switzerland in March 1999, while further revisions were discussed at the Berlin meeting (August 1999) and thereafter. In 2001 it was decided finally to use the old title. Ai^hr^edition the whole text and all tables have been revised, many chapters substantially. Thitwor\ was carried out mainly at ETH Zürich, where Jürgen Stohner coordinated the various contributions and corrections from the current project group members and prepared the print-ready electronic document. Larger changes compared to previous editions concern acomplete^dsubstantial update of recently available improved constants, sections on uncertainty in physical quantities, dimension-less quantities, mathematical symbols and numerous other sections. At the end of this historical survey we might refer also to what might be called the tradition of this manual. It is not the aim to present a list of recommendations in form of commandments. Rather we have always followed the principle that this manual should help the user in what may be called "good practice of scientific language". If there are several well established uses or conventions, these have been mentioned, giving preference to one, when this is useful, but making allowance for variety, if such variety is not harmful to clarity. In a few cases possible improvements of conventions or language are mentioned with appropriate reference, even if uncommon, but without specific recommendation. In those cases where certain common uees are deprecated, there are very strong reasons for this and the reader should follow the corresponding advice. The membership of the Commission during the period 1963 to 2006, during which the successive editions of this manual were ru^parelT, was as follows: Titular members Chairman: 1963-1967 GQjSddington (USA); 1967-1971 M.L. McGlashan (UK); 1971-1973 M.A. Paul (USA); 1973-1977 D.H. Whiffen (UK); 1977-1981 D.R. Lide Jr (USA); 1981-1985 K. Kuchitsu (Japan); 1985-1989 I.M. Mills (UK); 1989-1993 T. Cvitas (Croatia); 1993-1999 H.L. Strauss (USA); 2000-2007 J.G. Frey (UK). Secretary: 1963-lWTj. Brusset (France); 1967-1971 M.A. Paul (USA); 1971-1975 M. Fayard (France); 1975-1979 K.G. Weil (Germany); 1979-1983 I. Ansara (France); 1983-1985 N. Kallay (Croatia); 1985-1987 K.H. Homann (Germany); 1987-1989 T. Cvitas (Croatia); 1989-1991 I.M. Mills (UK); 1991-1997, 2001-2005 M. Quack (Switzerland); 1997-2001 B. Holmstrom (Sweden). Zürich, 2007 Martin Quack xn Other titular members 1975-1983 I. Ansara (France); 1965-1969 K.V. Astachov (Russia); 1963-1971 R.G. Bate^USA); 1963-1967 H. Brusset (France); 1985-1997 T. Cvitas (Croatia); 1963 F. Daniels (USA); 1^9-J981 D.H.W. den Boer (Netherlands); 1981-1989 E.T. Denisov (Russia); 1967-1975 M. FayardpW&ce); 1997-2005 J. Frey (UK); 1963-1965 J.I. Gerassimov (Russia); 1991-2001 B. Holmstrom (Sweden); 1979-1987 K.H. Homann (Germany); 1963-1971 W. Jaenicke (Germany); 1967-1971 F. Jellinek (Netherlands); 1977-1985 N. Kallay (Croatia); 1973-1981 V. Kello (Czechoslova«^ 1989-1997 I.V. Khudyakov (Russia); 1985-1987 W.H. Kirchhoff (USA); 1971-1979 J. Koefoed (Denmark); 1979-1987 K. Kuchitsu (Japan); 1971-1981 D.R. Lide Jr (USA); 1997-2001, 2006- R. Mar-quardt (France); 1963-1971 M.L. McGlashan (UK); 1983-1991 I.M. Mills/^FT963-1967 M. Milone (Italy); 1967-1973 M.A. Paul (USA); 1991-1999, 2006- F. Pavese (ifely); 1963-1967 K.J. Pedersen (Denmark); 1967-1975 A. Perez-Masia (Spain); 1987-1997 and 2002-2005 M. Quack (Switzerland); 1971-1979 A. Schuyff (Netherlands); 1967-1970 L.G. Sfli^^weden); 1989-1999 and 2002-2005 H.L. Strauss (USA); 1995-2001 M. Takami (Japan); 1987>Og^[ M. Tasumi (Japan); 1963-1967 G. Waddington (USA); 1981-1985 D.D. Wagman (USA>; 1^/^979 K.G. Weil (Germany); 1971-1977 D.H. Whiffen (UK); 1963-1967 E.H. Wiebenga Qrf^rfands). Associate members 1983-1991 R.A. Alberty (USA); 1983-1987 I. Ansara (FrancejQ^79*-1991 E.R. Cohen (USA); 1979-1981 E.T. Denisov (Russia); 1987-1991 G.H. Findenegg (Germany); 1987-1991 K.H. Homann (Germany); 1971-1973 W. Jaenicke (Germany); 198fr-Ji8e'N. Kallay (Croatia); 1987-1989 and 1998-1999 I.V. Khudyakov (Russia); 1979-1980 J. Koefoed (Denmark); 1987-1991 K. Kuchitsu (Japan); 1981-1983 D.R. Lide Jr (USA); 1971-1979 M.L. McGlashan (UK); 1991-1993 I.M. Mills (UK); 1973-1981 M.A. Paul (USA); 1999-2005 F^aVse (Italy); 1975-1983 A. Perez-Masia (Spain); 1997-1999 M. Quack (Switzerland); 1979-1987 A. Schuyff (Netherlands); 1963-1971 S. Seki (Japan); 2000-2001 H.L. Strauss (USA); 1991-1995 M. Tasumi (Japan); 1969-1977 J. Terrien (France); 1994-2001 A J Thor (Sweden); 1975-19*78^fallena (Spain); 1967-1969 G. Waddington (USA); 1979-1983 K.G. Weil (Germany); 1977-1985 D.H. Whiffen (UK). National representatives Numerous national representatives have served on the commission over many years. We do not provide this long list here. 1 v O xiii xiv PHYSICAL QUANTITIES AND UNITS 1.1 PHYSICAL QUANTITIES AND QUANTITY CALCULUS The value of a physical quantity Q can be expressed as the product of a numerical value {Q} and a unit [Q] Q = {Q} [Q] (i) Neither the name of the physical quantity, nor the symbol used to denote it, implies a particular choice of unit (see footnote 1, p. 4). Physical quantities, numerical values, and units may all be manipulated by the ordinary rules of algebra. Thus we may write, for example, for the wavelength A of one of the yellow sodium lines A = 5.896 x 1(T7 m = 589.6 nm (2) where m is the symbol for the unit of length called the metre (or meter, sel^Sections 3.2 and 3.3, p. 86 and 87), nm is the symbol for the nanometre, and the units metre and nanometre are related by 1 nm = 10 9 m or nm = 10 9 rg (3) The equivalence of the two expressions for A in Equation (2) follow^ at^bnce when we treat the units by the rules of algebra and recognize the identity of 1 nm anc^Mw9 m in Equation (3). The wavelength may equally well be expressed in the form A/m = 5.896 x 10 -7 (4) or A/nm = 589.6 \ (5) It can be useful to work with variables that are defined by dividing the quantity by a particular unit. For instance, in tabulating the numerical values of physical quantities or labeling the axes of graphs, it is particularly convenient to use the quotient of a physical quantity and a unit in such a form that the values to be tabulated are numerical values, as in Equations (4) and (5). Example ln(p/MPa) a + b/T = a + 6'(103 K/T) (6) T/K 103 K/T p/MPa ln(p/MPa) 216.55 273.15 304.19 4.6179 3.6610 3.2874 0.5180 3.4853 7.3815 -0.6578 1.2486 1.9990 4.0 103 K/T Algebraically equivalent forms may be used in place of 103 K/T, such as kK/T or 103 (T/K)-1. Equations between numerical values depend on the choice of units, whereas equations between quantities have the advantage of being independent of this choice. Therefore the use of equations between quantities should generally be preferred. The method described here for handling physical quantities and their units is known as quantity calculus [11-13]. It is recommended for use throughout science and technology. The use of quantity calculus does not imply any particular choice of units; indeed one of the advantages of quantity calculus is that it makes changes between units particularly easy to follow. Further examples of the use of quantity calculus are given in Section 7.1, p. 131, which is concerned with the problems of transforming from one set of units to another. 3 1.2 BASE QUANTITIES AND DERIVED QUANTITIES By convention physical quantities are organized in a dimensional system built upon seven base quantities, each of which is regarded as having its own dimension. These base quantities in the International System of Quantities (ISQ) on which the International System of units (SI) is based, and the principal symbols used to denote them and their dimensions are as follows: Base quantity Symbol for quantity Symbol for dimension length I L mass m Ml time t T electric current I i thermodynamic temperature T e amount of substance n N luminous intensity j All other quantities are called derived quantities and are regarded as having dimensions derived algebraically from the seven base quantities by multiplication andyfcWision. Example dimension of energy is equal to dimension of M Lj^}2 This can be written with the symbol dim for dimension (see footnote 1, below) dim(£0 = dim(m • I2 ■ £~2) = M L2 T 2 The quantity amount of substance is of special importance to chemists. Amount of substance is proportional to the number of specified elementary empties of the substance considered. The proportionality factor is the same for all substances; its reciprocal is the Avogadro constant (see Section 2.10, p. 47, Section 3.3, p. 88, and Chapter 5, p. 111). The SI unit of amount of substance is the mole, defined in Section 3.3, p. 88. The physical quantity "amount of substance" should no longer be called "number of moles", just as the physical quantity "mass" should not be called "number of kilograms". The name "amount of substance", sometimes also called "chemical amount", may often be usefully abbreviated to the single word "amount", particularly in such phrases as "amount concentration" (see footnote 2, below), and "amount of N2". A possible name for international usage has been suggested: "enplethy" [10] (from Greek, similar to enthalpy and entropy). The number and choice of base quantities is pure convention. Other quantities could be considered to be more fundamental, sj^h as electric charge Q instead of electric current /. Q = jldt (7) However, in the ISQ, electric current is chosen as base quantity and ampere is the SI base unit. In atomic and molecular physics, the so-called atomic units are useful (see Section 3.9, p. 94). 1 The symbol [Q] was formerly used for dimension of Q, but this symbol is used and preferred for unit of Q. 2 The Clinical Chemistry Division of IUPAC recommended that "amount-of-substance concentration" be abbreviated "substance concentration" [14]. 4 1.3 SYMBOLS FOR PHYSICAL QUANTITIES AND UNITS [5.a] A clear distinction should be drawn between the names and symbols for physical quantities, and the names and symbols for units. Names and symbols for many quantities are given in Chapter 2, p. 11; the symbols given there are recommendations. If other symbols are used they should be clearly defined. Names and symbols for units are given in Chapter 3, p. 83; the symbols for units listed there are quoted from the Bureau International des Poids et Mesures (BIPM) and/fr% mandatory. 1.3.1 General rules for symbols for quantities The symbol for a physical quantity should be a single letter (see footnote 1, below) of the Latin or Greek alphabet (see Section 1.6, p. 7). Capital or lower case letters may both be used. The letter should be printed in italic (sloping) type. When necessary the symboNufoy be modified by subscripts and superscripts of specified meaning. Subscripts and superscripts that are themselves symbols for physical quantities or for numbers should be printed in italic type; other subscripts and superscripts should be printed in Roman (upright) type. Examples Cp for heat capacity at constant pressure Pi for partial pressure of the ith substange\ but Cb for heat capacity of substance B /iBa for chemical potential of substance B in phase a -Ek for kinetic energy fiT for relative permeability ATH^ for standard reaction enthalpy Vm for molar volume Aiq for decadic absorbance The meaning of symbols for physical quantities may be further qualified by the use of one or more subscripts, or by information contained in parentle^gsV' Examples AflS^(HgCl2, cr, 25 °C) = -154.3 J K-1 moL1 Vectors and matrices may be printed in bold-face italic type, e.g. A, a. Tensors may be printed in bold-face italic sans serif type, e.g. S, T. Vectors may alternatively be characterized by an arrow, A, a and second-rank tensors by a double arrow, S,T. 1.3.2 General rules for symbols for units Symbols for units should be printed in Roman (upright) type. They should remain unaltered in the plural, and should not be followed by a full stop except at the end of a sentence. Example r = 10 cm, not cm. or cms. An exception is made for certain characteristic numbers or "dimensionless quantities" used in the study of transport processes for which the internationally agreed symbols consist of two letters (see Section 2.15.1, p. 82). Example Reynolds number, Re; another example is pH (see Sections 2.13 and 2.13.1 (viii), p. 70 and 75). When such symbols appear as factors in a product, they should be separated from other symbols by a space, multiplication sign, or parentheses. 5 Symbols for units shall be printed in lower case letters, unless they are derived from a personal name when they shall begin with a capital letter. An exception is the symbol for the litre which may be either L or 1, i.e. either capital or lower case (see footnote 2, below). Examples m (metre), s (second), but J (joule), Hz (hertz) Decimal multiples and submultiples of units may be indicated by the use of prefixes as^fefTned in Section 3.6, p. 91. Examples nm (nanometre), MHz (megahertz), kV (kilovolt) 1.4 USE OF THE WORDS "EXTENSIVE", "INTENSIVE", "SPECIFIC", AND "MOLAR" A quantity that is additive for independent, noninteracting subsystems is called extensive; examples are mass m, volume V, Gibbs energy G. A quantity that is independent of the extent of the system is called intensive; examples are temperature T, pressure p, chemical potential (partial molar Gibbs energy) p. The adjective specific before the name of an extensive quantitjTisfused to mean divided by mass. When the symbol for the extensive quantity is a capital le^ter^ the symbol used for the specific quantity is often the corresponding lower case letter. Examples volume, V, and specific volume, v = V/m = l/p (where p is mass density); heat capacity at constant pressure, Cp, and specific heat capacity at constant pressure, cp = Cp/m ISO [5.a] and the Clinical Chemistry Division of IUPAC recommend systematic naming of physical quantities derived by division with mass, volume, area, and length by using the attributes massic or specific, volumic, areic, and lineic, respectively. In addition the Clinical Chemistry Division of IUPAC recommends the use of the attjj^tHje entitic for quantities derived by division with the number of entities [14]. Thus, for exa^nle^the specific volume could be called massic volume and the surface charge density would be areic charge. The adjective molar before the name t)f^llyextensive quantity generally means divided by amount of substance. The subscript m on the symbol for the extensive quantity denotes the corresponding molar quantity. Examples volume, V molar volume, Vm = V/n (Section 2.10, p. 47) enthalpy, H molar eWialpy, Hm = H/n If the name enplethy (see Section L2Npr 4) is accepted for "amount of substance" one can use enplethic volume instead of molar volume, for instance. The word "molar" violates the principle that the name of the quantity should not be mixed with the name of the unit (mole in this case). The use of enplethic resolves this problemiflt is sometimes convenient to divide all extensive quantities by amount of substance, so that all quantities become intensive; the subscript m may then be omitted if this convention is stated and there is no risk of ambiguity. (See also the symbols recommended for partial molar quantities in Section 2.11, p. 57, and in Section 2.11.1 (iii), p. 60.) There are a few cases where the adjective molar has a different meaning, namely divided by amount-of-substance concentration. Examples absorptionTcoefficient, a molar absorption coefficient, e = a/c (see Section 2.7, note 22, p. 37) conductivity, re molar conductivity, A = re/c (see Section 2.13, p. 73) 2 However, only the lower case 1 is used by ISO and the International Electrotechnical Commission (IEC). 6 1.5 PRODUCTS AND QUOTIENTS OF PHYSICAL QUANTITIES AND UNITS ^ Products of physical quantities may be written in any of the ways a b or ab or a • b or a x b and similarly quotients may be written a lb or - or by writing the product of a and b 1 , e.g. ab 1 b ^^^^^ Examples F = ma, p = nRT/V Not more than one solidus (/) shall be used in the same expression unless parentheses are used to eliminate ambiguity. Example (a/b)/c or a/(b/c) (in general different), not a/b/c In evaluating combinations of many factors, multiplication written»%i^iout a multiplication sign takes precedence over division in the sense that a/bc is interpreted as a/'(be) and not as (a/b)c; however, it is necessary to use parentheses to eliminate ambiguity under all circumstances, thus avoiding expressions of the kind a/bed etc. Furthermore, a/b + c is interpreted as (a/b) +c and not as a/(b + c). Again, the use of parentheses is recommended (required for a/(b + c)). Products and quotients of units may be written in a similar way, except that the cross (x) is not used as a multiplication sign between units. When a product of units is written without any multiplication sign a space shall be left between the unit symbols. Example 1 N = 1 m kg s~2 = 1 m-kg-s~2 = 1 m kg/s2, not 1 mkgs~2 1.6 THE USE OF ITALIC AND ROMAN FONTS FOR SYMBOLS IN SCIENTIFIC PUBLICATIONS Scientific manuscripts should follow the accepted conventions concerning the use of italic and Roman fonts for symbols. An italic font is generally used for emphasis in running text, but it has a quite specific meaning when used for symbols in scientific text and equations. The following summary is intended to help in the correct use of italiCT^ts in preparing manuscript material. The general rules concerning the use of italic (sloping) font or Roman (upright) font are presented in Section 1.3.2, p. 5 and in Section 4.1, p. 103 in relation to mathematical symbols and operators. These rules are also presented in the International Standards ISO 31 (successively being replaced by ISO/IEC 80000) [5], ISO 1000 [6], and in the SI Brochure [3]. 2. The overall rule is that symbols representing physical quantities or variables are italic, but symbols representing units, mathematical constants, or labels, are roman. Sometimes there may seem to be doubt as to whether a symbol represents a quantity or has some other meaning (such as label): E^§^d rule is that quantities, or variables, may have a range of numerical values, but lajpelscannot. Vectors, tensors and matrices are denoted using a bold-face (heavy) font, but they shall be italic since they are quantities. ExamplesQflk Planck constant h = 6.626 068 96(33) xl0~34 J s. OThe electric field strength E has components Ex,Ey, and Ez. |The mass of my pen is m = 24 g = 0.024 kg. 7 3. The above rule applies equally to all letter symbols from both the Greek and the Latin alphabet, although some authors resist putting Greek letters into italic. Example When the symbol fi is used to denote a physical quantity (such as permegcilityj or reduced mass) it should be italic, but when it is used as a prefix in a^ajr such as microgram, jjtg, or when it is used as the symbol for the muon, |jl (see paragraph 5 below), it should be roman. 4. Numbers, and labels, are roman (upright). Examples The ground and first excited electronic state of the CH2 molecule ake denoted .. .(2ai)2(lb2)2(3ai)1(lbi)1, X 3Bi, and .. .(2ai)2(lb2)2(3ai)2, a 1A1, respectively. The it-electron configuration and symmetry of the benzene molecule in its ground state are denoted: .. .(a2U)2(eig)4, X 1Aig. All these symbols are labels and are roman. 5. Symbols for elements in the periodic system shall be roman. Similarly the symbols used to represent elementary particles are always roman. (See, however, paragraph 9 below for use of italic font in chemical-compound names.) Examples H, He, Li, Be, B, C, N, O, F, Ne, ... for atoms; e for the electron, p for the proton, n for the neutron, |jl for the muon, a forTB^alpha particle, etc. 6. Symbols for physical quantities are single, or exceptionally two letters of the Latin or Greek alphabet, but they are frequently supplemented with subscripts, superscripts or information in parentheses to specify further the quantity. Further symbols used in this way are either italic or roman depending on what they represent. Examples H denotes enthalpy, but Hm denotes moiar enthalpy (m is a mnemonic label for molar, and is therefore roman). Cp and Cy denote the heat capacity at constant pressure p and volume V, respectively (note the roman m but italic p and V). The chemical potential of argon might be denoted /iAr or /x(Ar), but the chemical potential of the z^LComponent in a mixture would be denoted fii, where i is italic because it is a variable subscript. 7. Symbols for mathematical operators are always roman. This applies to the symbol A for a difference, 8 for an infinitesimal variation, d for an infinitesimal difference (in calculus), and to capital S and n for sum!n%fion and product signs, respectively. The symbols k (3.141 592...), e (2.718 281...^ase of natural logarithms), i (square root of minus one), etc. are always roman, as are the symbols for specified functions such as log (lg for logio, m for loge, or lb for log2), exp, sin, cos, tan, erf, div, grad, rot, etc. The particular operators grad and rot and the corresponding symbols V for grad, Vx for rot, and V- for div are printed in bold-face to indicate the vector or tensor character following [5.k]. Some of these letters, e.g. e for elementary charge, are also sometimes used to represent physical quantities; then of course they shall be italic, to distinguish them from the corresponding mathematical symbol. Examples AH = H(final) — //"(initial); (dp/dt) used for the rate of change of pressure; 8x used to denote an infinitesimal variation of x. But for a damped linear oscillator the amplitude F as a function of time t might be expressed by the equation F = Fq exp(—5t) sin(tjt) where 5 is the decay coefficient (SI unit: Np/s) and u is the angular frequency (SI unit: rad/s). Note the use of roman ^8 foiythe operator in an infinitesimal variation of x, 8x, but italic 5 for the ^Ie3ay coefficient in the product 5t. Note that the products 5t and ut are both dimensionless, but are described as having the unit neper (Np = 1) and radian (rad = 1), respectively. O1 8 8. The fundamental physical constants are always regarded as quantities subject to measurement (even though they are not considered to be variables) and they should accordingly always be italic. Sometimes fundamental physical constants are used as though they were units, but they are still given italic symbols. An example is the hartree, (see Section 3.9.1, p. 95). However, the electronvolt, eV, the dalton, Da, or the unified atomic mass unit, u, and the astronomical unit, ua, have been recognized as units by the Comite International des Poids et Mesures (CIPM) of the BIPM and they are accordingly given Roman symbols. Examples cq for the speed of light in vacuum, me for the electron mass, h for the Planck constant, N\ or L for the Avogadro constant, e for the elementary charge, ao for the Bohr radius, etc. The electronvolt 1 eV = e-1 V = 1.602 176 487(40) x 10"19 J% 9. Greek letters are used in systematic organic, inorganic, macromolecular, and biochemical nomenclature. These should be roman (upright), since they are^o^eymbols for physical quantities. They designate the position of substitution in side#ch^i^f^igating-atom attachment and bridging mode in coordination compounds, end groups in structure-based names for macromolecules, and stereochemistry in carbohydrates and natural products. Letter symbols for elements are italic when they are locants in chemical-comj^md names indicating attachments to heteroatoms, e.g. 0-, N-, S-, and P-. The italic symbol H denotes indicated or added hydrogen (see reference [15]). Examples oc-ethylcyclopentaneacetic acid ß-methyl-4-propylcyclohexaneethanol tetracarbonyl(T)4-2-methylidenepropane-l,3-diyl)chromium a- (trichloromethyl)-co-chloropoly (1,4-phenylenemethylene) a-D-glucopyranose 5oc-androstan-3ß-ol ./V-methylbenzamide O-ethyl hexanethioate 3-ff-pyrrole naphthalen- 2(1 H) -one o 9 10 2 TABLES OF PHYSICAL QUANTITIES The following tables contain the internationally recommended names and symbols for the physical quantities most likely to be used by chemists. Further quantities and symbols may be found in recommendations by IUPAP [4] and ISO [5]. Although authors are free to choose any symbols they wish for the quantities tnfcw discuss, provided that they define their notation and conform to the general rules indicated in Chapter 1, it is clearly an aid to scientific communication if we all generally follow a standard notation. The symbols below have been chosen to conform with current usage and to minimize conflict so far as possible. Small variations from the recommended symbols may often be desirable in particular situations, perhaps by adding or modifying subscripts or superscripts, or by th^^ternative use of upper or lower case. Within a limited subject area it may also be possibk^o simplify notation, for example by omitting qualifying subscripts or superscripts, without introducing ambiguity. The notation adopted should in any case always be defined. Major deviations from the recommended symbols should be particularly carefully defined. The tables are arranged by subject. The five columns in each table give the name of the quantity, the recommended symbol(s), a brief definition, the symbol for the coherent SI unit (without multiple or submultiple prefixes, see Section 3.6, p. 91), and note references. When two or more symbols are recommended, commas are used to separate symbols that are equally acceptable, and symbols of second choice are put in parentheses. A semicolon is used to separate symbols of slightly different quantities. The definitions are given primarily for identification purposes and are not necessarily complete; they should be regarded as useful relations rather than formal definitions. For some of the quantities listed in this chapter, the definitions given in various IUPAC documents are collected in [16]. Useful definitions of physical quantities in physical organic chemistry can be found in [17] and those in polymer chemistry in [18]. For dimensionless quantities, a 1 is entered in the SI unit column (see Section 3.10, p. 97). Further information is added in notes, and in text inserts between the tables, as appropriate. Other symbols used are defined within the same table (not necessarily in the order of appearance) and in the notes. o 11 12 2.1 SPACE AND TIME The names and symbols recommended here are in agreement with those recommended hj^UPAP [4] and ISO [5.b]. Name Symbols Definition SI unit Notes cartesian space coordinates cylindrical coordinates spherical polar coordinates generalized coordinates position vector length special symbols: height breadth thickness distance radius diameter path length length of arc area volume plane angle solid angle time, duration period frequency angular frequency characteristic time interval, relaxation time, time constant angular velocity velocity speed acceleration x; y; z p; 0; r; Q, Qi r I h b d, Ö d r d s s V,{v) a, ß, 7, , t T ",f ijj X Sx\y ^y\Z S z a ■ Q v ijj c, r a> u, w, c, c V a s/r -- t/N l/T \dt/d In x\ dip/dt dr/dt \v\ dv/dt m m, lj/m m, (varies) m m m m3 rad, 1 sr, 1 s s Hz, s-1 rad s_1 s -1 rad s_1, s_1 m s_1 m s_1 m s-2 2 2 3 2,4 2, 5 6 7 (1) An infinitesimal area may be regarded as a vector endA, where en is the unit vector normal to the plane. (2) The units radian (rad) and steradian (sr) for plane angle and solid angle are derived. Since they are of dimension one (i.e. dimensionless), they may be included in expressions for derived SI units if appropriate, or omitted if clarity and meaning is not lost thereby. (3) ./V is the number of identical (periodic) events during the time t. (4) The unit Hz is not to be used for angular frequency. (5) Angular velocity can be treated as a vector, (j, perpendicular to the plane of rotation defined by v = uj x r. (6) For the speeds of light and sound the symbol c is customary. (7) For acceleration of free fall the symbol g is used. 13 2.2 CLASSICAL MECHANICS The names and symbols recommended here are in agreement with those recommended by IUPAP [4] and ISO [5.c]. Additional quantities and symbols used in acoustics can be found in [4%g|-V Name Symbol Definition 5/ unit Notes mass m kg reduced mass P p = 11111112/ (mi + m2) kg density, mass density P p = m/V kg m~3 relative density d d = p/p* 1 1 surface density PA, PS PA = ml A kg m~2 specific volume V v = V/m = 1/p m3 kg-1 momentum P p = mv kg m s_1 angular momentum L L = rx p J s 2 moment of inertia I, J I = ^2 min2 kg m2 3 force F i F = dp/dt = ma moment of force, M,(T) M= rx F m torque energy E J potential energy EP, V, <2> Ep = -f F-dr J 4 kinetic energy Ek, T, K Ek = (l/2)mv2 J work W, A, w W = f F-dr J power P p = F-v = dw*^r W generalized coordinate q (varies) generalized momentum p (varies) Lagrange function L J Hamilton function H H(V, P) ^SfiQi - L(q, q) J action S S = f L dt J s 5 pressure P, (P) p -1^^ Pa, N m~2 surface tension 7,«t 7 =jg&W/dA N m-1, J m~2 weight G,(W,P) £J1= rrfg N gravitational constant G l*W?mim2/r2 N m2 kg"2 (1) Usually p* = p(H20, 4 °C). (2) Other symbols are customary in atomic and molecular spectroscopy (see Section 2.6, p. 25). (3) In general / is a tensor quantity: Iaa = J2imi ifil + 7«2)> and lap = —J2imiaiPi if a 7^ where a, (3, 7 is a permutation aS a:\yf z. For a continuous mass distribution the sums are replaced by integrals. (4) Strictly speaking, only pfctentiiil energy differences have physical significance, thus the integral is to be interpreted as a definite integral, for instance Ep(r1,r2) = - F-dr J or possibly with the upper limit infinity /•oo Ep(r) = - F-dr J r (5) Action is the time integral over the Lagrange function L, which is equivalent to f pdq — f Hdt (see [19]). 14 Name Symbol Definition 5/ unit Notes normal stress a a = F/A Pa shear stress t t = F/A Pa linear strain, s, e E = Al/l 1 relative elongation modulus of elasticity, E E = a/e Pa 6 Young's modulus shear strain 7 7 = Ax/d 1 6, 7 shear modulus, g g = rh Pa 6 Coulomb's modulus volume strain, r3 rß = AV/Vo 1 6 bulk strain bulk modulus, K K = —Vq (dp/dV) Pa 6 compression modulus viscosity, v,(p) Txz =r](dvx/dz) Pa s dynamic viscosity fluidity

V = dW/dQ V, electric potential difference, U,AV, A0 U = V2 — V± electric tension electromotive force E, E = §(F/Q)-dr V (Emf, Emk) 3 electric field strength E E = F/Q = -W V m"1 electric flux & & = f D ■ en dA \~J 2 electric displacement D V • D = p C m~2 capacitance C C = Q/U ^F,CV4 permittivity e D = eE F m_1 4 electric constant, £0 £0 = /^o lcQ. F m_1 5 permittivity of vacuum relative permittivity er er = e/eo dielectric polarization, P P = D — sqE C m~2 electric polarization (electric dipole moment per volume) electric susceptibility Xe Xe = £r — 1 1 1st hyper-susceptibility Xe(2) Xe(2) = ^cT1 {d2P/ dE2) C m J"1, m V"1 7 2nd hyper-susceptibility Xe(3) Xe(3) = ^cT1 (d3P/dE3) C2 m2 J"2, m2 V~2 7 (1) The electric current / is a base quantity in ISQ. (2) endA is a vector element of area (see Section 2.1, note 1, p. 13). (3) The name electromotive force and flhe jymbol emf are no longer recommended, since an electric potential difference is not a force (see Section 2.13, note 14, p. 71). (4) e can be a second-rank tensor. (5) co is the speed of light in vacuum. (6) This quantity was formerj^callea dielectric constant. (7) The hyper-susceptibilities are the coefficients of the non-linear terms in the expansion of the magnitude P of the dielectric polarization P in powers of the electric field strength E, quite related to the expansion of the dipole moment vector described in Section 2.5, note 17, p. 24. In isotropic media, the expansion of the component i of the dielectric polarization is given by P, = £0[Xe(1)^ + (^e(2)£? + (V6)Xe(3)£f + • • • ] where Ei is the i-m lOtfeponent of the electric field strength, and Xe^ is the usual electric susceptibility Xe; equal to er — 1 in the absence of higher terms. In anisotropic media, Xe^\ Xe^2\ and Xe^ are tensors of rank 2, 3, and 4. For an isotropic medium (such as a liquid), or for a crystal with a centrosymmetric unit cell, Xe^ is zero by symmetry. These quantities are macroscopic 16 Name Symbol Definition SI unit Notes electric dipole moment P = YlQ%r% C m 8 magnetic flux density, B F = QvxB T magnetic induction magnetic flux <2> 3> = / B ■ en dA Wb 2 magnetic field strength, II V x H = j A m-1 ^\ \ magnetizing field strength permeability P B = pH N A-VH 10 magnetic constant, Po Pq = Akx 1(T7 H m"1 H m^ permeability of vacuum ^^^^^ relative permeability pT Pr = p/po 1 magnetization M M= B/po - H A m-1 (magnetic dipole moment per volume) magnetic susceptibility X, k, (Xm) X = Pr ~ 1 11 molar magnetic susceptibility Xm Xm = VmX m3 mol-1 magnetic dipole moment m. p E = -m - B ?A m2, J T 1-1 electric resistance R R = U/I n 12 conductance G G = 1/R s 12 loss angle 5 ö = ) j 3 wavefunction spherical harmonic Ylm = NllmlPlM (cos (fi^i 1 4 function probability density P P = tp*tp (m-3) 3, 5 charge density P p = -eP (C m-3) 3, 5, 6 of electrons probability current S S = -(ih/Žh^0 (m-V1) 3 density, probability flux electric current density 3 3 = -eS (A m-2) 3, 6 of electrons integration element dr dr = da; dy dz (varies) matrix element Aij, (i\A\j) Aíj/^pipŕÄipjdT (varies) 7 of operator A expectation value (A), A ^&ty= Ji>*Ai>dT (varies) 7 of operator A (1) The circumflex (or "hat"), ", serves distinguish an operator from an algebraic quantity. This definition applies to a coordinata^presentation, where V denotes the nabla operator (see Section 4.2, p. 107). (2) Capital and lower case ip are commonly used for the time-dependent function H/(x, t) and the amplitude function ip{x) respectivelj^Thus for a stationary state &(x,t) = ip(x)exp(—iEt/fi). (3) For the normalized wavefunction of a single particle in three-dimensional space the appropriate SI unit is given in parentheses. Results in quantum chemistry, however, are commonly expressed in terms of atomic units (see Section 3.9.2, p. 95 and Section 7.3 (iv), p. 145; and reference [22]). If distances, energies, angular momenta, charges and masses are all expressed as dimensionless ratios r/ao, E/Eh, etc., then all quantities are dimensionless. (4) P;lml denotes the associated Legendre function of degree I and order \m\. is a normalization factor. (5) ip* is the complex conjugate of ip. For an anti-symmetrized n electron wavefunction \P {r\, • • • ,rn), the total probability density of electrons is f2 ■ ■ ■ fn W* ÍP dr2 • • • drn, where the integration extends over the coordinates of all electrons but one. (6) — e is the charge of an electron. (7) The unit is the same as for the physical quantity A that the operator represents. 18 Name Symbol Definition SI unit Notes hermitian conjugate A] a = W (varies) 7 of operator A J commutator \A,B], \A,B\- \A,B] = AB-BA (varies) of A and B anticommutator \A,B} + &B] + =AB+BÄ (varies) ^ of A and B angular momentum see SPECTROSCOPY, ! section 2.6.1, p. 30 operators spin wavefunction a; ß 1 9 Hiickel molecular orbital theory (HMO) atomic-orbital basis Xr • m" -3/2 3 function molecular orbital i = y] Xrcri i m -3/2 3, 10 coulomb integral Hyy 5 Giy Hrr-- r ^ J 3, 10, 11 resonance integral Bjsi ßrs Hrs - = f Xr*Hxsdr J 3, 10, 12 energy parameter X —x = --(a-E)fi* 1 13 overlap integral Srs, s Srs — ~- jf Xr/fedV- 1 10 charge order qr qr = n T be 2 i=l 1 14 , 15 bond order Prs Prs — 1 15 , 16 i=l (8) The unit is the same as for the product of the rtt|^§£al quantities A and B. (9) The spin wavefunctions of a single electron, are defined by the matrix elements of the z component of the spin angular momentum, sz, by the relations (a \ sz| a) = +(1/2), ((3 \ sz| (3) = — (1/2), {(3 | sz| a) = (a \ sz| (3) = 0 in units of h. The total electron spin wavefunctions of an atom with many electrons are denoted by Greek letters a, (3, 7, etc. according to the value of ^rris, starting from the greatest down to the least. (10) H is an effective hamiltonian for a single electron, i and j label the molecular orbitals, and r and s label the atomic orbitals. In HucJ^lMO theory, Hrs is taken to be non-zero only for bonded pairs of atoms r and s, and all Srs are assumed to be zero for r 7^ s. (11) Note that the name "coulomb integral" has a different meaning in HMO theory (where it refers to the energy of the orbital Xr in the field of the nuclei) from Hartree-Fock theory discussed below (where it refers to a two-electron rep/lsion integral). (12) This expression describes a bonding interaction between atomic orbitals r and s. For an anti-bonding interaction, the corresponding resonance integral is given by the negative value of the resonance integral for the bonding interaction. (13) In the simplest application of Hiickel theory to the k electrons of planar conjugated hydrocarbons, a is taken to be the same for all carbon atoms, and (3 to be the same for all bonded pairs of carbon atoms; lr^then customary to write the Hiickel secular determinant in terms of the dimensionless parameter x. (14) — eqr is the electronic charge on atom r. qr specifies the contribution of all n k electrons to the total charge at center r, with J2qr = n. (15) bi gives the number of electrons which occupy a given orbital energy level £«; for non-degenerate orbitals, 6« can take the values 0, 1, or 2. (16) prs is the bond order between atoms r and s. 19 2.4.1 Ab initio Hartree-Fock self-consistent field theory (ab initio SCF) Results in quantum chemistry are typically expressed in atomic units (see Section 3.9.1, jt94 and Section 7.3 (iv), p. 145). In the remaining tables of this section all lengths, energies, masses, charges and angular momenta are expressed as dimensionless ratios to the corresponding atomic units, ao, Eh, me, e and h respectively. Thus all quantities become dimensionless, and the SI unit column is therefore omitted. Name Symbol Definition Notes molecular orbital molecular spin orbital total wavefunction core hamiltonian of a single electron one-electron integrals: expectation value of H, the core hamiltonian two-electron repulsion integrals: coulomb integral exchange integral one-electron orbital energy a total electronic energy E coulomb operator J,L exchange operator K i Fock operator F 4>i (aO 4>i (fi)a(fi); 9 h core = -(1/2)V2 - J2 Za/t^a Jij K, Hu = /0i*(l)F1core0i(l)dT1 Jij = //0i*(l)«2)^0i(l)^(2)dridr2 Kij = //0i*(l)^*(2)^^(l)0i(2)dridr2 £j = Hu + Yl (^ij^r Kij) J^j(2) = (0,(1) ^ 0,(1)) 0,(2) F=H-+Z 2Ji~K 17 17 17, 18 17, 19 17, 19 17, 20 17, 20 17, 21 17, 21, 22 17 17 17, 21, 23 (17) The indices i and j label the mol^jilar orbitals, and either /x or the numerals 1 and 2 label the electron coordinates. (18) The double vertical bars denote an anti-symmetrized product of the occupied molecular spin orbitals 4>ia and iß (sometimes denoted 4>i and for a closed-shell system IF would be a normalized Slater determinant, (n!)-1/2 is the normalization factor and n the number of electrons. (19) Za is the proton number (charge number) of nucleus A, and rßa is the distance of electron fi from nucleus A. Ha is the energy of an electron in orbital 4>i m the field of the core. (20) The inter-electron repulsion integral is written in various shorthand notations: In J,Lj = the first and third indices refer to the index of electron 1 and the second and fourth indices to electron 2. In = the first two indices refer to electron 1 and the second two indices to electron 2. Usually the functions are real and the stars are omitted. The exchange integral is written in various shorthand notations with the same index convention as described: = or = (21) These relations apply to closed-shell systems only, and the sums extend over the occupied molecular orbiOll^ (22) The sum over j includes the term with j = i, for which J a = Ku, so that this term in the sum simplifies toVdve/Jji — Ka = Ju. 20 2.4.2 Hartree-Fock-Roothaan SCF theory, using molecular orbitals expanded as linear combinations of atomic-orbital basis functions (LCAO-MO theory Name Symbol Definition Notes tes atomic-orbital basis function molecular orbital Xr 4>i = Y. XrCri 724 overlap matrix element Srs Srs — J Xr Xsďr, E^ Cri SrsCsj — Sjj r,s density matrix element p 1 rs occ Prs — 2 E^ Cri Csi 25 integrals over the basis functions: one-electron integrals Hra = fXr*{l)H1COKXs{l)dT1 two-electron integrals (rs\tu) (rs\tu) = ff Xr* (1) Xs (1) ^Xt* (2) Xu (2) dndr2 26, 27 total electronic energy E E = J2Y1 PrsHrs r s +(1/2) EEEE PrsPtuWlpk- (1/2) (ru\ts)] 25, 27 matrix element of the Fock operator 1 rs Frs =F„ + EEPtu [(rs\tu) - (1/2) (ru\ts)} t u 25, 28 (Notes continued) (23) The Hartree-Fock equations read (F — £j)4>j = 0. Note that the definition of the Fock operator involves all of its eigenfunctions (pi through the coulomb and exchange operators, J-L and If j. (24) The indices r and s label the basis functions. In numerical computations the basis functions are either taken as Slater-type orbitals (STO) or as Gaussian-type orbitals (GTO). An STO basis function in spherical polar coordinates has the general form x{r,®,{ Nrn-LeM-(nir)Ylm(e,4>), where (ni is a shielding parameter representing the effective charge in the state with quantum numbers n and I. GTO functions are typically expressed in cartesian space coordinates, in the form x(x,y,z) = Nxaybzc exp (-ar2). Commonly, a linear combination of such functions with varying exponents a is used in such a way as to model an STO. ./V denotes a normalization factor. (25) For closed-shell species with two elections per occupied orbital. The sum extends over all occupied molecular orbitals. Prs may Jffll^pe called the bond order between atoms r and s. (26) The contracted notation for two-electron integrals over the basis functions, (rs\tu), is based on the same convention outlined in note 20. (27) Here the two-electron integral is expressed in terms of integrals over the spatial atomic-orbital basis functions. The matrix elements Ha , J,Lj , and K,-Lj may be similarly expressed in terms of integrals over the spatial atomic-orbital basis functions according to the following equations: Ha — E/ ^2 Cri CsiHrs^*. J, K, i*i\fj) = EEEE cn*cstctj*cUJ (r*s\t*u) r s t u = EEEEcri*csictj*cuj (r*u\ťs) r t u (28) The Hartree-Fock-Roothaan SCF equations, expressed in terms of the matrix elements of the Fock operator Frs, and the overlap matrix elements Srs, take the form: (.E-rs s'iSrs) Csi — 0 21 2.5 ATOMS AND MOLECULES The names and symbols recommended here are in agreement with those recommended h^tUPAP [4] and ISO [5.i]. Additional quantities and symbols used in atomic, nuclear and plasma pl^sicsi;an be found in [4,5.j]. Name Symbol Definition 5/jflfcř Notes nucleon number, mass number A ]/ proton number, atomic number Z A neutron number N N = A-Z 1 electron mass me kg 1, 2 mass of atom, atomic mass kg atomic mass constant mu mu = ma(12C)/12 1, 3 mass excess A A = ma — Amu Tkg elementary charge e proton charge C 2 Planck constant h J s Planck constant divided by 2k h h = h/2K J s 2 Bohr radius a0 ao = 4tc£o^2/mee2 Eh = h2/meaQ m 2 Hartree energy Eh J 2 Rydberg constant Roo = Eh/2M^ m-1 fine-structure constant a a = e2/AkeqHc 1 ionization energy Ei,I J 4 electron affinity Eea,A t J X = (1/2) (Ei + £ea) J 4 electronegativity X J 5 dissociation energy Ed,D J from the ground state D0 J 6 from the potential minimum De J 6 (1) Analogous symbols are used for other particles with subscripts: p for proton, n for neutron, a for atom, N for nucleus, etc. (2) This quantity is also used as an atomic unit (see Section 3.9.1, p. 94 and Section 7.3 (iv), p. 145). (3) mu is equal to the unified atomic mass unit, with symbol u, i.e. mu = 1 u (see Section 3.7, p. 92). The name dalton, with symbol Da, is used as an alternative name for the unified atomic mass unit [23]. (4) The ionization energy is frequently called the ionization potential (Ip). The electron affinity is the energy needed to detach an electron. (5) The concept of electronegativity was introduced by L. Pauling as the power of an atom in a molecule to attract electrons to itself. There are several ways of defining this quantity [24]. The one given in the table has a clear physical meaning of energy and is due to R. S. Mulliken. The most frequently used scale, due to Pauling, is based on bond dissociation energies E<± in eV and it is relative in the sense that the values are dimensionless and that only electronegativity differences are defined. For atoms A and B Xr,A - Xr,B Eg (AB) 1 [Eg (AA) + Eg (BB) 2 eV 2 eV where Xr denotes the Pauling relative electronegativity. The scale is chosen so as to make the relative electronegativity of hydrogen Xr,H = 2.1. There is a difficulty in choosing the sign of the square root, which determines the sign of Xr,A — Xr,B- Pauling made this choice intuitively. (6) The symbols Dq and De are used for dissociation energies of diatomic and polyatomic molecules. 22 Name Symbol Definition SI unit Notes principal quantum number n E = hcZ2ROQ/n2 1 Cj (hydrogen-like atom) angular momentum see SPECTROSCOPY, Section 2.6 quantum numbers magnetic dipole moment m. fi Ep = -mB J T-1 V8 of a molecule magnetizability m = £B jr2 of a molecule Bohr magneton hb Hb = eh/2me J T-1 nuclear magneton H~n Hn = eh/2mp = (me/mp) hb J T-1 gyromagnetic ratio, 7 7e = -QeHB/h s-1 T-1 9 (magnetogyric ratio) ^-factor 9,9e 9e = -7e(2me/e) Vi 10 nuclear ^-factor on 9N = 7n(2mp/e) I1 10 Larmor angular frequency wl ljl = — 7-B s-1 11 Larmor frequency vl, = u^/2-k Hz relaxation time, longitudinal Ti s 12 transverse T2 s 12 electric dipole moment Ep = -p-E C m 13 of a molecule quadrupole moment Q,e Ep = (1/2)Q:V" = (1/3)6 :V" Cm2 14 of a molecule (7) For an electron in the central coulomb field of an infinitely heavy nucleus of atomic number Z. (8) Magnetic moments of specific particles may be denoted by subscripts, e.g. He, Hp, Hn for an electron, a proton, and a neutron. Tabulated values usually refer to the maximum expectation value of the z component. Values for stable nuclei are gStjapn Section 6.3, p. 121. (9) The gyromagnetic ratio for a nucleus is 7n t^^Mn/^- (10) For historical reasons, ge > 0. e is the (positive) elementary charge, therefore 7e < 0. For nuclei, 7n and #n have the same sign. A different sign convention for the electronic g-factor is discussed in [25]. (11) This is a vector quantity with magnitucrfctekL- This quantity is sometimes called Larmor circular frequency. (12) These quantities are used in the^y^text of saturation effects in spectroscopy, particularly spin-resonance spectroscopy (see Se^no!y2.6, p. 27—28). (13) See Section 2.6, note 9, p. 26. (14) The quadrupole moment of a molecule may be represented either by the tensor Q, defined by an integral over the charge c^nsitv p: Qaß = J rarßp dV in which a and ß denote x, y or z, or by the tensor & of trace zero defined by Oaß = (1/2) j Kß r2) pdV = (1/2) [3Qaß - Kß (.Qxx Qyy Qzz)] V" is the second derivative of the electronic potential: Vaß" = -qaß = d2V/dadß The contribution to the potential energy is then given by Ep = (1/2)4^" = (1/2) £ £ QaßVaß" a ß 23 Name Symbol Definition SI unit Notes quadrupole moment eQ eQ = 2(Gzz) C m2 c of a nucleus electric field gradient Q Qa/3 = -d2V/dad[3 V m-2 tensor quadrupole interaction X Xaf3 = eQQaf3 J y 16 energy tensor electric polarizability a &ab = dpa/dEb C2 nA^S 17 of a molecule 1st hyper-polarizability 0 Pabc = d2pa/dEbdEc C3 m3 J-2 17 2nd hyper-polarizability 7 labcd = d3pa/dEbdEcdEd C4ni%-3 17 activity (of a radio- A A = -dNB/dt 18 active substance) decay (rate) constant, A, k A = XNB s-1 18 disintegration (rate) constant half life *l/2) ^1/2 NB(t1/2)=NB(0)/2 s 18, 19 mean life, lifetime T r=l/A s 19 level width r r = h/r J disintegration energy Q J cross section a m2 electroweak charge of a Qw Qw « Z(l -4sin2#w) - N 1 20 nucleus (15) Nuclear quadrupole moments are conventionally/ffllfinted in a different way from molecular quadrupole moments. Q has the dimension of an area^wlfeJis the elementary charge. eQ is taken to be twice the maximum expectation value of tfje iA^ensor element (see note 14). The values of Q for some nuclei are listed in Section 6.3, p. 121. (16) The nuclear quadrupole interaction energy tensor % is usually quoted in MHz, corresponding to the value of eQq/h, although the h is usually omitted. (17) The polarizability a and the hyper-polarizabilities /5./>. • • • are the coefficients in the expansion of the dipole moment p in powers of the electric field strength E (see Section 2.3, note 7, p. 16). The expansion of the component a is given by Pa = Pa°] + E aabEb + (1/2) Z PabcEbEc + (1/6) £ labcdEbEcEd + ■■■ b be bed in which aab, (3abc, and ^abed are elements of the tensors a, 0, and 7 of rank 2, 3, and 4, respectively. The components of these tensors are distinguished by the subscript indices abc- • • , as indicated in the definitions, the first index a always denoting the component of p, and the subsequent indices the components of the electric field. ThQjpolarizability and the hyper-polarizabilities exhibit symmetry properties. Thus a is commonly a symmetric tensor, and all components of 0 are zero for a molecule with a centre of symmetry, etc. Values of the polarizability are commonly quoted as the value a/4-neo, which is a volume. The value is commonly expressed in the unit A3 (A should not be used, see Section 2.6, note lLpr 27/or in the unit ao3 (atomic units, see Section 3.9.1, p. 94). Similar comments apply to the hyper-polarizabilities with (3/(4k6o)2 in units of ao5e_1, and 7/(4rceo)3 m units of ao7e~2, etc. (18) NB is the nunJserof decaying entities B (1 Bq = 1 s_1, see Section 3.4, p. 89). (19) Half lives and nrean lives are commonly given in years (unit a), see Section 7.2, note 4, p. 137. *i/2 = t In 2 for exponential decays. (20) The electroweak charge of a nucleus is approximately given by the neutron number ./V and the proton i/amM Z with the weak mixing angle #w (see Chapter 5, p. 111). It is important in calculations of atomic and molecular properties including the weak nuclear interaction [26]. 24 2.6 SPECTROSCOPY This section has been considerably extended compared with the original Manual [l.a—l.c] and with the corresponding section in the IUPAP document [4]. It is based on the recommendations of the ICSU Joint Commission for Spectroscopy [27, 28] and current practice in the field which is well represented in the books by Herzberg [29-31]. The IUPAC Commission on Molecular Structure and Spectroscopy has also published recommendations which have been taken into account [32-43]. Name Symbol Definition iSIwait Notes total term T T = Etot/hc m-1 1, 2 transition wavenumber V v = T - T" m-1 1 transition frequency V v = {E'- E") /h^ Hz electronic term Te Te = Ee/hc m-1 1, 2 vibrational term g g = Evib/hc m-1 1, 2 rotational term F F = ETOt/Jic ( m-1 1, 2 spin-orbit coupling constant a Tso = a < m-1 1, 3 principal moments Ia; Ib; Ic I a < Ib ^Sp kg m2 of inertia rotational constants, in wavenumber a-b-C a = h/8K2dA m-1 1, 2 in frequency a-b-C a = h/8K2IA Hz inertial defect A [A= = Mjhip, ancn^i^' = Khip.) I is frequently called the azimuthal quantum number and mi the magnetic quantum number. (2) Some authors, notably Herzberg [29-31], treat the component quantum numbers A,Q,l and K as taking positive or zero values only, so that each non-zero value of the quantum number labels two wavefunctions with opposite signs for the appropriate angular momentum component. When this is done, lower case k is commonly regarded as a signed quantum number, related to K by K = \k\. However, in theoretical discussions all component quantum numbers are usually treated as signed, taking both positive and negative values. (3) There is no uniform convention for denoting the internal vibrational angular momentum; j, tv, p and G have all been used. For symmetric top and linear molecules the component of j in the symmetry axis is always denoted by the quantum number I, where I takes values in the range —v ^ I ^ +v in steps of 2. The corresponding component of angular momentum is actually l(h, rather than lh, where ( is the Coriolis C-constant (see note 7, p. 26). (4) Asymmetric top rotational states are labeled by the value of J (or TV" if S ^ 0), with subscripts Ka, Kc, where the latter correlate with the K = \k\ quantum number about the a and c axes in the prolate and oblate symmetric top limits respectively. Exampll ^tKa,Kc = 52,3 for a particular rotational level. 30 2.6.2 Symbols for symmetry operators and labels for symmetry species (i) Symmetry operators in space-fixed coordinates [41,48] identity E permutation P, p space-fixed inversion E*, (P) permutation-inversion P* (= PE*), p* The permutation operation P permutes the labels of identical nuclei. Example In the NH3 molecule, if the hydrogen nuclei are labeled 1, 2 and 3, then P = (123) would symbolize the permutation where 1 is replaced by 2, 2 by 3, and 3 by 1. The inversion operation E* reverses the sign of all particle coordinates in the space-fixed origin, or in the molecule-fixed centre of mass if translation has been separated. It is also called the parity operator and then frequently denoted by P, although this cannot^^done in parallel with P for permutation, which then should be denoted by lower case p. In field-free space and in the absence of parity violation [26], true eigenfunctions of the hamiltonian are either of positive parity + (unchanged) or of negative parity — (change sign) under E*. The label may be used to distinguish the two nearly degenerate components formed by A-doubling (in a degenerate electronic state) or /-doubling (in a degenerate vibrational state) in linear molecules, or by If-doubling (asymmetry-doubling) in slightly asymmetric tops. For linear molecules, A- or /-doubled components may also be distinguished by the labels e or f [49]; for singlet states these correspond respectively to parity + or — for J even and vice versa for J odd (but see [49]). Fo/lineat molecules in degenerate electronic states the A-doubled levels may alternatively be labeled II (A') or II (A") (or A (A') , A (A") etc.) [50]. Here the labels A' or A" describe the symmetryyor^he electronic wavefunction at high J with respect to reflection in the plane of rotation (bi©3#e) [50] for further details). The A' or A" labels are particularly useful for the correlation of states of molecules involved in reactions or photodissociation. In relation to permutation-inversion symmetry species the superscript + or — may be used to designate parity, whereas a letter is used to designate symmetry with respect to the permutation group. One can also use the systematic not^tk^l^om the theory of the symmetric group (permutation group) Sn [51], the permutation inversion group being denoted by Sn* in this case, if one considers the full permutation group. The species is then given by the partition P(Sn) [52-54]. The examples give the species for S4*, where the partition is conventionally given in square brackets [ ]. Conventions with respect to these symbols still vary ([2.b] and [39-41,51-54]). Examples Af totally symmetric species with respect to permutation, positive parity, [4]+ A^" totally symmetric species with respect to permutation, negative parity, [4]~ E+ doubly degenerate species with respect to permutation, positive parity, [22]+ E~ doubly degenerate species with respect to permutation, negative parity, [22]~ F^ triply dege/eratejspecies with respect to permutation, positive parity, [2,12]+ triply degenerate species with respect to permutation, negative parity, [2,12]~ The Hermann-Mauguin symbols of symmetry operations used for crystals are given in Section 2.8.1 (ii) , p. 44. (ii) Symmetry operators in molecule-fixed coordinates (Schonfiies symbols) [29-31] identity^ E rotation by 2k/'n Cn reflection a, o~h inversion i rotation-reflection Sn (= Cn : áQ/át W 3 per time) 3 radiant intensity h h = YáP/áQ W sr-1 3,4 radiant excitance M = dP/ dAsource W m-2 3,4 (1) When there is no risk of ambiguity the subscript denoting vacuum is often omitted, n denotes the refraction index of the medium. (2) The unit cm-1 is widely used for the quantity wavenumber in vacuum. (3) The symbols for the quantities such as radiant energy and intensity are also used for the corresponding quantities concerning visible radiation, i.e. luminous quantities and photon quantities. Subscripts e for energetic, v for visible, and p for photon may be added whenever confusion between these quantities might otherwise occtyf. The units used for luminous quantities are derived from the base unit candela (cd) (see Section 3.3, p. 87). Examples radiant intensity Ie, Ie = dP/dfi, SI unit: W sr-1 luminous intensity Iv, SI unit: cd photon intensity Ip, SI unit: s-1 sr-1 The radiant intensity^l^should be distinguished from the plain intensity or irradiance / (see note 5). Additional subscripts to distinguish absorbed (abs), transmitted (tr) or reflected (refl) quantities may 'Beladded, if necessary. (4) The radiant intensity is the radiant power per solid angle in the direction of the point from which the source is being observed. The radiant excitance is the total emitted radiant power per area ^source of the radiation source, for all wavelengths. The radiance is the radiant intensity per area of 34 Name Symbol Definition SI unit Notes radiance L h = J Leos & dAsource W sr 1 m 2 3,4 intensity, irradiance I,E I -- = dP/dA W m-2 3, 5 spectral intensity, h = dl/dv W m"1 6 spectral irradiance fluence F = jldt = = J(dP/dA)dt J m-2 V Einstein coefficient, 8, 9 spontaneous emission Aij dNj/dt = - ■ E AijNj s-1 stimulated or induced emission Bij dNj/dt = - s kg^^ absorption Bji dNi/dt = - i s kg-1 emissivity, emittance £ £ = = M/Mhh 3 10 Stefan-Boltzmann constant a Mbb = aT4 • W m~2 K~4 10 etendue (throughput, E,(e) E = AQ = P/L m2 sr 11 light gathering power) (4) (continued) radiation source; & is the angle between the normal to the area element and the direction of observation as seen from the source. (5) The intensity or irradiance is the radiation power per area that is received at a surface. Intensity, symbol /, is usually used in discussions involving collimated beams of light, as in applications of the Beer-Lambert law for spectrometric analysis. Intensity of electromagnetic radiation can also be defined as the modulus of the Poynting vector (see Section 2.3, p. 17 and Section 7.4, p. 148). In photochemistry the term intensity is sometimes used as^xj&as for radiant intensity and must not be understood as an irradiance, for which the symbol E is preferred [60]. (6) Spectral quantities may be defined with respect to frequency u, wavelength A, or wavenumber v\ see the entry for spectral radiant energy density in this table. (7) Fluence is used in photochemistry to specify the energy per area delivered in a given time interval (for instance by a laser pulse); fluence is the time integral of the fluence rate. Sometimes distinction must be made between irradiance and fluence rate [60]; fluence rate reduces to irradiance for a light beam incident from a single direction perpendicularly to the surface. The time integral of irradiance is called radiant exposure. (8) The indices i and j refer to individual states; Ej > Ei, Ej — E,-L = hcVij, and Bji = B,Lj in the defining equations. The coefficients jp are defined here using energy density py in terms of wavenumber; they may alternatively be defined using energy density in terms of frequency pv, in which case B has SI units m kg-1, and Bv = cqB„ where Bv is defined using frequency and By using wavenumber. The defining equations refer to the partial contributions to the rate of change. (9) The relation between the Einstein coefficients A and By is A = 8Khcou3By. The Einstein stimulated absorption or emission coefficient B may also be related to the transition moment between the states i and j; for an electric dipole transition the relation is B^ = 3^0(4^) ? K'l^>l2 in which the summer p runs over the three space-fixed cartesian axes, and pp is a space-fixed component of the dipole moment operator. Again, these equations are based on a wavenumber definition of the Einstein coefficient B (i.e. By rather than Bv). (10) The emittance of a sample is the ratio of the radiant excitance emitted by the sample to the radiant excitance emitted by a black body at the same temperature; Mbb is the latter quantity. See Chapter 5, p. 112 for the value of the Stefan-Boltzmann constant. 35 Name Symbol Definition SI unit Notes resolution m-1 2, 12^13 resolving power R R = 1 13 ( free spectral range Av Av = 1/21 m-1 2, 14 finesse f f = Av/hv 1 14 quality factor Q Q = w ■2KU-dw/dt 1 14, 15 first radiation constant Cl Cl = - 2-nhcQ2 W m2 second radiation constant c2 c2 = -- hco/kB K m transmittance, transmission r,T T = Ptr/Po 1 17, 18 factor absorptance, absorption a a = 1 17, 18 factor reflectance, reflection p,R P = Preü/Po 17, 18 factor (decadic) absorbance Aw, A A10 = -lg(l-a;)# 17-20 napierian absorbance Ae,B Ae~- = — ln(l — a;) V 17-20 absorption coefficient, (linear) decadic a, K a = Aw/l 18, 21 (linear) napierian a a = Ae/l m-1 18, 21 molar (decadic) £ £ = a/c =^A$v)f<} m2 mol-1 18, 21, 22 molar napierian k k = a/cf= AgTcl m2 mol-1 18, 21, 22 (11) Etendue is a characteristic of an optical instrumental is a measure of the light gathering power, i.e. the power transmitted per radiance of the source. A is the area of the source or image stop; fi is the solid angle accepted from each point of the source by the aperture stop. (12) The precise definition of resolution depends on the lineshape, but usually resolution is taken as the full line width at half maximum intensity (FWHM) on a wavenumber, hv, or frequency, 8^, scale. Frequently the use of resolving power, of dimension 1, is preferable. (13) This quantity characterizes the performance of a spectrometer, or the degree to which a spectral line (or a laser beam) is monochromatic. It ma^also be defined using frequency u, or wavelength A. (14) These quantities characterize a Fabry-Perot cavity, or a laser cavity. I is the cavity spacing, and 21 is the round-trip path length. The free spectral range is the wavenumber interval between successive longitudinal cavity modes. I (15) W is the energy stored in the cavity, and —AW/At is the rate of decay of stored energy. Q is also related to the linewidth of a single cavity mode: Q = v/hv = v/W. Thus high Q cavities give narrow linewidths. (16) &b is the Boltzmann constant (see Section 2.9, p. 45 and Chapter 5, p. 111). (17) If scattering and luminekcenca can be neglected, r + a + p = 1. In optical spectroscopy internal properties (denoted by subscript i) are defined to exclude surface effects and effects of the cuvette such as reflection losses, so that r; + a; = 1, if scattering and luminescence can be neglected. This leads to the customary form of the Beer-Lambert law, Ptv/Po = hr/Io = r; = 1 — a; = exp {—net). Hence Ae = - ln(n), ^ff/^- lg(n). (18) In spectroscopj^ill of these quantities are commonly taken to be defined in terms of the spectral intensity, I^iv), hence they are all regarded as functions of wavenumber v (or frequency v) across the spectrum. Thus, for example, the absorption coefficient a(v) as a function of wavenumber v defines the absorption spectrum of the sample; similarly T{u) defines the transmittance spectrum. Spectroscopists use I{u) instead of Iy{v). (19) The definitions given here relate the absorbance A\q or Ae to the internal absorptance a; (see 36 Name Symbol Definition SI unit Notes net absorption cross section Onet Onet = k/Na m2 23 absorption coefficient integrated over u A,Ä A = I du m mol-1 23, 24 S S = A/Na m m 24 S S = (l/p/)/ln(/0//) du Pa"1 m~2 23- -25 integrated over In u r r = f k(^)^-1 du m2 mol-1 23, 24 integrated net absorption cross section GQet GQet = J Onet^)^1 du m2 , 23, 24 absorption index, imaginary refractive index k k = a/Aiiu 1 26 complex refractive index h h = n + \k 1 molar refraction, f 2 1 \ molar refractivity R R = ( n — 1 \ y \nz + 2 J angle of optical rotation a 1, rad 27 specific optical rotatory power w 9 = a hi rad m2 kg-1 27 molar optical rotatory power am - = a/cl \rad m2 mol-1 27 (19) (continued) note 17). However the subscript i on the absoro^^e a is often omitted. Experimental data must include corrections for reflections, scattering and luminescence, if the absorbance is to have an absolute meaning. In practice the absorbance is measured as the logarithm of the ratio of the light transmitted through a reference cell (with solvent only) to that transmitted through a sample cell. (20) In reference [57] the symbol A is used for decadic absorbance, and B for napierian absorbance. (21) I is the absorbing path length, and c is the amount (of substance) concentration. (22) The molar decadic absorption coefficient e iS sometimes called the "extinction coefficient" in the published literature. Unfortunately numerical values of the "extinction coefficient" are often quoted without specifying units; the absence of units usually means that the units are mol-1 dm3 cm-1 (see also [61]). The word "extinction" should properly be reserved for the sum of the effects of absorption, scattering, and luminescence. A (23) Note that these quantities give the net absorption coefficient re, the net absorption cross section (Tnet, and the net values of A, S, S, T, and Gnet; in the sense that they are the sums of effects due to absorption and induced emission (see the discussion in Section 2.7.1, p. 38). (24) The definite integral defining thes kT the temperature dependence is small and for v > 1000 cm-1 at ordinary temperatures induced emission can generally be neglected. In a more fundamental approach one defines the cross section Oyt{y) for an induced radiative transition from the state i to the state j (in either absorption or emission). For air ideal absorption experiment with only the lower state i populated the integrated absorption cross section for the transition j <— i is given by Gji = J Oji{py0^xdv = J cjji (v) v~x&v ft If the upper and lower energy levels are degenerate, the observed line strength is given by summing over transitions between all states i in the lower energy level j^ana all states j in the upper energy level n, multiplying each term by the fractional population p,L in the appropriate initial state. Neglecting induced emission this gives Gnet (n <- m) = yiJk^W (Notes continued) (25) The quantity S is only used for gases; it is defined in a manner similar to A, except that the partial pressure p of the gas replaces the concentration c. At low pressures pi « CiRT, so that S and A are related by the equation S ~ A/RT. Thus if S is used to report line or band intensities, the temperature should be specified. Iq is the incident, / the transmitted intensity, thus ln(/0A0 = -ln(J/J0) = -ln(l - Pabs/^o) = Ae (see also notes 17 and 19, p. 36). (26) a in the definition is the napierian absorption coefficient. (27) The sign convention for the angle of optical rotation is as follows: a is positive if the plane of polarization is rotated clockwise as viewed looking towards the light source. If the rotation is anti clockwise, then a is negative. The optical rotation due to a solute in solution may be specified by a statement of the type a(589.3 nm, 20 °C, sucrose, 10 gdm~3 in H20, 10 cm path) = +0.6647° The same information may flLcon^eyed by quoting either the specific optical rotatory power a/7/, or the molar optical rotareora^power a/cl, where 7 is the mass concentration, c is the amount (of substance) concentration, and I is the path length. Most tabulations give the specific optical rotatory power, denoted [a}\6. The wavelength of light used A (frequently the sodium D line) and the Celsius temperatunrt^re conventionally written as a subscript and superscript to the specific rotatory power [a]. For pure liquids and solids [a]\e is similarly defined as [a]\e = a/pi, where p is the mass density. Specific optical rotatory powers are customarily called specific rotations, and are unfortunately usually quoted without units. The absence of units may usually be taken to mean that the units are 0 cm3 gT1- dfllK^ for pure liquids and solutions, or 0 cm3 g_1 mm-1 for solids, where 0 is used as a symbol for degrees of plane angle. 38 If induced emission is significant then the net integrated cross section becomes °net {n^m) = Y^ (Pi " Pj) Gji = (j1 - j^jY, Gji . . \ iim processed to yield n and k. For non-isotropic solids all intensity properties must be defined with respect to specific crystal axes. If the n and k spectra are known, spectra of any optical property or measured quantity of the sample may be obtained from them. Physicists prefer to use the complex relative permittivity (see Section 2.3, p. 16), er = sT' + i eT" = Re er + i Im er instead of the complex refractive index n. sT' = Re er denotes the real part and eT" = Im er the imaginary part of er, respectively. They are related through er = re2, so that e'T = re2 — k2 and w^^^2nk. The refractive indices and relative permittivities are properties of the bulk phase. In order to obtain information about the molecules in the liquid free from dielectric effects of the bulk, the local field that acts on the molecules, E\oc, must be determined as a function of the applied field E. A simple relation is the Lorentz local field, i?ioc = E + P/3eo, where P is the dielectric polarization. This local field is based on the assumption that long-range interactions are isotropic, so it is realistic only for liquids and isotropic solids. Use of fhis^cal field gives the Lorentz-Lorenz formula (this relation is usually called the Clausius-Mossotti formula when applied to static fields) generalized to treat absorbing materials at 40 any wavenumber £r(^j + 2 3£0 Vm Here is the molar volume, and am is the complex molar polarizability (see Section 2.3, note 7, p. 16). The imaginary part am" of the complex molar polarizability describes the absorption by molecules in the liquid, corrected for the long-range isotropic dielectric effects but influenced by the anisotropic environment formed by the first few nearest neighbor molecules. The real part am' is the molar polarizability (see sections 2.3 and 2.5) in the limit of infinite frequency. The integrated absorption coefficient of a molecular absorption band in the condensed phase is described by [67,68] (see note 1, below) 1 4TC£0 7band j Cj = —— j ua^\u)du Theoretical analysis usually assumes that the measured band includes the transition j <— i with band centre wavenumber v$ and all of its hot-band transitions. Then I where uog(\Mji\2) is the population-weighted sum over all contributing transitions of the wavenumber times the square of the electric dipole moment of the transition (g is in general a temperature dependent function which includes effects from the inhomqyreneojis band structure). The traditional relation between the gas and liquid phase values of the absorption coefficient of a given band j is the Polo-Wilson equation [69] _(n'+2)Q/ liq " 9n ^SpS where n is the estimated average value of n through the absorption band. This relation is valid under the following conditions: (i) bands are sufficiently weak, i.e. 2nk + k2 AT 1 5 structure factor F(h,k,l) F = 1 6 with indices h, k, I Tl— 1 lattice plane spacing d m Bragg angle e nX = 2dsinö rad, 1 7 order of reflection n 1 order parameters, • short range a 1 long range s 1 Burgers vector b m particle position vector m 8 equilibrium position Ro m vector of an ion displacement vector u u = = R — Rq m of an ion . B,D Debye-Waller factor D -- = e-2<(q-u)2> 1 9 (1) ni,n2 and n3 are integers, a, b and c are also called the lattice constants. (2) Reciprocal lattice vectors are sometimes defined by 0« • bk = 8^- (3) m is an integer witl^k =^mih + n2k + n3l. (4) X denotes the coordinate of dimension length. (5) Ea and Ee denote the scattering amplitudes for the atom and the isolated electron, respectively. (6) ./V is the number of atoms in the unit cell. (7) A is the wavelength of the incident radiation. (8) To distinguish between electron and ion position vectors, lower case and capital letters are used respectively. The subscript j relates to particle j. (9) hq is the momentum transfer in the scattering of neutrons, <> denotes thermal averaging. 42 Name Symbol Definition SI unit Notes Debye angular wavenumber kD -- = (C,6^)1/3 m-1 10 Debye angular frequency wd : = c0 s-1 10 Debye frequency uD ~- = ujd/2t: s-1 Debye wavenumber uD ~- = Wco m-1 10 Debye temperature eD @d = hvv/kn K Griineisen parameter 7,r 7 = aV 1 nCy 1 11 aN\ z^Z-e2 12 Madelung constant ^ $ = : Eoc — Ey J 21 electron work function angular wave vector, k, q k = 2k/\ m-1 22 propagation vector m-3/2 Bloch function Uk{r) v(r ) = uk(r)exp(\k- r) 23 charge density of electrons P P{r) i = -eip*{r)ip{r) Cm"3 23, 24 effective mass m* kg 25 mobility P ^drift — PE m2 V-1 s-1 25 mobility ratio b b = Pn/Pp 1 diffusion coefficient D 3 = -dvc 2 — 1 mz s 1 25, 26 diffusion length l L = ■ Vd^ m 25, 26 characteristic (Weiss) Mw K temperature Curie temperature TC K Neel temperature K (10) Ci is the ion density, cq is the speed of light in vacuum, is equal to 2k times the inverse of the Debye cut-off wavelength of the elastic lattice wave. (11) a is the cubic expansion coefficient, V the volume, re the isothermal compressibility, and Cy the heat capacity at constant volume. (12) -Ecoul is the electrostatic interaction energy per mole of ion pairs with charges z+e and — Z-e. 43 2.8.1 Symbols for planes and directions in crystals Miller indices of a crystal face, or of a single net plane (hkl) or (hih2h%) indices of the Bragg reflection from the set of parallel net planes (hkl) hkl or hih2h% I indices of a set of all symmetrically equivalent crystal faces, {hkl} or {hih2h%} or net planes indices of a lattice direction (zone axis) [uvw] indices of a set of symmetrically equivalent lattice directions < uvwj^S In each of these cases, when the letter symbol is replaced by numbers it is customary to omit the commas. For a single plane or crystal face, or a specific direction, a negative number is indicated by a bar over the number. Example (110) denotes the parallel planes h = —1, k = 1, I = 0. (i) Crystal lattice symbols primitive P face-centred F body-centred I base-centred A;B;C rhombohedral R (ii) Hermann-Mauguin symbols of symmetry operations Operation Symbol Examples ra-fold rotation n 1; 2; 3; 4; 6 ra-fold inversion ra 1; 2; 3; 4; 6 ra-fold screw ra^ 2i; 3i; 32;... reflection rra glide a; b; c; ra; d (Notes continued) (13) N(E) is the total number of states of electronic energy less than E, divided by the volume. (14) N(uj) is the total number of vibrational modes with circular frequency less than uj, divided by the volume. (15) Tensors may be replaced by their corresponding scalar quantities in isotropic media. Jq is the heat flux vector or thermal current derkifey. (16) The definition applies to electrons in metals; I is the mean free path, and v-p is the electron velocity on the Fermi sphere. (17) A and a are the thermal and electrical conductivities in isotropic media. (18) The substances to which the symbol applies are denoted by subscripts. The thermoelectric force is an electric potential difference induced by the gradient of the chemical potential. (19) Specific number densities are denoted by subscripts: for electrons ran, ra_,(ra); for holes rap, ra+, (p); for donors ra^; for acceptors raa; for the intrinsic number density n,i(n2 = ra+ra_). (20) The commonly useifcnit for this quantity is eV. fi is the chemical potential per entity. (21) Eoo is the electron energy at rest at infinite distance [71]. (22) k is used for particles, q for phonons. Here, A is the wavelength. (23) ip(r) is a one-electron wavefunction. (24) The total charge density is obtained by summing over all electrons. (25) Subscripts n and p or — and + may be added to denote electrons and holes, respectively. (26) j is the particle flux density. D is the diffusion coefficient and r the lifetime. 44 2.9 STATISTICAL THERMODYNAMICS The names and symbols given here are in agreement with those recommended by IUPAP [4] and by ISO [5.h]. Name Symbol Definition SI unit Notes number of entities number density of entities, number concentration Avogadro constant Boltzmann constant (molar) gas constant molecular position vector molecular velocity vector molecular momentum vector velocity distribution function speed distribution function average speed generalized coordinate generalized momentum volume in phase space probability statistical weight, degeneracy (cumulative) number of states density of states partition function, sum over states, single molecule canonical ensemble, (system, or assembly) N C L, NA k, kB R r(x,y,z) c(cj; , Cy , cz) , 1l(ux 1 , Uz) 1 V(VX,Vy,VZ) P(Px,Py,Pz) f(cx) F(c) Č, Ü, V, (c), (u), (v) q p n p,p g, d, W, to, ß W,N p(E) Q\z\ C = N/V L = N/n R = Lk c = dr/di mol-1 ■5 k-1 J KT1 moh m s m -l P mc kg m s 1 1/2 /=(2^r) exp 2kT F = Akc2 m v3/ 2xkT exp mc 2kŤ m m xs ms -l p = dL/dq Í?;*27M fpdq W(E) =EH(£- Ei) i p{E) =áW{E)/áE 1 = Y,9ie*p{-Ei/kT) Q = 52giexp{-Ei/kT) (varies) (varies) 1 1 1 1 J-1 1 1 4 5, 6 (1) n is the amount of substance (or the chemical amount, enplethy). While the symbol Na is used to honour Amedeo Avogadro, the symbol L is used in honour of Josef Loschmidt. (2) m is the mass of the particle. (3) If q is a length the^^j^ii momentum. In the definition of p, L denotes the Lagrangian. (4) (3 is sometimes used for a spin statistical weight and the degeneracy is also called polytropy. It is the number of linearly independent energy eigenfunctions for the same energy. (5) H(x) is the Heaviside function (see Section 4.2, p. 107), W or W(E) is the total number of quantum states with energy less than E. (6) E-i denotes the energy of the ith level of a molecule or quantum system under consideration and Qi denotes its degeneracy. 45 Name Symbol Definition SI units Notes microcanonical fi, z, Z z 1 0 ensemble i=l partition function, sum over states, grand canonical 1 ensemble symmetry number a, s 1 reciprocal energy parameter ß ß = 1/kT J-1 to replace temperature characteristic temperature &, e Ko 7 absolute activity A AB = exp(pB/RT) 1- 8 density operator p, a P = Y.Pk\^k)(^k\ h 9 density matrix P,P P—{Pmn\ 10 element Pram Pinn Pmn — (4>m\p\4>n) 10 statistical entropy s S = -kJ2Pi^Pi > i JK-1 11 (7) Particular characteristic temperatures are denoted with subscripts, e.g. rotational 0r = hcB/k, vibrational @v = hcu/k, Debye @d = hcu^/k, Einstein @e = hcv^/k. & is to be preferred over 6 to avoid confusion with Celsius temperature. (8) The definition applies to entities B. pb is the chemical potehtial (see Section 2.11, p. 57). (9) refers to the quantum state k of the system and pk to the probability of this state in an ensemble. If pk = 1 for a given state k one speaks of a iWe^tate, otherwise of a mixture. (10) The density matrix P is defined by its matrix elements Pmn in a set of basis states 4>m. Alternatively, one can write Pmn = £fcPfc cm^ cn^ *, where cm^ is the (complex) coefficient of 4>m in the expansion of \H/k) in the basis states {4>i}. (11) In the expression for the statistical entrop^^^f the Boltzmann constant and pi the average population or probability for a quantum level. The equilibrium (maximum) entropy for a microcanonical ensemble results then as S = klnZ. A variety of other ensembles beyond the microcanonical, canonical, or grand canonical can be defined. o 46 2.10 GENERAL CHEMISTRY The symbols given by IUPAP [4] and by ISO [5.d,5.h] are in agreement with the recommendations given here. Name Symbol Defin ition SI unit Notes number of entities N 1 (e.g. molecules, atoms, ions, formula units) >^ amount of substance, amount n nB = NB/L mol 1, 2 (chemical amount) ^^^^ Avogadro constant L,NA mol-1 3 mass of atom, ma,m kg atomic mass mass of entity m, rrif kg 4 (molecule, formula unit) atomic mass constant mu mu = :ma(12C)/12 kg 5 molar mass M MB = - m/n-Q kg mol-1 2, 6, 7 molar mass constant Mu Mu = -- muNA ^ g mol-1 7, 8 relative molecular mass, MT MT = mf/mu 8 (relative molar mass, molecular weight) relative atomic mass, j4j> - "vFlu 1 8 (atomic weight) molar volume Vm Vm:b m3 mol-1 2, 6, 7 mass fraction w WB = 1 2 (1) The words "of substance" may be replaced bp tflWpecincation of the entity, e.g. "amount of oxygen atoms" or "amount of oxygen (or dioxygen, O2) molecules". Note that "amount of oxygen" is ambiguous and should be used only if the meaniag^^clear from the context (see also the discussion in Section 2.10.1 (v), p. 53). Example When the amount of O2 is equal to 3 mol, n(02) = 3 mol, the amount of (1/2) 02 is equal to 6 moj^id^((l/2) 02) = 6 mol. Thus n((l/2) 02) = 2n(02). (2) The definition applies to entities B wfl^h should always be indicated by a subscript or in parentheses, e.g. n-Q or n(B). When the chemical composition is written out, parentheses should be used, n(02). (3) The symbol Na is used to honoj^^prhedeo Avogadro, the symbol L is used in honour of Josef Loschmidt. (4) Note that "formula unit" is not a/init (see examples in Section 2.10.1 (iii), p. 50). (5) mu is equal to the unified atomic mass unit, with symbol u, i.e. mu = 1 u (see Section 3.7, p. 92). The dalton, symbol Da, is used as an alternative name for the unified atomic mass unit. (6) The definition applies to pure substance, where m is the total mass and V is the total volume. However, corresponding quantities may also be defined for a mixture as m/n and V/n, where n = Yli ni - These quantities are called the mean molar mass and the mean molar volume respectively. (7) These names, which include the word "molar", unfortunately use the name of a unit in the description of a quantity, which in principle is to be avoided. (8) For historical reasons the terms "molecular weight" and "atomic weight" are still used. For molecules Mr is the relative molecular mass or "molecular weight". For atoms Mr is the relative atomic mass or "atomic weight", and the symbol AT may be used. Mr may also be called the relative molar mass,/Mj^fc= Mb/Mu, where Mu = 1 g mol-1. The standard atomic weights are listed in Section 6.2, p. 117. 47 Name Symbol Definition SI unit Notes volume fraction 0 b 1 T J 2, 10 mole fraction, amount-of- x,y xB = nB/ ni 1 substance fraction, i amount fraction (total) pressure p, (P) Pa 2, 11 partial pressure pb pb : = vbp Pa 12 mass concentration, 7.P 7B : = mB/V kg m^3 2, 13, 14 (mass density) number concentration, C,n CB = NB/V m~3 2, 13, 15 number density of entities amount concentration, c, [B] CB -- = nB/V mol m~3 2, 13, 16 concentration solubility s SB : = cB (saturated solution) / 2, 17 molality m, b mB = nB/mA mol kg-1 2, 7, 14 surface concentration r rB = nB/A mol m~2 2, 18 stoichiometric number V 1 19 extent of reaction, i z = (nB - nBfi)/vB mol 19 advancement degree of reaction a a = 1 £/(max 1 20 (9) Here, VB and V-i are the volumes of appropriate components prior to mixing. As other definitions are possible, e.g. ISO 31 [5], the term should not be used in accurate work without spelling out the definition. (10) For condensed phases x is used, and for gaseous mixtures y may be used [72]. (11) Pressures are often expressed in the non-SI unit bar, where 1 bar = 105 Pa. The standard pressure = 1 bar = 105 Pa (see Section 2.11.1^|^jp. 62, Section 7.2, p. 138, and the conversion table on p. 233). Pressures are often expressed iifll^Iubar or hectopascal, where 1 mbar = 10~3 bar = 100 Pa = 1 hPa. (12) The symbol and the definition apply to^iolecules B, which should be specified. In real (non-ideal) gases, care is required in defining the partial pressure. (13) V is the volume of the mixture. Quantities that describe compositions of mixtures can be found in [72]. (14) In this definition the symbol m isVug^a with two different meanings: mB denotes the molality of solute B, mA denotes the mass of solvent A (thus the unit mol kg-1). This confusion of notation is avoided by using the symbol b for molality. A solution of molality 1 mol kg-1 is occasionally called a 1 molal solution, denoted 1 m solution; however, the symbol m must not be treated as a symbol for the unit mol kg~inh combination with other units. (15) The term number concentration and symbol C is preferred for mixtures. Care must be taken not to use the symbol n where it may be mistakenly interpreted to denote amount of substance. (16) 'Amount concentration" is an abbreviation of "amount-of-substance concentration". (The Clinical Chemistry Division of IUPAC recommends that amount-of-substance concentration be abbreviated to "substance concentration" [14,73].) The word "concentration" is normally used alone where there is no risk of confusion, as in conjunction with the name (or symbol) of a chemical substance, or as contrast to moialfly (see Section 2.13, p. 70). In polymer science the word "concentration" and the symbol c is normally used for mass concentration. In the older literature this quantity was often called m^arHfJa usage that should be avoided due to the risk of confusion with the quantity molality. Umts^fcnmonly used for amount concentration are mol L_1 (or mol dm-3), mmol L_1, |i.mol L_1 etc., often denoted m, mM, |jlm etc. (pronounced molar, millimolar, micromolar). 48 2.10.1 Other symbols and conventions in chemistry (i) The symbols for the chemical elements The symbols for the chemical elements are (in most cases) derived from their Latin nmies^ind consist of one or two letters which should always be printed in Roman (upright) type. A complete list is given in Section 6.2, p. 117. The symbol is not followed by a full stop except at the end of a sentence. Examples I, U, Pa, C The symbols have two different meanings (which also reflects on their use in chemical formulae and equations): (a) On a microscopic level they can denote an atom of the element. For example, CI denotes a chlorine atom having 17 protons and 18 or 20 neutrons (giving a mass nlimper of 35 or 37), the difference being ignored. Its mass is on average 35.4527 u in terrestrial samples. (b) On a macroscopic level they denote a sample of the element. For example, Fe denotes a sample of iron, and He a sample of helium gas. They may also be used^s a shorthand to denote the element: "Fe is one of the most common elements in the Earth's atiistA The term nuclide implies an atom of specified atomic /rumbef (proton number) and mass number (nucleon number). A nuclide may be specified by attaching the mass number as a left superscript to the symbol for the element, as in 14C, or added after the name of the element, as in carbon-14. Nuclides having the same atomic number but different mass numbers are called isotopic nuclides or isotopes, as in 12C, 14C. If no left superscript is attached, the symbol is read as including all isotopes in natural abundance: n(Cl) = n(35Cl) + n(37Cl). Nuclides having the same mass number but different atomic numbers are called isobaric nuclides or isobars: 14C, 14N. The atomic number may be attached as a left subscript: 1gC, 14N. The ionic charge number is denoted by a right superscript, by the sign alone when the charge number is equal to plus one or minus one. Examples Na+ ^^,sodium positive ion (cation) 79Br~ a bromine-79 negative ion (anion, bromide ion) Al3+ or Al+3 Y aluminium triply positive ion 3 S2~ or 3 S~2 three sulfur doubly negative ions (sulfide ions) Al3+ is commonly used in chemistry finjrecommended by [74]. The forms A1+3 and S~2, although widely used, are obsolete [74], as^jeTrW the old notation Al+++, S=, and S~~. (Notes continued) (16) (continued) Thus m is often treated as a symbol for mol L_1. (17) Solubility may also be expressed in any units corresponding to quantities that denote relative composition, such as mass fraction, amount fraction, molality, volume fraction, etc. (18) A denotes the surface area. (19) The stoichioiqu^ric number is defined through the stoichiometric equation. It is negative for reactants and positive for products. The values of the stoichiometric numbers depend on how the reaction equation is written (see Section 2.10.1 (iv), p. 52). riB,o denotes the value of n-Q at "zero time", when £ = 0 mol. (20) £max is/he^Wue of £ when at least one of the reactants is exhausted. For specific reactions, terms such as "degree of dissociation" and "degree of ionization" etc. are commonly used. 49 The right superscript position is also used to convey other information. Excited electronic states may be denoted by an asterisk. Examples H*, CI* Oxidation numbers are denoted by positive or negative Roman numerals or by zero (see also Section 2.10.1 (iv), p. 52). Examples Mnvn, manganese(VII), 0~n, Ni° The positions and meanings of indices around the symbol of the element are summarized as follows: left superscript left subscript right superscript right subscript mass number atomic number charge number, oxidation number, excitation , number of atoms per entity (see Section 2.10.1 (iii) below) (ii) Symbols for particles and nuclear reactions proton p, p+ positron e+ antiproton p positive muon (j.-1 neutron n negative muon |jT antineutron n photon y electron e, e~, p~ deuteron d ß+ triton helion a-particle (electron) neutrino electron antineutrino t h (3He24 a (4He2^ Ve Ve Particle symbols are printed in Roman (upright) type (HuTsee Chapter 6, p. 113). The electric charge of particles may be indicated by adding the superscript +, —, or 0; e.g. p+, n°, e~, etc. If the symbols p and e are used without a charge, they refer to the positive proton and negative electron respectively. A summary of recommended names for muonium and hydrogen atoms and their ions can be found in [75]. The meaning of the symbolic expression indicating a nuclear reaction should be as follows: initial / incoming particles^' outgoing particles \ final nuclide y or quanta or quanta j nuclide Examples 14N(a, p)170, ^N^> y)60Co 23Na(T, 3n)20Na, Jgtf, pn)29Si One can also use the standard notation from kinetics (see Section 2.12, p. 63). Examples 14,N + a -> 1780 + p n -> p + e^VeJ (iii) Chemical formulaV^ As in the case of chemical symbols of elements, chemical formulae have two different meanings: (a) On a microsco/mQ_level they denote one atom, one molecule, one ion, one radical, etc. The number of atoms m 'anlentity (always an integer) is indicated by a right subscript, the numeral 1 being omitted. Groups of atoms may be enclosed in parentheses. Charge numbers of ions and excitation symbols are added as right superscripts to the formula. The radical nature of an entity may be expressed by adding a dot to the symbol. The nomenclature for radicals, ions, radical ions, and related species is described in [76,77]. 50 Examples Xe, N2, C6H6, (CH3)3COH so42- NO NO* (a 2-methyl-2-propanol molecule) (an excited nitrogen dioxide molecule) (a nitrogen oxide molecule) (a nitrogen oxide molecule, stressing its free radical character) In writing the formula for a complex ion, spacing for charge number can be added (staggered arrangement), as well as parentheses: S042~, (S04)2~. The staggered arrangement is now recommended [74]. Specific electronic states of entities (atoms, molecules, ions) can be denoted by giving the electronic term symbol (see Section 2.6.3, p. 32) in parentheses. Vibrational and rotational states can be specified by giving the corresponding quantum numbers (see Section 2.6, p*: 25 and 33). Examples Hg(3Pi) a mercury atom in the triplet-P-one state HF(v = 2, J = 6) a hydrogen fluoride molecule in the vibrational state v = 2 and the rotational state J = 6 H20+(2Ai) a water molecule ion in the doublet-A-one state (b) On a macroscopic level a formula denotes a sample of a chemical substance (not necessarily stable, or capable of existing in isolated form). The chemical crJH^osition is denoted by right subscripts (not necessarily integers; the numeral 1 being omitted). A "formula unit" (which is not a unit of a quantity!) is an entity specified as a group of atoms (see (iv) and (v) below). Examples Na, Na+, NaCl, Fe0.9iS, XePtF6, NaCl The formula can be used in expressions like p(H2S04), mass density of sulfuric acid. When specifying amount of substance the formula is often multiplied wiilr^i factor, normally a small integer or a fraction, see examples in (iv) and (v). Less formally,Qfljrformula is often used as a shorthand ("reacting with H2S04"). Chemical formulae may be written in different ways according to the information they convey [15,74,78,79]: Formula Information conveyed Example Notes empirical molecular structural connectivity stereochemical stoichiometric proportion only in accord with molecular mass structural arrangemtfANcf, atoms connectivity stereochemical configuration CH20 C3H6O3 CH3CH(OH)COOH H H H- -C-I H O OH OH 0 Fischer projection resonance structure electronic arrangement H- H OH COO H -OH CH3 2,4 (1) Molecules differing only in isotopic composition are called isotopomers or isotopologues. For example, CH20, CHDO, CD20 and CH2170 are all isotopomers or isotopologues of the formaldehyde molecule. It has been suggested [16] to reserve isotopomer for molecules of the same isotopic 51 (iv) Equations for chemical reactions (a) On a microscopic level the reaction equation represents an elementary reaction (an event involving single atoms, molecules, and radicals), or the sum of a set of such reactions. Stoioiiomejtric numbers are ±1 (sometimes ±2). A single arrow is used to connect reactants and produ^SMn an elementary reaction. An equal sign is used for the "net" reaction, the result of a set of elementary reactions (see Section 2.12.1, p. 68). H + Br2 —> HBr + Br one elementary step in HBr formation H2 + Br2 = 2 HBr the sum of several such elementary steps (b) On a macroscopic level, different symbols are used connecting the reactants and products in the reaction equation, with the following meanings: H2 + Br2 = 2 HBr stoichiometric equation H2 + Br2 —> 2 HBr net forward reaction H2 + Br2 ^=> 2 HBr reaction, both directions H2 + Br2 f± 2 HBr equilibrium The two-sided arrow <-> should not be used for reactions to avoid confusion with resonance structures (see Section 2.10.1 (hi), p. 50). Stoichiometric numbers are not unique. One and the same reaction can be expressed in different ways (without any change in meaning). Examples The formation of hydrogen bromide from the elements can equally well be written in either of these two ways (1/2) H2 + (1/2) Br2 = HBr H2 + Br2 = 2 HBr Ions not taking part in a reaction ("spectator ions") are often removed from an equation. Redox equations are often written so that the value of the stoichiometric number for the electrons transferred (which are normally omitted from the overall equation) is equal to ±1. Example (1/5) KMnVII04 + (8/5) HC1 = (1/5) MnnCl2 + (1/2) Cl2 + (1/5) KC1 + (4/5) H20 The oxidation of chloride by permanganate in acid solution can thus be represented in several (equally correct) ways. Examples KMn04 + 8 HC1 = MnC^^(5/2) Cl2 + KC1 + 4 H20 MnO^ + 5 CI- + 8 Mh2+ + (5/2) Cl2 + 4 H20 (1/5) MnOT + CI" +0^) H+ = (1/5) Mn2+ + (1/2) Cl2 + (4/5) H20 Similarly a reaction in an electrochemical cell may be written so that the electron number of an electrochemical cell reaction, z, (see Section 2.13, p. 71) is equal to one: Example (1/3) In°(s) + (1/2) HgI2S04(s) = (1/6) Inni2(S04)3(aq) + Hg°(l) where the symbols in parentheses denote the state of aggregation (see Section 2.10.1 (vi), p. 54). (Notes continued) (1) (continued) composition but different structure, such as CD3CH(OH)COOH and CH3CD(OD)COOD for which one also uses isotope-isomer. (2) The lines in th#^onnectivity, stereochemical, or resonance structure formula represent single, double, or triple bonasJThey are also used as representing lone-pair electrons. (3) In the Fischer projection the horizontal substituents are interpreted as above the plane of the paper whereas the vertical substituents are interpreted as behind the plane of the paper. (4) The two-sided arrow represents the arrangement of valence electrons in resonance structures, such as benzene CqHq and does not signify a reaction equation (see Section 2.10.1 (iv), this page). 52 Symbolic Notation: A general chemical equation can be written as where Bj denotes a species in the reaction and Vj the corresponding stoichiometric number (negative for reactants and positive for products). The ammonia synthesis is equally well expressed in these two possible ways: (i) (1/2) N2 + (3/2) H2 = NH3 z/(N2) = -1/2, z/(H2) = -3/2, ^(N5*^Tl (ii) N2 + 3 H2 = 2 NH3 z/(N2) = -1, i/(H2) = -3, ^(NH3)^+2 The changes Arij = rij — n^o in the amounts of any reactant and product j during the course of the reaction is governed by one parameter, the extent of reaction £, through the equation rij = iijm + /'; £ The extent of reaction depends on how the reaction is written, but it is independent of which entity in the reaction is used in the definition. Thus, for reaction (i), when £ = 2 mol, then An(N2) = — 1 mol, An(H2) = —3 mol and An(NH3) = +2 mol. For reaction (ii), when An(N2) = — 1 mol then £ = 1 mol. Matrix Notation: For multi-reaction systems it is convenient to write the chemical equations in matrix form Av = where A is the conservation (or formula) matrix with elements A,-Lj representing the number of atoms of the ith. element in the jth reaction species (reactant or product entity) and v is the stoichiometric number matrix with elements Ujk being the stoichiometric numbers of the jth reaction species in the fcth reaction. When there are Ns reacting sp&ies'involved in the system consisting of Ne elements A becomes an Ne x Ns matrix. Its nullity, N(A) = Ns— rank(^4), gives the number of independent chemical reactions, Nr, and the Ns x A^sllttchiometric number matrix, u, can be determined as the null space of A. 0 is an Ne x Nr £^,o matrix [80]. (v) Amount of substance anl^ihe/Bpecification of entities The quantity "amount of substance" or "chemical amount" ("Stoffmenge" in German, "quantité de matiěre" in French) has been used by chemists for a long time without a proper name. It was simply referred to as the "number of moles". This practice should be abandoned: the name of a physical quantity should not contain the name of a unit (few would use "number of metres" as a synonym for "length"). The amount of substance is proportional to the number of specified elementary entities of that substance; the proportionality constant is the same for all substances and is the reciprocal of the Avogadro conltaM. The elementary entities may be chosen as convenient, not necessarily as physically real individual particles. In the examples below, (1/2) Cl2, (1/5) KMnGj, etc. are artificial in the sense that no such elementary entities exists. Since the amount of substance and all physical quantities derived from it depend on this choice it is essential to specify the entities to avoid ambiguities. 53 Examples nci, n(Cl) n(Cl2) n(H2S04) n((l/5) KMn04) M(P4) cci-,c(Cl-),[Cl-] P(H2S04) A(MgS04) A((l/2) MgS04) A(Mg2+) A((l/2) Mg2+) amount of CI, amount of chlorine atoms amount of Cl2, amount of chlorine molecules amount of (entities) H2S04 amount of (entities) (1/5) KMn04 molar mass of tetraphosphorus P4 amount concentration of Cl_ mass density of sulfuric acid molar conductivity of (entities of) MgS04 molar conductivity of (entities of) (1/2) MgS04 ionic conductivity of (entities of) Mg2i ionic conductivity of (entities of) (1/2) m£1/ Using definitions of various quantities we can derive equations like n((l/5) KMn04) = 5n(KMn04) A((l/2) Mg2+) = (l/2)A(Mg2+) [(1/2) H2S04] = 2 [H2S04] (See also examples in Section 3.3, p. í i-) Note that "amount of sulfur" is an ambiguous statement, because it might imply n(S), ra(Ss), or n(S2), etc. In most cases analogous statements are less ambiguous. Thus for compounds the implied entity is usually the molecule or the common formula entity, and for solid metals it is the atom. Examples "2 mol of water" implies n(H20) = 2 mol "0.5 mol of sodium chloride" implies ra(NaCl) "3 mmol of iron" implies ra(Fe) = 3 mmol 0.5 mol Such statements should be avoided whenever there might be ambiguity. In the equation pV = nRT and in equations involving colligative properties, the entity implied in the definition of n should be an independently translating particle (a whole molecule for a gas), whose nature is unimportant. Quantities that describe compositions of mixtures can be found in [72]. (vi) States of aggregation The following one-, two- or three-lemlr symbols are used to represent the states of aggregation of chemical species [l.j]. The lasers are appended to the formula symbol in parentheses, and should be printed in Roman (uprigl^tyye without a full stop (period). a, ads am aq aqj si cr f oo species adsorbed on a surface g amorphous solid 1 aqueous solution lc aqueous solution at infinite dilution mon /condensed phase n (i.e., solid or liquid) pol crystalline s fluid phase sin (i.e., gas or liquid) vit gas or vapor liquid liquid crystal monomeric form nematic phase polymeric form solid solution vitreous substance 54 Examples HCl(g) CV(f) Vm(lc) U(cr) Mn02(am) Mn02(cr, I) NaOH(aq) NaOH(aq, oo) Af^(H20, 1) hydrogen chloride in the gaseous state heat capacity of a fluid at constant volume molar volume of a liquid crystal internal energy of a crystalline solid manganese dioxide as an amorphous solid manganese dioxide as crystal form I aqueous solution of sodium hydroxide aqueous solution of sodium hydroxide at infinite dilution standard enthalpy of formation of liquid water The symbols g, 1, etc. to denote gas phase, liquid phase, etc., are also sometimes used as a right superscript, and the Greek letter symbols a, p, etc. may be similarly used to denote phase a, phase P, etc. in a general notation. Examples Vml, Vms molar volume of the liquid phase, ... of the solid phase Sma, Sm$ molar entropy of phase a, ... of phase p 4* 1 v O 55 2.11 CHEMICAL THERMODYNAMICS The names and symbols of the more generally used quantities given here are also recommended by IUPAP [4] and by ISO [5.d,5.h]. Additional information can be found in [l.d,l.j] and [8l]L Name Symbol Definition SI unit Notes heat Q,q J 1 work W,w J 1 internal energy U dU=dQ+dW J 1 enthalpy H H = U + PV J thermodynamic T,(G) K temperature International T90 2 temperature Celsius temperature 9, t e/°C = T/K - 273.15 °C 3 entropy S dS = dQrev/T JK-1 Helmholtz energy, A,F A = U - TS 4 (Helmholtz function) Gibbs energy, G G = H -TS J (Gibbs function) Massieu function J j = -A/T JK-1 Planck function Y Y = -G/T JK-1 surface tension 7, a 7 = (dG/dAsCp^ \ J m-2, N m-1 molar quantity X Xm, (X) Xm = Xj n [X]/mol 5, 6 specific quantity X X x = X/m [X]/kg 5, 6 pressure coefficient ß ß = (dpßT)v Pa K -1 relative pressure coefficient ap ap = (l/p)(dp/dT)v K-1 compressibility, isothermal KT = -(l/V){dV/dp)T Pa"1 isentropic KS Ks £^/V)(dV/dP)s Pa-1 linear expansion coefficient m ai = a/i)(di/dT) K-1 cubic expansion coefficient a, ay, 7 a = {\/V){dV/dT)p K-1 7 heat capacity, at constant pressure Cp Cp = (dHßT)p JK-1 at constant volume Cv Cv = (dU/dT)v JK-1 (1) In the differential form, d denotes an inexact differential. The given equation in integrated form is AU = Q + W. Q > 0 and W > 0 indicate an increase in the energy of the system. (2) This temperature is defined by the "International Temperature Scale of 1990 (ITS-90)" for which specific definitions are prescribed [82-84]. However, the CIPM, in its 94th Meeting of October 2005, has approved a Recommendation T3 (2005) of the Comite Consultatif de Thermometrie, where the ITS-90 becomes one of several "mises en pratique" of the kelvin definition. The Technical Annex (2005) is available as Doc. CCT_05_33 [85]. It concerns the definition of a reference isotopic composition of hydrogen and water, when used for the realization of fixed points of the "mises en pratique". (3) This quantity is sometimes misnamed "centigrade temperature". (4) The symbol F is sometimes used as an alternate symbol for the Helmholtz energy. (5) The definition applies to pure substance. However, the concept of molar and specific quantities (see Section 1.4, p. 6) may also be applied to mixtures, n is the amount of substance (see Section 56 Name Symbol Deft nition SI unit Notes ratio of heat capacities 7.(«) 7 = Cp/Cy 1 Joule-Thomson coefficient P, PJT P = (dT/dP)H K Pa"1 thermal power <2>,P $ = dQ/dt W virial coefficient, second B pVm = RT(1 + B/Vm+ m3 mol"J^ V8 third C C/Vm2 + ---) m6 mol-2 8 van der Waals a (P + a/Vm2)(Vm -b)=RT J m3 mol-2 9 coefficients b 9 compression factor, Z z = pVm/RT 1A (compressibility factor) partial molar Xb, {Xb) XB - = (dX/driB)T,p,nJ7,B 10 quantity X chemical potential, P PB = = (9G/9nB)T,p,n37,B J mor1 11 (partial molar Gibbs • energy) standard chemical J mor1 12 potential absolute activity A AB = = exp(fiB/RT) 1 11 (relative) activity a aB = 1 11, 13 standard partial Hb* = PB*+^B*\ J mor1 11, 12 molar enthalpy standard partial Sb* Sb* = -(dpB^T)p J mor1 KT1 11, 12 molar entropy (Notes continued) (5) (continued) 2.10, notes 1 and 2, p. 47). (6) X is an extensive quantity, whose SI unit isjpii^ In the case of molar quantities the entities should be specified. Example Vm:b = ^m(B) = V/ub denotes the molar volume of B (7) This quantity is also called the coefficient of thermal expansion, or the expansivity coefficient. (8) Another set of "pressure virial coefficients" may be defined by PVm = RT(1 + BpP + Cpp2 + ■■■) (9) For a gas satisfying the van der Waals equation of state, given in the definition, the second virial coefficient is related to the parameters a and b in the van der Waals equation by B = b-a/RT (10) The symbol applies to ettities|B which should be specified. The bar may be used to distinguish partial molar X from X when necessary. Example The partial molar volume of Na2SO"4 in aqueous solution may be denoted F(Na2S04, aq), in order to distinguish it from the volume of the solution F(Na2S04, aq). (11) The definition applies to entities B which should be specified. The chemical potential can be defined equivalently by the corresponding partial derivatives of other thermodynamic functions (U, H, A). (12) The symbol ^ or ° is used to indicate standard. They are equally acceptable. Definitions of standard states are discussed in Section 2.11.1 (iv), p. 61. Whenever a standard chemical potential 57 Name Symbol Definition 5/ unit Notes standard reaction Gibbs ATG^ J mor1 12, 14 energy (function) B 15, 16 affinity of reaction A, A A = -(dG/dt)PtT B J mor1 15 ^ \ standard reaction ATH* ATH* = £ vbHb* J mor1 Zl2, 14 enthalpy B 7l5, 16 standard reaction ATS* J mor1 K-1 12, 14 entropy B reaction quotient q q = U aBu» B K* = exp(-ATG*/KT) 1 17 equilibrium constant K*,K 1 12, 15: equilibrium constant, pressure basis kp kp = n pbub KC = U cb"b PaEl/B 15, 19 concentration basis Kc (mol m~ 15, 19 molality basis k I i Km = U TTlBVB B /B = AB lim(pB/AB)T (mol kg" -1)5>b 15, 19 fugacity f,p ^Pa 11 fugacity coefficient 4>b = Ib/pb 1 Henry's law constant fcH,B = lim (/efe) J Pa 11, 20 = (3/b/^\=o (Notes continued) (12) (continued) /T or a standard equilibrium constant or other standard quantity is used, the standard state must be specified. (13) In the defining equation given here, the pressure dependence of the activity has been neglected as is often done for condensed phases at atmospheric pressure. An equivalent definition is aB = Ab/Ab^, where Ab^ = ex.p(pB^/RT). The definition of /T depends on the choice of the standard state (see Section 2.11.1 (iv), p. 61). (14) The symbol r indicates reaction in^ureral. In particular cases r can be replaced by another appropriate subscript, e.g. AfH^ denotes the standard molar enthalpy of formation; see Section 2.11.1 (i), p. 59 below for a list of subscripts. Ar can be interpreted as operator symbol Ar = d/d£. (15) The reaction must be specified for which this quantity applies. (16) Reaction enthalpies (and reaction energies in general) are usually quoted in kJ mor1. In the older literature kcal mor1 is also common, however, various calories exist. For the thermochemical calorie, 1 kcal = 4.184 kJ (see^SectioTi 7.2, p. 137). (17) This quantity applies in general to a system which is not in equilibrium. (18) This quantity is equal to q in equilibrium, when the affinity is zero. It is dimensionless and its value depends on the choice of standard state, which must be specified. ISO [5.h] and the IUPAC Thermodynamics Commission [81] recommend the symbol and the name "standard equilibrium^MSffifnt". Many chemists prefer the symbol K and the name "thermodynamic equilibrium const any'. (19) These quantities are not in general dimensionless. One can define in an analogous way an equilibrium constant in terms of fugacity Kf, etc. At low pressures Kp is approximately related to by the equation K*~ k, Kp/(p^)T'1/B, and similarly in dilute solutions Kc is approximately related to by k, Kc/(c^)^ub; however, the exact relations involve fugacity coefficients or activity coefficients [81]. 58 Name Symbol Definition SI unit Notes activity coefficient referenced to Raoult's law / referenced to Henry's law molality basis 7m concentration basis 7C mole fraction basis ^x ionic strength, molality basis Im,I concentration basis Ic, I osmotic coefficient, molality basis (f)m mole fraction basis (f)x osmotic pressure H (Notes continued) (19) (continued) The equilibrium constant of dissolution of an electrolyte (describing the equilibrium between excess solid phase and solvated ions) is often called a solubility product, denoted Kso\ or Ks (or Kso\^ or Ks^ as appropriate). In a similar way the equilibrium constant for an acid dissociation is often written Ka, for base hydrolysis K^, and fco^w^ter dissociation Kw. (20) Henry's law is sometimes expressed in terms of molalities or concentration and then the corresponding units of Henry's law constant are Pa kg mol-1 or Pa m3 mol-1, respectively. (21) This quantity applies to pure phases, substances in mixtures, or solvents. (22) This quantity applies to solutes. (23) A is the solvent, B is one or more sondes, (24) The entities B are independent solute molecules, ions, etc. regardless of their nature. Their amount is sometimes expressed in osm^hWmeaning a mole of osmotically active entities), but this use is discouraged. (25) This definition of osmotic pressure applies to an incompressible fluid. V 2.11.1 Other symbols and conventions in chemical thermodynamics A more extensive descripfira^f this subject can be found in [81]. (i) Symbols used as subscripts to denote a physical chemical process or reaction These symbols should b^Winted in Roman (upright) type, without a full stop (period). adsorption ads atomization at combustion reaction c dilution (of a solution) dil ^placement dpi RTMAZ mB b 1 21 1 ii, 22 1 >! ii, 22 10, 21 /b = liquid) fus mixing of fluids mix reaction in general r solution (of solute in solvent) sol sublimation (solid —> gas) sub transition (between two phases) trs triple point tp vaporization, evaporation (liquid - -> gas) vap (ii) Recommended superscripts activated complex, transition state |, 7^ apparent app excess quantity E ideal id infinite dilution 00 pure substance * standard o (iii) Examples of the use of the symbol A The symbol A denotes a change in an extensive thermodynamic quantity for a process. The addition of a subscript to the A denotes a change in the propertx^ Examples AVap# AVap# A\gH = H(g) — H(l) for the molar enthalpy of vaporization. 40.7 kJ mol-1 for water at 100 °C under its own vapor pressure. This can also be written AH^p, but this usage is not recommended. The subscript r is used to denote changes associated with a chemical reaction. Symbols such as ATH are defined by the equation ArF = 5>b#b = (9#/90t,p b It is thus essential to specify the stoichiometric reaction equation when giving numerical values for such quantities in order to define^textent of reaction £ and the value of the stoichiometric numbers ub ■ Example N2(g) + 3 H2(g) = 2 NH3(g), ATH* (298.15 K) = -92.4 kJ mol-1 ATS*(298.15 K) = -199 J mol-1 K-1 The mol-1 in the units identifies the quantities in this example as the change per extent of reaction. They may be called the molar enthalpy and entropy of reaction, and a subscript m may be added to the symbol to emphasize the difference from the integral quantities if desired. The standard reaction quantities are particularly important. They are defined by the equations b ATS* = Y,^S^ b b 60 It is important to specify notation with care for these symbols. The relation to the affinity of the -A = ATG = ATG* + RT In (jj aB "B and the relation to the standard equilibrium constant is ATG* = — RT In K*~. The product of the activities is the reaction quotient Q, see p. 58. The term combustion and symbol c denote the complete oxidation of a substance. For the definition of complete oxidation of substances containing elements other than C, H and O see [86]. The corresponding reaction equation is written so that the stoichiometric number zTof the substance is -1. Example The standard enthalpy of combustion of gaseous methane is AcfP(CH4, g, 298.15 K) = -890.3 kJ mol"1, implying the reaction CH4(g) + 2 02(g) ^ C02(g) + 2 H20(1). The term formation and symbol f denote the formation of the suwteince from elements in their reference state (usually the most stable state of each element at the chosen temperature and standard pressure). The corresponding reaction equation is written so that the stoichiometric number v of the substance is +1. Example The standard entropy of formation of crystalline mercury(II) chloride is AflS^(HgCl2, cr, 298.15 K) = -154.3 J mol^K^implying the reaction Hg(l) + Cl2(g) - HgCl2(cr). The term atomization, symbol at, denotes a procesj^n\vhich a substance is separated into its constituent atoms in the ground state in the gas phase. The corresponding reaction equation is written so that the stoichiometric number v of the substance is —1. Example The standard (internal) energy of atomization of liquid water is Aat^(H20, 1) = 625 kJ mol-1, implying the reaction H20(1) ^ 2 H(g) + 0(g). (iv) Standard states [l.j] and [81] The standard chemical potential of substance B at temperature T,/iB&(T), is the value of the chemical potential under standard conditions, specified as follows. Three differently defined standard states are recognized. For a gas phase. The standard state for a gaseous substance, whether pure or in a gaseous mixture, is the (hypothetical) state of the pure substance B in the gaseous phase at the standard pressure p = p^ and exhibiting ideal gas behavior. The standard chemical potential is defined as PbKT) J lim [/iB(T,p,yB, •••) - RTln{yBp/p*)} For a pure phase, or a mixture, or a solvent, in the liquid or solid state. The standard state for a liquid or solid substance, whether pure or in a mixture, or for a solvent, is the state of the pure substance B in the liquid or solid phase at the standard pressure p = p^. The standard chemical potential is defined as /iB(r)=/4(r,p-) For a solute in solution. For a solute in a liquid or solid solution the standard state is referenced to the ideal dilute behavior of the solute. It is the (hypothetical) state of solute B at the standard molality m^/sta^dard pressure , and behaving like the infinitely dilute solution. The standard chemical potential is defined as 61 /iB*(T) = \pB(T,p*,mB,...) - RTln(mB/m«)] oo The chemical potential of the solute B as a function of the molality mB at constant pressdrep J p^ is then given by the expression Sometimes (amount) concentration c is used as a variable in place of molality m;'br34h of the above equations then have c in place of m throughout. Occasionally mole fraction x is used in place of m; both of the above equations then have x in place of m throughout, aiKN^ = 1. Although the standard state of a solute is always referenced to ideal dilute behavior, the definition of the standard state and the value of the standard chemical potential are different depending on whether molality m, concentration c, or mole fraction x is used as a variable. (v) Standard pressures, molality, and concentration In principle one may choose any values for the standard pressure pVtl^ standard molality , and the standard concentration c6', although the choice must be specified. For example, in tabulating data appropriate to high pressure chemistry it may be 4wwtenient to choose a value of p* = 100 MPa (= 1 kbar). In practice, however, the most common choice is These values for rrf and are universally accepted. The value for p^ = 100 kPa, is the IUPAC recommendation since 1982 [l.j], and is recommended for tabulating thermodynamic data. Prior to 1982 the standard pressure was usually Ml»enao be p^ = 101 325 Pa (= 1 atm, called the standard atmosphere). In any case, the value fj^iy should be specified. The conversion of values corresponding to different p^ is described in [87-89]. The newer value of , 100 kPa is sometimes called the standard state pressure. (vi) Biochemical standard states Special standard states that are close to physiological conditions are often chosen. The biochemical standard state is often chosen at [H+] = 10~7 mol dm~3. The concentrations of the solutes may be grouped together as for example, the total phosphate concentration rather than the concentration of each component, (H3PO4, H2P04~, HP042~, P043~), separately. Standard and other reference states must be specified with care [90,91]. (vii) Thermodynamic properties Values of many thermodynamic quantities represent basic chemical properties of substances and serve for further calculations. Extensive tabulations exist, e.g. [92-96]. Special care has to be taken in reporting the data and their uncertainties [97,98]. (viii) Reference state (of an element) The state in which the element is stable at the chosen standard state pressure and for a given temperature [16]. HB(mB) = Hb^ + RTln(mB^m,B/'m^) m -©- 0.1 MPa = 100 kPa ( = 1 bar) 1 mol kg-1 1 mol dm~3 62 2.12 CHEMICAL KINETICS AND PHOTOCHEMISTRY The recommendations given here are based on previous IUPAC recommendations [l.c,l.k] and [17], which are not in complete agreement. Recommendations regarding photochemistry are given in [60] and for recommendations on reporting of chemical kinetics data see also [99]. A glossary of terms used in chemical kinetics has been given in [100]. ^\ y Notes Name rate of change of quantity X rate of conversion rate of concentration change (due to chemical reaction) rate of reaction (based on amount concentration) rate of reaction (based on number concentration), (reaction rate) partial order of reaction overall order of reaction rate constant, rate coefficient Symbol Definition X X = dX/dt £ £ = d£/dt tb,vb rB = dcB/d£ v,vc v = v-Q^dc-Q/dt = £/V v,vc vc = ub^dCB/dt me,tiB v = k Y\ cb™-0 B m,n m = ^ mB B k, k(T) v = k ]1 cBmB B 5/ unit (varies) mol s-1 mol m~3 s_1 mol nw3fl5L m»3 sw 1 2 3,4 2, 4 (m3 mol ) l\m— 1 (1) E.g. rate of pressure change p = dp/dt, for which the SI unit is Pa s 1, rate of entropy change dS/dt with SI unit J K"1 s-1. (2) The reaction must be specified, for which this quantity applies, by giving the stoichiometric equation. (3) The symbol and the definition apply to entities B. (4) Note that rB and v can also be defined on the basis of partial pressure, number concentration, surface concentration, etc., with analogous definitions. If necessary differently defined rates of reaction can be distinguished by a subscript, e.g. vp = ^B_1dpB/dt, etc. Note that the rate of reaction can only be defined for a reaction of known and time-independent stoichiometry, in terms of a specified reaction equation; also the second equation for the rate of reaction follows from the first only if the volume V is constant and CBS^uniform throughout (more generally, the definition applies to local concentrations). The derivatives must be those due to the chemical reaction considered; in open systems, such as flow systems^p^ts due to input and output processes must be taken into account separately, as well as transport processes in general by the equation (dcB/dt)total = (dCB/dt)jJtetion + (dcB/d£)tranSp0rt (5) The symbol applies to reactant B. The symbol m is used to avoid confusion with n for amount of substance. The order of reaction is only defined if the particular rate law applies, m is a real number. (6) Rate constants k and pre-exponential factors A are usually quoted in either (dm3 mol_1)m_1 s_1 or on a molecular scale in (cm3 )m_1 s_1 or (cm3 molecule-1)"1-1 s_1. Note that "molecule" is not a unit, but is often included for clarity, although this does not conform to accepted usage. k(T) is written to stress temperature dependence. Rate constants are frequently quoted as decadic logarithms. Example second order reaction k = 1082 cm3 mol-1 s-1 or lg(fc/cm3 mol-1 s-1) = 8.2 or alternatively k = 10~12'6 cm3 s_1 or lg(fc/cm3 s_1) = —12.6 63 Name Symbol Definition SI unit Notes rate constant of kuni, kuni(T, cm) unimolecular reaction at high pressure k^ at low pressure &o Boltzmann constant k, k-Q half life t1/2 relaxation time, r lifetime, mean life (Arrhenius) activation EA,Ea energy pre-exponential factor, A frequency factor hard sphere radius r collision diameter dAB collision cross section a mean relative speed cab between A and B collision frequency of A with A zA(A) of A with B zA(B) (7) The rates of unimolecular reactions show a dependence upon the concentration cm of a collision partner M. One writes kuni(T, cm) to emphasize the temperature and "pressure" dependence. At high cm(^ oo) the dependence vanishes. At low cm(^ j^itVives a partial order 1 of reaction with respect to cm- In this case one defines the second order rate constant ko. (8) r is defined as the time interval in which a concentration perturbation Ac falls to 1/e of its initial value Ac(0). If some initial concentration-oN^substance decays to zero as t —> oo, as in radioactive decay, then the relaxation time is the average lifetime of that substance (radioisotope). This lifetime or decay time must be distinguished from the half life. (9) One may use as defining equation EA = -i2d(lnfe)/d(l/T) The term Arrhenius activation energy is to^he used only for the empirical quantity defined in the table. Other empirical equations with different "activation energies", such as k(T) = A' Tn exp(—Ea'/RT) are also being used. In such expressions A', n, and EJ are taken to be temperature independent parameters. The term activation energy is also used for an energy threshold appearing in the electronic potential (the height of the electronic energy barrier). For this "activation energy" the symbol Eq and the term threshold energy is preferred, but E& is also commonly used. Furthermore, E$ may or may not include a correction for zero point energies of reactants and the transition state. It is thus recommended to specify in any given context exactly which activation energy is meant and to reserve (Arrhenius) activation energy only and exactly for the quantity defined in the table. EA depends on temperature and may be written EA(T). (10) A is dependent on temperature and may be written A(T). (11) The collision cross section a is a constant in the hard sphere collision model, but generally it is energy dependent. One may furthermore define a temperature dependent average collision cross section (see note 16). C denotes the number concentration. (12) fi is the reduced mass, fi = mAmB/(mA + tti-q). v = kunicB s 7 kum(cM oo) s_1 7 v = &ocmcb m3 m°l 1 s X^ 7 J K-1 c(*i/2) = c(0)/2 s Ac(r) = Ac(0)/e s 8 EA = i?T2d(ln k)/dT J mol^ 9 A = kexp(EA/RT) (m3*^-1)™-1 s"1 9, 10 m oab = rA + rB m a = tcoab2 m2 11 cab (8kT/sip)1/2 m s-1 12 zA(A) = V2CAac* s-1 11 zA(B) = Cb i 0 and N$(E) is the corresponding number of states of the transition state. In the framework of the statistical adiabatic channel model one has V ; hp(E,J) ymax -g ^e maximum of the potential of adiabatic channel a, H(x) is the Heaviside function (see Section 4.2, p. 107) and J is the angular momentum quantum number [102]. There may be further constants of the motion beyond J appearing here. (20) The Michaelis constant arises in the treatment of a mechanism of enzyme catalysis ki k2 E + S *± (ES) *± E + P k-i k-2 where E is the enzyme, S the substrate, ES the enzyme-substrate complex and P the product. V is called limiting rate [100]. (21) In generalized first-order kinetics, the rate equation can be written as a matrix equation, with the concentration vector c = (c^tfs^W 'C«)T and the first-order rate coefficients Kji as matrix elements. (22) The quantum yield 0 is defined in general by the number of defined events divided by number of photons absorbed [60]. Fo/a photochemical reaction it can be defined as d£/di dra7/d£ which is the rate of conversion divided by the rate of photon absorption. (23) For exponential decay by spontaneous emission (fluorescence) one has for the decay of the excited state concentration c* dlhu] dc* ' >^ dt dt The full widtl^aPl|a(i maximum T of a lorentzian absorption line is related to the rate constant of fluorescence. For a predissociation Tp is related to the predissociation rate constant kp. However, linewidths may also have other contributions in practice. 67 2.12.1 Other symbols, terms, and conventions used in chemical kinetics Additional descriptions can be found in [100]. (i) Elementary reactions Reactions that occur at the molecular level in one step are called elementary reactions. It is conventional to define them as unidirectional, written with a simple arrow always from left to right. The number of relevant reactant particles on the left hand side is called the molecularity of the elementary reaction. Examples A —> B + C unimolecular reaction (also monomolecular) A + B —> C + D bimolecular reaction A + B + C^D + E trimolecular reaction (also termolecular) (ii) Composite mechanisms A reaction that involves more than one elementary reaction is said to occur by a composite mechanism. The terms complex mechanism, indirect mechanism, and stepwise mechanism are also commonly used. Special types of mechanisms include chain-reaction mechanisms, catalytic reaction mechanisms, etc. Examples A simple mechanism is composed of forward and reverse reactions A B + C B + C A It is in this particular case conventional t<| writa these in one line A *± B + C However, it is useful in kinetics to distinguish this from a net reaction, which is written either with two one-sided arrows or an "equal" sign A f= B + C A = B + C When one combines a composite mechanism to obtain a net reaction, one should not use the simple arrow in the resulting equation. Example A —> B + C unimolecular elementary reaction B + C —> D + E bimolecular elementary reaction A = D + E net reaction (no elementary reaction, no molecularity) It is furthermore useful to distinguish the stoichiometric equation defining the reaction rate and rate constant from the equation defining the elementary reaction and rate law. Example Recombination ormeVkyl radicals in the high pressure limit The elementary reaction is CH3 + CH3 —> C2H6 This has a seikmd H2 with the same stoichiometric equation is a different elementary reaction (a very unlikely one), not the same as the trimolecular reaction. Other examples include catalyzed reactions such as X + A ^ + 3 with the stoichiometric equation A = B and the rate vc = —d[A]/d£ = fc[A][X] (^ d[X]/dt = 0). Again, the unimolecular reaction A —> B exists, but would be a different elementary reaction with the same stoichiometry. The elementary reaction isH + H + H^H2 + H This has a third order rate law. The stoichiometric equation is 2H = H2 and the definition of the reaction rate gives o 69 2.13 ELECTROCHEMISTRY Electrochemical concepts, terminology and symbols are more extensively described in [l.i] and [103-107]. For the field of electrochemical engineering see [108], for semiconductor electrochemistry and photo-electrochemical energy conversion see [109], for corrosion nomenclature see [110], for impedances in electrochemical systems see [111], for electrokinetic phenomena see [112], for electrochemical biosensors see [113], and for chemically modified electrodes (CME) see [114]. Name Symbol Definition SI umhs. Notes elementary charge e proton charge Faraday constant F F = eNA C mor1 charge number of an ion z zb = Qv/e 1 ionic strength, molality basis I 171} I Im = (1/2) ^rriiZi2 mol kg-1 concentration basis Ic,I /c = (l/2) Y.clZl2 mol m~3 2 mean ionic molality m± mol kg-1 3 mean ionic activity 7± 1 3 coefficient mean ionic activity a± a± = m±^f±/m^ 1 3, 4,5 activity of an electrolyte a(kv+Bv_) a{Av+Bv_) = cgyy) 1 3 pH pH pH = - lgp^+ 1 5, 6 outer electric potential v> V 7 surface electric potential X V 8 inner electric potential 4> = x + V> V Volta potential difference Aj'ip f^fji-7- V 9 (1) The definition applies to entities B. (2) To avoid confusion with the cathodic current, symbol Ic (note roman subscript), the symbol / or sometimes fi (when the current is denoted by /) is used for ionic strength based on concentration. (3) v+ and v- are the numbers of cations and anions per formula unit of an electrolyte Av+Bv-. Example For Al2(804)3, v+ = 2 anj£p% ^3. m+ and m_, and 7+ and 7_, are the molalities and activity coefficients of cations and anions. If the molality of A.v+ B^_ is m, then m+ = u+m and m_ = v-m. A similar definition is used on a concentration scale for the mean ionic concentration c±. (4) rrf = 1 mol kg-1. (5) For an individual ion, neither the activity a+,a_ nor the activity coefficient 7+,7- is experimentally measurable. (6) The definition of pH is discussed in Section 2.13.1 (viii), p. 75. The symbol pH is an exception to the general rules for the symbols of physical quantities (Section 1.3.1, p. 5) in that it is a two-letter symbol and it is always pwrnea in Roman (upright) type. (7) is the electrostatic potential of phase p due to the electric charge Q of the phase. It may be calculated from classical electrostatics. For example, for a conducting sphere with excess charge Q and radius r, placed^in vacuo, i\) = Q/AK£or. (8) The surface potential is defined as the work (divided by the charge) necessary to transfer an ideal (i.e. massless) positive charge through a dipole layer at the interphase between phase a and p. The absolute value of x cannot be determined, only differences are measurable. (9) Aip is the outer potential difference due to the charge on phases a and p. It is a measurable quantity. 70 Name Symbol Defi nition SI unit Notes Galvani potential difference A.P. V electrochemical potential PBa PBa = (dG/dnBa)T,p,njjtB = /iBa* + RT In aB + zBF*18 potential (of an electro- i chemical reaction) ^/ formal potential Eeq = E»' -{RT/zF^Vihiici/c* ) v 1 15, 19 liquid junction potential E] 20 electric current I I = dQ/dt AA J21 electric current density 3,3 3 = I/A A mX *22 faradaic current h h = = h+h A J 23 reduction rate constant kc Ic = -- -nFAkcU(cB')nB 1, 15, 24 oxidation rate constant K h = -- nFAK UM nB (varies) 1, 15, 24 transfer coefficient a, ac ac = = -(RT/nF)d(lnkc)/dE 1 15, 25 overpotential V, Er, V = E — Eeq Tafel slope b b = (dE/dln\IF\)c^p V 26 mass transfer coefficient kd kd,b = Wb\ hm,b/nFcA m s_1 1, 15, 27 (17) (continued) transfer equilibria across phase boundaries that are represented in the cell diagram (except at possible electrolyte-electrolyte junctions) and local chemical equilibria within the phases are established. (18) This is the Nernst equation. Yl vim ai refers to the electrode reaction, where a« are the activities of the species taking part in it; v,t are the stoichiometric numbers of these species in the equation for the electrode reaction, written as a reduction, v,t is positive for products and negative for reactants. The equilibrium potential is also called Nernst potentialfayStreversible potential. (19) It is E*~' which is calculated in electrochemical experiments when the concentrations of the various species are known, but not their activities. ^tf^raTue depends on the composition of the electrolyte solution. The argument of In is dimensionless, the concentration c,L is normalized through division by the standard concentration c6', usually c6' = 1 mol dm~3 for soluble species. (20) Ej is the Galvani potential difference between two electrolyte solutions in contact. (21) Q is the charge transferred through the external circuit of the cell. (22) Formally, the current density is a vector, dl = j ■ endA (see Section 2.3, note 2, p. 16). (23) If is the current through the electrode (solution interface, resulting from the charge transfer due to an electrode reaction proceeding as reactants + ne~ —> products. Ic is the cathodic partial current due to the reduction reaction, 7a is the anodic partial current due to the oxidation reaction. By definition Ic is negative and 7a positive. At the equilibrium potential, 7a = —Ic = Iq (the exchange current) and ja = — jc = jo (the exchange current density). I or j may achieve limiting values, indicated by the subscript lim, in addition to the subscripts F, c, or a. (24) For a first-order reaction the SI unit is m s_1. Here n (or z) is the number of electrons transferred in the electrochemical! reaction, cb' is the concentration at the interphase, nB is the order of the reaction with respect to entity B. The formerly used symbols kTed,k{ and k (for kc) and kox, k^ and k (for kg) are not recommended. (25) a or ac is also named cathodic transfer coefficient. Analogously the anodic transfer coefficient is defined as aa = (RT/nF)d(]n k&)/dE. At the same potential aa = 1 — ac. The symmetry factor, (3, is the cathodic transfer coefficient, a,L of an elementary reaction step i, in which only one electron is transferred. (26) The Tafel slope b is an experimental quantity from which kinetic information can be derived. (27) The mass transfer coefficient is the flux divided by the concentration. For steady-state mass transfer, k^t =^Ib/^b where 5b is the diffusion layer thickness (which is model dependent) and Db is the diffusion coefficient of entity B. For more information see [17]. 72 Name Symbol Definition SI unit Notes electrokinetic potential c V (C-potential) conductivity k, (a) 3 = kE S m-1 22 ,28 electric mobility u, (m) ub = \vb\/\E\ m2 V-1 s-1 ^ 1, 29 ionic conductivity, A Ab = \zb\Fub S m2 mol^L 1, ,30, 31 molar conductivity of an ion molar conductivity A A{AU+BU-) = i/+A_ S m2/^J 1, 30, 31 transport number t £b = Abcb/X^ä i i 1 1 (28) Conductivity was formerly called specific conductance. E is the electric field strength vector. (29) vb is the migration velocity of entities B and \E\ is the electric field strength within the phase concerned. (30) The unit S cm2 mol 1 is often used for molar conductivity. The conductivity is equal to k = AjCj. (31) It is important to specify the entity to which molar conductivity refers. It is standard practice to choose the entity to be 1/zb of an ion of charge number zb, in atadit to normalize ion charge, so that for example molar conductivities for potassium, barium and lanthanum ions would be quoted as A(K+), A((l/2) Ba2+), or A((l/3) La3+) the so-called molecular conductivity of equivalent entities (formerly called equivalent conductivity) [115]. 2.13.1 Sign and notation conventions in electrochemistry 1 (i) Electrochemical cells Electrochemical cells consist of at least two (usually metallic) electron conductors in contact with ionic conductors (electrolytes). The current flow^jrpugh electrochemical cells may be zero or non-zero. Electrochemical cells with current flo!N^n operate either as galvanic cells, in which chemical reactions occur spontaneously and chemical energy is converted into electrical energy, or as electrolytic cells, in which electrical energy is converted into chemical energy. In both cases part of the energy becomes converted into (positive or negative) heat. (ii) Electrode There are at present two usages for the term electrode, namely either (i) the electron conductor (usually a metal) connected to the external circuits or (ii) the half cell consisting of one electron conductor and at least one ionic conductor. The latter version has usually been favored in electrochemistry. (iii) Representation of electrochemical cells Electrochemical cells are represented by diagrams such as those in the following examples: Examples Pt(s) | H2(g) | HCl(aq) | AgCl(s) | Ag(s) Cu(s/^CugD4(aq) j ZnS04(aq) | Zn(s) Cu(s^ViS04(aq) i! KCl(aq, sat) i! ZnS04(aq) | Zn(s) A single vertical bar ( | ) should be used to represent a phase boundary, a dashed vertical bar (j) to represent a junction between miscible liquids, and double dashed vertical bars (jj) to represent a liquid junction, in which the liquid junction potential is assumed to be eliminated. 1 These are in accordance with the "Stockholm Convention" of 1953 [103]. 73 (iv) Potential difference of an electrochemical cell The potential difference of an electrochemical cell is measured between a metallic conductor attached to the right-hand electrode of the cell diagram and an identical metallic conductor attached to the left-hand electrode. Electric potential differences can be measured only between two pieces of material of the same composition. In practice, these are almost always two pieces of copper, attached to the electrodes of the cell. At a boundary copper | electrode material there is a contact potential differja^y its value is incorporated in the constant to which the electrode potential is referred. Due to the different rates of diffusion of anions and cations from one solution to the other, liquid junction potentials, Ej, appear whenever two solutions of different composition are immiscible or are separated by a diaphragm or some other means to avoid mixing. If equilibrium i^^ft established at these junctions, the cells may include unknown junction potentials. Salt bridges (see, for example, the third cell in the previous Section 2.13.1 (hi)) are commonly employed to minimize or to stabilize the contributions of (liquid) junction potentials at the interface of two miscible electrolyte solutions to the measured potential difference. At the interface of two immiscible electrolyte solutions, a thermodynamic distribution potential can be established due to equilibrium charge partition. (v) Standard potential of an electrochemical cell reaction If no current flows through the cell, and all local charge transfer and local chemical equilibria of each electrode reaction are established, the potential differencel^iie cell is related to the Gibbs energy of the overall cell reaction by the equation ArG = -zFEcen. eq assuming that junction potentials are negligible. If the reacN^pn of the electrochemical cell as written takes place spontaneously, ArG is negative and Ece\\ positive. If the cell is written with the electrodes in the inverse order ArG is positive and Ece\\ is negative. The equilibrium potential of the cell, i.e. when no current flows, is given by j~>^p _ -Ecell,eq = -E%ell,eq--- > vr m ai a,t are the activities of the species taking part in the cell reaction and v,t are the stoichiometric numbers of these species in the equation written for the cell reaction (see also notes 15 and 16, p. 71). (vi) Standard electrode potential (of an electrochemical reaction) The standard potential of an electrochemical reaction, abbreviated as standard potential, is defined as the standard potential of a hypothetical cell, in which the electrode (half-cell) at the left of the cell diagram is the standard hydrogen electrode (SHE) and the electrode at the right is the electrode in question. This implies thatjfche cell reaction always involves the oxidation of molecular hydrogen. The standard hydrogen electrode consists of a platinum electrode in contact with a solution of H+ at unit activity and saturated with H2 gas with a fugacity referred to the standard pressure of 105 Pa (see Section 2.11.1 (v), p. 62). For a metallic electrode in equilibrium with solvated ions the cell diagram is Pt I H2 I H+ J J JVP+ I M and relates to the reaction JVP+ + (z/2) H2(g) = M + z H+ This diagram may be abbreviated E(MZ+/M), but the order of these symbols should not be reversed. Note that the standard hydrogen electrode as defined is limited to aqueous solutions. For more information on measuring electrode potentials in aqueous and non-aqueous systems, see [103,107]. 74 (vii) Anode, cathode The terms anode and cathode may only be applied to electrochemical cells through which a net current flows. In a cell at equilibrium the terms plus pole and minus pole are used. An anode is an electrode at which the predominating electrochemical reaction is an oxidation; electrons are produced (in a galvanic cell) or extracted (in an electrolytic cell). A cathode is an electrode at which the predominating electrochemical reaction is a reduction which consumes the electrons from the anode, and that reach the cathode through the external circuit. Since electrons fljj^om lower to higher (more positive) electric potentials, in a galvanic cell they flow from the anode (the negative electrode) to the cathode (the positive electrode), while in an electrolytic cell th^^glectiWs extracted from the anode (the positive electrode) by the external source flow to the cathode (the negative electrode). Note that in rechargeable batteries such as the lead-acid battery (lead accumulator), the positive electrode is a cathode during discharge and an anode during charge, and the negative electrode is the anode during discharge and the cathode during charge. In order to avoid confusion, it is recommended that in rechargeable batteries only the terms positive and negative electrodes (or plates) be used. For more detailed information on electrode potentials, see [109]. (viii) Definition of pH [116] The quantity pH is defined in terms of the activity of hydrogen(l+) ions (hydrogen ions) in solution: pH = paH+ = -lg(GH+) = -lg("^H+7m, H+/m&) where an+ is the activity of the hydrogen(l+) (hydrogen ifcn) involution, H+(aq), and 7m H+ is the activity coefficient of H+(aq) on the molality basis at molality mH+. The symbol p is interpreted as an operator (px = — Igx) [116,117] (see Section 4.1.^3^.03) with the unique exception of the symbol pH [117]. The symbol pH is also an exception to the rules for symbols and quantities (see Section 1.3.1, p. 5). The standard molality is chosen to be equal to 1 mol kg-1. Since pH is defined in terms of a quantity that cannot be measured independently, the above equation can be only regarded as a notional definition. The establishment of primary pH standards requires the application of the concept of the "primary method of measurement" [116], assuring full traceability of the results of all measurements and their uncertainties. Any limitation in the theory or determination of experimental variables must be included in the estimated uncertainty of the method. The primary method for the measuremlfcjf of pH involves the use of a cell without transference, known as the Earned cell [118]: Pt(s) | H2(giiWffer S, Cr(aq) | AgCl(s) | Ag(s) Application of the Nernst equation to the above leads to the relationship RT In 10 E =--- ]g[(mH+7H+/™*)(™ci-7ci-/"0] where E is the potential difference of the cell and the standard potential of the AgCl|Ag electrode. This equation can be rearranged to yield ^>^lg(«H+7cr) = (i?Tlnl0)/F+lg(mci-/^) Measurements of E 'ara made and the quantity — lg(aH+7ci^) *s obtained by extrapolation to mcl-/m^ = 0. The value of 7C1- is calculated using the Bates-Guggenheim convention [117] based on Debye-Hiickel theory. Then — lg(aH+) is calculated and identified as pH(PS), where PS signifies primary standard. The uncertainties in the two estimates are typically ±0.001 in — lg(aH+7cl-)^, and ±0.003 in pH, respectively. 75 Materials for primary standard pH buffers must also meet the appropriate requirements for reference materials including chemical purity and stability, and applicability of the Bates-Guggenheim convention to the estimation of lg(7cl-). This convention requires that the ionic strength must be ^ 0.1 mol kg-1. Primary standard buffers should also lead to small liquid junction potentials when used in cells with liquid junctions. Secondary standards, pH(SS), are also available, but carry a greater uncertainty in measured values. Practical pH measurements generally use cells involving liquid junctions in which, consequently, liquid junction potentials, Ej, are present [116]. Measurements of pH are not normally performed using the Pt|H2 electrode, but rather the glass (or other H+-selective) electrfjtie, wlose response factor (d-E/dpH) usually deviates from the Nernst slope. The associated uncertainties are significantly larger than those associated with the fundamental measurements using the Harned cell. Nonetheless, incorporation of the uncertainties for the primary method, and for all subsequent measurements, permits the uncertainties for all procedures to be linked to the primary standards by an unbroken chain of comparisons. Reference values for standards in D2O and aqueous-organic solvent mixtures exist [119]. o 76 2.14 COLLOID AND SURFACE CHEMISTRY The recommendations given here are based on more extensive IUPAC recommendations/pTe—l.h] and [120-123]. Catalyst characterization is described in [124] and quantities related to macro-molecules in [125]. Name Symbol Definition SI unit Y Notes specific surface area as = - A/m m2 kg* X 1 surface amount of B nBs m01A 2 adsorbed amount of B nBa mol 2 surface excess amount of B mol 3 surface excess rB,(rBCT) rB: = nB°/A mol m~ 2 3 concentration of B total surface excess r, cn r = ^mol m~ -2 concentration i area per molecule a, a aB - = A/NBa %n2 4 area per molecule in a ^m> Cm am,B = A/Nm,B m2 4 filled monolayer surface coverage e e = NBa/Nm,B 1 4 contact angle e rad, 1 film thickness t, h, ö C ) m thickness of (surface or t, S,t m interfacial) layer surface tension, 7,«t 7 = {dG/dAs)T,p,ni N m-1, J m-2 interfacial tension film tension Zf Ef = = (dG/dAf)T^ni N m-1 5 Debye length of the LD LD m 6 diffuse layer average molar masses number-average Mn Mn J2niMi/J2ni kg mol -l mass-average = J2n,tMt2/j:^M,t kg mol -l z- average yMz = X>,M,3/X>M,2 kg mol -l sedimentation coefficient s = v/a s 7 van der Waals constant J retarded van der Waals J constant van der Waals-Hamaker/* J constant surface pressure 7T tt = 7°-7 N in"1 8 (1) The subscript s designates any surface area where absorption or deposition of species may occur. m designates the mass of a solid absorbent. (2) The value of nBs depends on the thickness assigned to the surface layer (see also note 1, p. 47). (3) The vd&ue&jyP^or and rB depend on the convention used to define the position of the Gibbs surface. They are given by the excess amount of B or surface concentration of B over values that would apply if each of the two bulk phases were homogeneous right up to the Gibbs dividing surface. 77 Additional recommendations The superscript s denotes the properties of a surface or interfacial layer. In the presence of adsorption it may be replaced by the superscript a. Examples Helmholtz energy of interfacial layer As amount of adsorbed substance na amount of adsorbed 02 raa(0-2) or n(0-2, a) area per molecule B in a monolayer a(B), a-Q The superscript a is used to denote a surface excess property relative to the Gjfcbs surface. Example surface excess amount n-Qa (or Gibbs surface amount) of B In general the values of Ta and Tb depend on the position chosen for the Gibbs dividing surface. However, two quantities, and (and correspondingly n-Qa^ anH^^V11)), may be defined in a way that is invariant to this choice (see [l.e]). E-q^ is called the relative surface excess concentration of B with respect to A, or more simply the relative afisorowpy of B; it is the value of Tb when the surface is chosen to make Ta = 0. is called the reduced surface excess concentration of B, or more simply the reduced adsorption of B; i%.i!mhe value of Tb when the surface is chosen to make the total excess T = J2i A = 0- Properties of phases (oc,p,y) may be denoted by corresponding superscript indices. Examples surface tension of phase a 7a interfacial tension between phases a and p | 7^ Symbols of thermodynamic quantities divided by surface area are usually the corresponding lower case letters; an alternative is to use a circumflex. Example interfacial entropy per area ^s(= $s) = Ss/A The following abbreviations are used in colloid chemistry: ccc critical coagulation concerf^^ion cmc critical micellisation concentration iep isoelectric point pzc point of zero charge _/v_ (Notes continued) (3) (continued) See [l.e], and also additional recommendations on this page. (4) A^ba is the number of adsorbed molecules (A^ba = Ln-Q3-), and -/Vm,ba is the number of adsorbed molecules in a filled monolayer. The definition applies to entities B. (5) In the equation, AfnWae area of the film. (6) The characteriaii^Debye length [l.e] and [123] or Debye screening length [123] Ld appears in Gouy-Chapman theory and in the theory of semiconductor space-charge. (7) In the definition, v is the velocity of sedimentation and a is the acceleration of free fall or centrifugation. The symbol for a limiting sedimentation coefficient is [s], for a reduced sedimentation coefficient s^f aiSMor a reduced limiting sedimentation coefficient [s°]; see [l.e] for further details. (8) In the definition, 70 is the surface tension of the clean surface and 7 that of the covered surface. 78 2.14.1 Surface structure [126] (i) Single-crystal surface and vector designations 3D Miller indices (hkl) are used to specify a surface termination with a (hkl) bulk plane. Families of symmetry-equivalent surfaces are designated {hkl}. Vector directions are designated [hkl] and families of symmetry-equivalent vector directions < hkl >. (ii) Notations for stepped surfaces (for ideally terminated bulk lattices) Stepped surfaces (often consisting of mono-atomic height steps and terraces of arbitrary width) with general Miller indices (hkl) are viewed as composed of alternating low-Miller-index facets (hsksls) for the step faces and (htktlt) for the terrace planes. The step notation designates such a surface as (hkl) = n(htktlt) x (hsksls) where n measures the terrace width in units of atom rows. Example The (755) surface of an fee solid, with (111) oriented terraces and (100) oriented step faces is designated in the step notation as 6(111) x (100). The microfacet notation allows more complex surface terminations, using decomposition into three independent low-index facet orientations (h\kili), (h^k^l^) and (h^k^): (hkl) = a\(hikili) + a2ii(h permeability (p), electric conductivn^^s), and magnetic flux density (B). Name Symbol Deft Inition SI unit Notes flux of mass m Qm Qm " = ám/át kg s_1 1 flux density of mass m Jm Jm " = Qm/ A kg m~2 s_1 2, 3 heat flux, <2>, P <2> = dQ/dt W 1 thermal power heat flux density J, J,= = &/A W m~2 3 thermal conductance G G = W K-1 thermal resistance R R = : 1/G K W-1 thermal conductivity A, k A = W ni"1 K-1 coefficient of heat transfer h, (k, K, a) h = W m-2 KT1 thermal diffusivity a a = A/p^ 2 —i mz s 1 diffusion coefficient D D = = -Jn/{dc/dl) 2 —i mz s 1 4 viscosity, V V = -jSJpVx/dz)'1 Pa s 5 shear viscosity bulk viscosity K K = ctxiV • V)-1 Pa s 5 thermal diffusion coefficient DT DT = JxC^idT/dx)-1 m2 K-1 s-1 (1) A vector quantity, qm = (dm/di) e, where e is the unit vector in the direction of the flow, can be defined, sometimes called the "flow rate'/A^/n. Similar definitions apply to any extensive quantity. (2) The flux density of molecules, Jn, determines either the rate at which the surface would be covered if each molecule stuck, or the rate of effusion through a hole in the surface. In studying the exposure, f J^di, of a surface to a gas, surface scientists find it useful to use the product of pressure and time as a measure of the exposure, since this product is equal to the number flux density, J/v, times the time J^t = C(u/A)ptfTmíem C is the number density of molecules and u their average speed. The unit langmuir (symbol L) corresponds to the exposure of a surface to a gas at 10~6 Torr for 1 s. (3) In previous editions, the term "flux" was used for what is now called "flux density", in agreement with IUPAP [4]. The term "density" in the name of an intensive physical quantity usually implies "extensive quantity divided by volume" for scalar quantities but "extensive quantity divided by area" for vector quantities denoting flow or flux. (4) c is the amount concentration. (5) See also Section 2.2, p. 15; r is the shear stress tensor. O 81 2.15.1 Transport characteristic numbers: Quantities of dimension one Name Symbol Definition Reynolds number Re Re = pvl/r] Euler number Eu Eu = Ap/pv2 Froude number Fr Fr = v/(lg)1/2 Grashof number Gr Gr = l3gaATp2/r]2 Weber number We We = pv2l/^ Mach number Ma Ma = v/c Knudsen number Kn Kn = X/l Strouhal number Sr Sr = If /v Fourier number Fo Fo = at/I2 Peclet number Pe Pe = vl/a Rayleigh number Ra Ra = figaATp/r]a Nusselt number Nu Nu = hl/k Stanton number St St = h/pvcp Fourier number for Fo* Fo* = Dt/l2 mass transfer Pe* = vl/D Peclet number for Pe* mass transfer Grashof number for Gr* mass transfer Nusselt number for Nu* Nu* = kdl/D mass transfer Stanton number for St* *yst* = kd/v mass transfer Prandtl number Pr Pr = r]/pa Schmidt number Sc Sc = 7]/pD Lewis number Le Le = a/D magnetic Reynolds number Rm, Rem Rm = vpnl Alfven number AtS Al = v(ppYl2/B Hartmann number Ha\ Ha = Bl(n/n)1/2 Cowling number Co Co = B2/ppv2 Motes \ Axp (1) This quantity applies to the transport of matter in binary mixtures [72]. (2) The name Sherwood number and symbol Sh have been widely used for this quantity. 82 DEFINITIONS AND SYMBOLS FOR UNITS 83 84 3.1 THE INTERNATIONAL SYSTEM OF UNITS (SI) The International System of Units (SI) was adopted by the 11th General Conference ofr^eights and Measures (CGPM) in 1960 [3]. It is a coherent system of units built from seven 5/ oteeujtits, one for each of the seven dimensionally independent base quantities (see Section 1.2, p. 4); they are: metre, kilogram, second, ampere, kelvin, mole, and candela for the base quantities leWsth? mass, time, electric current, thermodynamic temperature, amount of substance, and luminous intensity, respectively. The definitions of the SI base units are given in Section 3.3, p. 87. The 5/ derived units are expressed as products of powers of the base units, analogous to the corresponding relations between physical quantities. If no numerical factors other than 1 are used in the defining equations for derived units, then the derived units defined in this way are called coherent derived units. The SI base units and the derived units without any multiple or sub-multiple prefixes7form a coherent set of units, and are called the coherent SI units. In the International System of Units there is only one coherent SI unit for each physical quantity. This is either the appropriate SI base unit itself (see Section 3.2, p. 86) or the appropriate SI derived unit (see sections 3.4 and 3.5, p. 89 and 90). However, any of the approved decimal prefixes, called SI prefixes, may be used to construct decimal multiples or submultiples of SI units (see Section 3.6, p. 91). The SI units and the decimal multiples and submultiples constructed with the SI prefixes are called the complete set of SI units, or simply the SI units, or the units of the SI. It is recommended that only units of the SI be used in science and technology (with SI prefixes where appropriate). Where there are special reasons for making an exception to this rule, it is recommended always to define the units in terms of SI units. O 85 3.2 NAMES AND SYMBOLS FOR THE SI BASE UNITS The symbols listed here are internationally agreed and shall not be changed in other languages or scripts. See sections 1.3 and 1.4, p. 5 and p. 6 on the printing of symbols for units. SI base unit Base quantity Name Symbol length metre m mass kilogram kg time second s electric current ampere A thermodynamic temperature kelvin K amount of substance mole mol luminous intensity candela cd 86 3.3 DEFINITIONS OF THE SI BASE UNITS The following definitions of the seven SI base units are adopted by the General Conference on Weights and Measures (CGPM) [3]. metre (symbol: m) The metre is the length of path traveled by light in vacuum during a time interval of 1/299 792 458 of a second. (17th CGPM, 1983) kilogram (symbol: kg) The kilogram is the unit of mass; it is equal to the mass of the international prototype of the kilogram. (3rd CGPM, 1901) 1 second (symbol: s) The second is the duration of 9192 631770 peri^rflWof the radiation corresponding to the transition between the two hyperfine levels of the ground state of the caesium 133 atom. (13th CGPM, 1967) This definition refers to a caesium atom at rest at a temperature of 0 K. (CIPM, 1997) In this definition it is understood that the Cs atom at a temperature of T = 0 K is unperturbed by black-body radiation. The frequency of primary freque*f^ standards should therefore be corrected for the frequency shift due to the ambient radiation, as stated by the Consultative Committee for Time and Frequency (CCTF, 1999). ampere (symbol: A) The ampere is that constant current which, if maintained in two straight parallel conductors of infinite length, of negligible circular cross-section, and placed 1 metre apart in vacuum, would produce between these conductors a force equal to 2x10 7 newton per metre of length. (9th CGPM, 1948) kelvin (symbol: K) The kelvin, unit of thermodynamic temperature, is the fraction 1/273.16 of the thermodynamic temperature of the triple point of water. (13th CGPM, 1967) This definition refers to water having the isotopic composition defined exactly by the following amount-of-substance ratios: 0.000 155 76 mole of 2H per mole of 1H, 0.000 379 9 mole of 170 per mole of 160, and 0.002 005 2 mole of 180 per mole of 160. (CIPM, 2005) 2 / 1 The kilogram is mefomy base unit which is not defined by a measurement on a system defined by natural microscopic constants or an experimental setup derived from such a system. Rather it is defined by a human artefact (the international prototype of the kilogram). Therefore, alternative definitions of the kilogram are under current discussion [127-129]. 2 See also Se\tiorj/?.ll, note 2, p. 56. 87 mole (symbol: mol) 1. The mole is the amount of substance of a system which contains as many elementary entities as there are atoms in 0.012 kilogram of carbon 12; its symbol is "mol". 2. When the mole is used, the elementary entities must be specified and may be atoms, molecules, ions, electrons, other particles, or specified groups of such particles. (14th CGPM, 1971) In this definition, it is understood that unbound atoms of carbon 12, at rest and in their ground state, are referred to. (CIPM, 1980) Examples of the use of the mole 1 mol of 1H2 contains about 6.022xl023 1H2 molecules, or 12.044xl023 XH atoms 1 mol of HgCl has a mass of 236.04 g 1 mol of Hg2Cl2 has a mass of 472.09 g 1 mol of Hg22+ has a mass of 401.18 g and a charge of 192.97 kC 1 mol of Feo.giS has a mass of 82.88 g 1 mol of e~ has a mass of 548.58 [ig and a charge of —96.49 kC 1 mol of photons whose frequency is 5xl014 Hz has an energy of about 199.5 kJ Specification of the entity does not imply that the entities are identical: one may have 1 mol of an isotope mixture or gas mixture. candela (symbol: cd) ^/ The candela is the luminous intensity, in a given direction, of a source that emits monochromatic radiation oj fimauency 540 xlO12 hertz and that has a radiant intensity in that direction watt per steradian. (16th CGPM, 1979) o 88 3.4 SI DERIVED UNITS WITH SPECIAL NAMES AND SYMBOLS SI derived unit Derived quantity Name Symbol Expressed in terms of other SI units Notes plane angle radian rad m m-1 = 1 1 solid angle steradian sr m2 m~2 = 1 1 frequency hertz Hz s"1 ' 2 force newton N m kg s~2 pressure, stress pascal Pa N m"2 = m-1 kg s~2 energy, work, heat joule J N m = m2 kg power, radiant flux watt W J s-1 = m2 kg s'^^ electric charge coulomb C A s electric potential, volt V j c-1 = m2 kg s~3 A-1 electromotive force, electric tension electric resistance ohm n V A"1 = m2 kg s~3 A~2 electric conductance Siemens s n-1 = m~2 kg-1 s3 A2 electric capacitance farad F c v-1 = V¥ kg-1 s4 A2 magnetic flux weber Wb V s ^^m^ kg s~2 A-1 magnetic flux density tesla T Wb m"2 ji^^kg s~2 A-1 inductance henry H V A"1 s = m2 kg s~2 A~2 Celsius temperature degree Celsius °C 3 luminous flux lumen lm cd sJL 1 = cd illuminance lux lx lm m~2 = cd m~2 activity, (radioactivity) becquerel Bq 4 referred to a radio- nuclide absorbed dose, gray Gy = m2 s~2 4 kerma dose equivalent sievert Sv J kg-1 = m2 s~2 4 (dose equivalent index) catalytic activity katal kat mol s-1 4, 5 (1) Radian and steradian are derived units. Since they are then of dimension 1, this leaves open the possibility of including them or omitting them in expressions of SI derived units. In practice this means that rad and sr may be used when appropriate and may be omitted if clarity is not lost. (2) For angular frequency and for angular velocity the unit rad s_1, or simply s_1, should be used, and this may not be replaced with Hz. The unit Hz shall be used only for frequency in the sense of cycles per second. (3) The Celsius temperature 1*js defined by the equation t/°C = T/K - 273.15 The SI unit of Celsius temperature is the degree Celsius, °C, which is equal to the kelvin, K. °C shall be treated as a sSlgl^symbol, with no space between the ° sign and the C. The symbol °K, and the symbol °, shall no longer be used for the unit of thermodynamic temperature. (4) Becquerel is the basic unit to be used in nuclear and radiochemistry; becquerel, gray, and sievert are admitted for rejgons of safeguarding human health [3]. (5) When the amount of a catalyst cannot be expressed as a number of elementary entities, an amount of substance, or a mass, a "catalytic activity" can still be defined as a property of the catalyst measured by a catalyzed rate of conversion under specified optimized conditions. The katal, 1/^alW^l mol s_1, should replace the "(enzyme) unit", 1 U = [imol min-1 « 16.67 nkat [130]. 1 89 3.5 SI DERIVED UNITS FOR OTHER QUANTITIES This table gives examples of other SI derived units; the list is merely illustrative. O SI derived unit Derived quantity Symbol Expressed in terms of SI base units mMgtes efficiency w w-1 = 1 \ * area m2 volume m3 speed, velocity m s-1 angular velocity rad s-1 acceleration m s~2 moment of force N m = m2 kg s~2 repetency, wavenumber m-1 1 density, mass density kg m~3 specific volume m3 kg-1 • amount concentration mol m~3 2 molar volume m3 mol-1 heat capacity, entropy J K-1 = m2 kg s~2 K7i^ molar heat capacity, J KT1 mor1 = m2 kg s~2 K71 mor1 molar entropy specific heat capacity, J KT1 kg"1 = m2s-2>K-^ specific entropy molar energy J mor1 = m2 kg s~2 mol-1 specific energy J kg-1 = m2 s/V energy density J m-3 = m_1 kg s~2 surface tension N in"1 = k^y^ heat flux density, W m-2 irradiance thermal conductivity W m-1 KT1 ^^^n kg s~3 Kr1 kinematic viscosity, mz s 1 diffusion coefficient dynamic viscosity, Pa s 7^= m_1 kg s_1 shear viscosity *N y electric charge density C m-3 > = m~3 s A electric current density A m~2 conductivity S m-1^ = m~3 kg-1 s3 A2 molar conductivity S ifWV _ jjg-i g3 ^2 moi-i permittivity F m-Ai = m^kg^sU2 permeability = m kg s~2 A-2 electric field strength lv 111) 1 = m kg s~3 A-1 magnetic field strength > A m-1 exposure (X and 7 rays^< Ckg-1 = kg-1 s A absorbed dose rate jSy s-1 = m2 s~3 (1) The word "wavenumber" denotes the quantity "reciprocal wavelength". Its widespread use to denote the unit cm-1 should be discouraged. (2) The words "amount Concentration" are an abbreviation for "amount-of-substance concentration". When there is not likely to be any ambiguity this quantity may be called simply "concentration". 90 3.6 SI PREFIXES AND PREFIXES FOR BINARY MULTIPLES The following prefixes [3] are used to denote decimal multiples and submultiples of SI unycs! Prefix Prefix Submultiple Name Symbol Multiple Name Symbol lo-1 deci d 101 deca 1 da io-2 centi c 102 hector Yh 1(T3 milli m 103 kilo^ k 1(T6 micro ¥■ 106 mega M nano n 109 giga G 10-12 pico P 1012 tera^ T 10-15 femto f 1015 peta P io-18 atto a 1018 exa E 10-21 zepto z 1021 zetta Z 10-24 yocto y 1024 yotta Y Prefix symbols shall be printed in Roman (upright) type with no space between the prefix and the unit symbol. Example kilometre, km When a prefix is used with a unit symbol, the combination is taken as a new symbol that can be raised to any power without the use of parentheses. Examples 1 cm3 = (10~2 m)3 = 10~6 m3 1 [is-1 = (1(T6 s)-1 = 106 s-1 1 V/cm = 1 V/(1(T2 m) = 102 V/m 1 mmol/dm3 = 10~3 mol/(10~3 m3) = 1 mol m~3 A prefix shall never be used on its own, and prefixll^r^not to be combined into compound prefixes. Example pm, not |i.|i.m The names and symbols of decimal muh^nlS^nd submultiples of the SI base unit of mass, the kilogram, symbol kg, which already coHfl^uas a prefix, are constructed by adding the appropriate prefix to the name gram and symbol g. Examples mg, not |i.kg; Mg, not kkj^ The International Electrotechnical Commission (IEC) has standardized the following prefixes for binary multiples, mainly used in information technology, to be distinguished from the SI prefixes for decimal multiples [7]. Prefix Multiple Name Symbol Origin (210)1 = (1024)1 kibi Ki kilobinary (210)2 = (1024)2 mebi Mi megabinary (210)3 = (1024)3 gibi Gi gigabinary (210)4 = (1024)4 ( tebi Ti terabinary (210)5 = (1024^ pebi Pi petabinary (210)6 = exbi Ei exabinary (210)7 = (1024)7 zebi Zi zettabinary (210)8 = (1024)8 yobi Yi yottabinary 91 3.7 NON-SI UNITS ACCEPTED FOR USE WITH THE SI The following units are not part of the SI, but it is recognized by the CGPM [3] that/fey will continue to be used in appropriate contexts. SI prefixes may be attached to some of these units, such as millilitre, mL; megaelectronvolt, MeV; kilotonne, kt. A more extensive list of non-siunits. with conversion factors to the corresponding SI units, is given in Chapter 7, p. 129. Unit accepted for use with the SI Physical quantity Name Symbol Value in SI units Notes time minute min = 60 s time hour h = 3600 s time day d = 86 400 s plane angle degree °,deg = (tc/180) rad ^ plane angle minute = (it/10 800) rad plane angle second a = (tc/648 000) rad volume litre 1, L = 1 dm3 = /»yyn3 1 mass tonne t = 1 Mg = 103 kg level of a field quantity, neper Np = lne =(1/2) Ine2 = 1 2 level of a power quantity level of a field quantity, bel B 2 level of a power quantity energy electronvolt eV (= e •IV) = 1.602 176 487(40) xl0~19 J 3 mass dalton, Da = 1.660 538 782(83) xl0~27 kg 3,4 unified atomic U (= TOa ,(12C)/12) =*k*a mass unit length nautical mile M ^✓1852 m 5 astronomical ua = 1.495 978 706 91(6) xlO11 m 6 unit (1) The alternative symbol L is the only exception q^rV general rule that symbols for units shall be printed in lower case letters unless they are derived from a personal name. In order to avoid the risk of confusion between the letter 1 and the nunro*^?, the use of L is accepted. However, only the lower case 1 is used by ISO and IEC. (2) For logarithmic ratio quantities and their units, see [131]. (3) The values of these units in terms of the corresponding SI units are not exact, since they depend on the values of the physical constantSy^lfor^electronvolt) and ma(12C) or Na (for the unified atomic mass unit), which are determined by experiment, see Chapter 5, p. 111. (4) The dalton, with symbol Da, and the unified atomic mass unit, with symbol u, are alternative names for the same unit. The dalton may be combined with SI prefixes to express the masses of large or small entities. (5) There is no agreed symbol for the nautical mile. The SI Brochure uses the symbol M. (6) The astronomical unit is a unit of length approximately equal to the mean Earth-Sun distance. Its value is such that, when used to describe the motion of bodies in the Solar System, the heliocentric gravitational constant is (0.017 202 098 95)2 ua3 d~2 (see also [3]). o 92 3.8 COHERENT UNITS AND CHECKING DIMENSIONS If equations between numerical values have the same form as equations between physical cu45htities, then the system of units defined in terms of base units avoids numerical factors between units, and is said to be a coherent system. For example, the kinetic energy T of a particle of mass m moving with a speed v is defined by the equation where it is to be noted that the factor (1/2) is omitted. In fact the joule, symbol J, is simply a special name and symbol for the product of units kg m2 s~2. The International System (SI) is a coherent system of units. The advtapajfs of a coherent system of units is that if the value of each quantity is substituted for the quantity symbol in any quantity equation, then the units may be canceled, leaving an equation between numerical values which is exactly similar (including all numerical factors) to the original ecnmtion between the quantities. Checking that the units cancel in this way is sometimes described as checking the dimensions of the equation. The use of a coherent system of units is not essential. In particular the use of multiple or submultiple prefixes destroys the coherence of the SI, but is nonetheless often convenient. T = (1/2) mv2 but the SI unit of kinetic energy is the joule, defined by the equation J = kg (m/s)2 = kg m2 s o 93 3.9 FUNDAMENTAL PHYSICAL CONSTANTS USED AS UNITS Sometimes fundamental physical constants, or other well defined physical quantities, ar^Tised as though they were units in certain specialized fields of science. For example, in astronomy it may be more convenient to express the mass of a star in terms of the mass of the sun, and to express the period of the planets in their orbits in terms of the period of the earth's orbit, rather tha^Tta use SI units. In atomic and molecular physics it is similarly more convenient to express masses in terms of the electron mass, me, or in terms of the unified atomic mass unit, 1 u = mu and to express charges in terms of the elementary charge e, rather than to use SLunnH One reason for using such physical quantities as though they were units is that the nature'^^jfi/experimental measurements or calculations in the specialized field may be such that results are naturally obtained in such terms, and can only be converted to SI units at a later stage. When physical quantities are used as units in this way their relation to the SI must be determined by experiment, which is subject to uncertainty, and the conversion factor may change as new and more precise experiments are developed. Another reason for using such units is that uncertainty in the conversion factor to the SI may be greater than the uncertainty in the ratio of the measurements expressed in terms of the physical constant as a unit. Both reasons make it preferable to present experimental results without converting to SI units. Three such physical quantities that have been recognized as units by the CIPM are the electronvolt (eV), the dalton (Da) or the unified atomic mass unit (u), and the astronomical unit (ua), listed below [3]. The electronvolt is the product of a fundamental constant (the elementary charge, e) and the SI unit of potential difference (the voLt!f*V7l*3Tie dalton is related to the mass of the carbon-12 nuclide, and is thus a fundamental corktantJ The astronomical unit is a more arbitrarily defined constant that is convenient to astronomers. However, there are many other physical quantities or fundamental constants that are sAneiimes used in this way as though they were units, so that it is hardly possible to list them all. Physical Name Symbol quantity of unit for unit l^ajue i n SI units Notes energy electronvolt eV 1 eV = 1.602 176 487(40) xl0~19 J 1 mass dalton, Da, u i Si Da = 1.660 538 782(83) xl0~27 kg 2 unified atomic mass unit length astronomical unit ua 1 ua = 1.495 978 706 91(6) xlO11 m 3 (1) The electronvolt is the kinetic energy acquired by an electron in passing through a potential barrier of 1 V in vacuum. (2) The dalton and the unified atomic mass unit are alternative names for the same unit. The dalton may be combined with the SI prefixes to express the masses of large molecules in kilodalton (kDa) or megadalton (MDa). (3) The value of the astronomical unit in SI units is defined such that, when used to describe the motion of bodies in the solar system, the heliocentric gravitational constant is (0.017 202 098 95)2 ua3 d~2. The value must be obtained from experiment, and is therefore not known exactly (see also [3])- 3.9.1 Atomic units [22] (see also Section 7.3, p. 143) One particular groMjvof physical constants that are used as though they were units deserve special mention. These are the so-called atomic units and arise in calculations of electronic wavefunctions for atoms and molecules, i.e. in quantum chemistry. The first five atomic units in the table below have special names and symbols. Only four of these are independent; all others may be derived by multiplication and division in the usual way, and the table includes a number of examples. The 94 relation between the five named atomic units may be expressed by any one of the equations Eh = h2/mea02 = e2/4rc£oao = mee4 / (4keq)2 h2 The relation of atomic units to the corresponding SI units involves the values of the fundamental physical constants, and is therefore not exact. The numerical values in the table are from the CODATA compilation [23] and based on the fundamental constants given in Chapter^ pxlll. The numerical results of calculations in theoretical chemistry are frequently quoted in atomic units, or as numerical values in the form physical quantity divided by atomic unit, so that the reader may make the conversion using the current best estimates of the physical constants. Name Symbol Physical quantity of unit for unit Value in SI units Notes mass electron mass me = 9.109 382 15(45) xl0~31 kg charge elementary charg ;e e = 1.602 176 487(40) xl0~19 C action, Planck constant h = 1.054 571 628(53) xl0~34 J s 1 (angular momentum) divided by 2k length Bohr radius «0 = 5.291 772 085 9(36) xlO"11 m 1 energy Hartree energy Eh = 4.359 743 94j^l^l(r18 J 1 time h/Eh = 2.418 884 326 505(16) xlO"17 s speed a0Eh/h = 2.187 691 254 1(15) xlO6 m s"1 2 force Eh/ «o = 8.238 722 06(41) xl0~8 N linear momentum h/ao = 1.992 851 565(99) xl0~24 N s electric current eEh/h = 6.623 617 63(17) xl0~3 A electric field Eh/ea0 = 5fi42 20ß 32(13)xlO11 V m"1 electric dipole eao = 8.478 352 81(21)xlO"30 C m moment electric quadrupole eao2 =01^551 07(11)xlO"40 C m2 moment electric polarizability e2a02/^h *^*648 777 253 6(34) xl0~41 C2 m2 J-1 1st hyper-polarizability e3a03/£2 = 3.206 361 533(81) xl0~53 C3 m3 J"2 2nd hyper-polarizability = 6.235 380 95(31)xlO"65 C4 m4 J"3 magnetic flux h/eao2 = 2.350 517 382(59)xlO5 T density magnetic dipole eh/me = 1.854 801 830(46) xl0~23 J T"1 3 moment magnetizability e ao /me = 7.891 036 433(27) xl0~29 J T~2 (1) h = Ji/2k; a0 = AkeqH2 / mee2; Eh = ■ H2/meaQ2. (2) The numerical value of the speed of light, when expressed in atomic units, is equal to the reciprocal of the fine-structure constant a; c/(au of speed) = ch/a0Eh = a'1 = 137.035 999 679(94). (3) The atomic unit of magnetic dipole moment is twice the Bohr magneton, /iß- 3.9.2 The equations of quantum chemistry expressed in terms of reduced quantities using atomic units It is customary to write the equations of quantum chemistry in terms of reduced quantities. Thus energies are expressed as reduced energies E*, distances as reduced distances r*, masses as reduced masses m*, charges as reduced charges Q*, and angular momenta as reduced angular momenta J*, where the reduced quantities are given by the equations E* = E/Eh, r* = r/a0, m* = m/me, Q* = Q/e, and J* = J/h (1) 95 The reduced quantity in each case is the dimensionless ratio of the actual quantity to the corresponding atomic unit. The advantage of expressing all the equations in terms ofjgduced quantities is that the equations are simplified since all the physical constants disappear from the equations (although this simplification is achieved at the expense of losing the advantage of dimensional checking, since all reduced quantities are dimensionless). For example the Schrodinger equation for the hydrogen atom, expressed in the usual physical quantities, has the Jm:irK -(h2/2me)\/,2 V(r, 6, 4>) + V{r)ip{r, 6, 0) = Eip{r, 9, 0) (2) Here r, 6, and are the coordinates of the electron, and the operator Vr involves derivatives d/dr,d/d9, and d/d(j). However in terms of reduced quantities the corresponding equation has the form -(l/2)Vr2 r(>(r*,6, 0) + V*(r*)4,(r* ,6, 0) = E*4,(r*^J^ (3) where Vr* involves derivatives d/dr*,d/d6, and d/d(f>. This may be shown by substituting the reduced (starred) quantities for the actual (unstarred) quantities in Equation (2) which leads to Equation (3). In the field of quantum chemistry it is customary to write all equations in terms of reduced (starred) quantities, so that all quantities become dimensionless, and all fundamental constants such as e,me,h, Eh, and ao disappear from the equations. As observed above this simplification is achieved at the expense of losing the possibility of dimensional checking. To compare the results of a numerical calculation with experiment it is of course necessary to transform the calculated values of the reduced quantities back to the values of the actual quantities using Equation (1). Unfortunately it is also customary not to use the star that has been used here, but instead to use exactly the same symbol for the dimensionless reduced quantities and the actual quantities. This makes it impossible to write equations such as (1). (It is analogous to the situation that would arise if we were to use exactly the same symbol for h and h, where h = h/2K, thus making it impossible to write the relation between h andfijj^ipis may perhaps be excused on the grounds that it becomes tedious to include a star on the symbol for every physical quantity when writing the equations in quantum chemistry, but it is important that readers unfamiliar with the field should realize what has been done. It is also important to realize how the values of quantities "expressed in atomic units", i.e. the values of reduced quantities, may be converted back to the values of the original quantities in SI units. It is also customary to make statements such as "in atomic units e,me, h, E^, and ao are all equal to 1", which is not a correct statement. The correct statement would be that in atomic units the elementary charge is equal to 1 e, {hejtass of an electron is equal to 1 me, etc. The difference between equations such as (3), which contain no fundamental constants, and (2) which do contain fundamental constants, concerns the quantities rather than the units. In (3) all the quantities are dimensionless reduced quantities, defined by (1), whereas in (2) the quantities are the usual (dimensioned) physical quantities with which we are familiar in other circumstances. Finally, many authors make no use of the symbols for the atomic units listed in the tables above, but instead use the symbol "a.u." or "au" for all atomic units. This custom should not be followed. It leads to confusion, just as it would if we were to write "SI" as a symbol for every SI unit, or "CGS" as a symbol for every CGS unit. Examples For the hydrogen molecule the equilibrium bond length re, and the dissociation energy De, are given by re = 2.1 ao not re = 2.1 a.u. De = 0.16 Eh not De = 0.16 a.u. e> 96 3.10 DIMENSIONLESS QUANTITIES Values of dimensionless physical quantities, more properly called "quantities of dimension/5ne", are often expressed in terms of mathematically exactly defined values denoted by special symbols or abbreviations, such as % (percent). These symbols are then treated as units, and are used as such in calculations. ^\ \ 3.10.1 Fractions (relative values, yields, and efficiencies) Fractions such as relative uncertainty, amount-of-substance fraction x (also called amount fraction), mass fraction w, and volume fraction ip (see Section 2.10, p. 47 for all these quantities), are sometimes expressed in terms of the symbols in the table below. Name Symbol Value Example percent % io-2 The isotopic abundance of carbon-13 expressed as an amount-of-substance fractfbnK x = 1.1 %. permille %o ID"3 The mass fraction of water in a sample is w = 2.3 %o. These multiples of the unit one are not part of the SI and ISO recommends that these symbols should never be used. They are also frequently used as units of "concentration" without a clear indication of the type of fraction implied, e.g. amount-of-substance fraction, mass fraction or volume fraction. To avoid ambiguity they should be used only in a context where the meaning of the quantity is carefully defined. Even then, the use of an appropriate SI unit ratio may be preferred. Examples The mass fraction w = 1.5xl0~6 = 1.5 mg/kg. The amount-of-substance fraction x = 3.7xl0~2 = 3.7 % or x = 37 mmol/mol. Atomic absorption spectroscopy shows the aqueous solution to contain a mass concentration of nickel p(Ni) = 2.6 mg dm~3, which is approximately equivalent to a mass fraction w(Ni) = %.6l*l&'6. Note the importance of using the recommended name and symbol for the quantity in each of the above examples. Statements such as "the concentration of nickel was 2.6xl0~6" are ambiguous and should be avoided. The last example illustrates the approximate equivalence of p/mg dm~3 and w/10~6 in aqueous solution, which follows from the fact that the mass density of a dilute aqueous solution is always approximately 1.0 g cm~3. Dilute ^Jhriions are often measured or calibrated to a known mass concentration in mg dm~3, and this unit is then to be preferred to using ppm (or other corresponding abbreviations, which are language dependent) to specify a mass fraction. 3.10.2 Deprecated usage Adding extra labels to % and similar symbols, such as % (V/V) (meaning % by volume) should be avoided. Qualifying labels may be added to symbols for physical quantities, but never to units. Example A mass fraction w = 0.5 %, but not 0.5 % (m/m). The symbol % should not be used in combination with other units. In table headings and in labeling the axes of graphs the use of % in the denominator is to be avoided. Although one would write x(13C) = 1.1 %, the notation 100 x is to be preferred to x/% in tables and graphs (see for example Section 6.3, column 5, p. 122). 97 The further symbols listed in the table below are also found in the literature, but their use is not recommended. Note that the names and symbols for 10~9 and 10~12 in this table are here based on the American system of names. In other parts of the world, a billion often stände for 1012 and a trillion for 1018. Note also that the symbol ppt is sometimes used for part per thousand, and sometimes for part per trillion. In 1948 the word billion had been proposed for 1012 and trillion for 1018 [132]. Although ppm, ppb, ppt and alike are widely used in various applications of analytical and environmental chemistry, it is suggested to abandon completely their use because of the ambiguities involved. These units are unnecessary and can be easily replaced by Si-compatible quantities such as pmol/mol (picomole per mole), which are unambiguous. The last column contains suggested replacements (similar replacements can be formulated as mg/g, |JLg/g, pg/g etc.). Name Symbol Value Examples ' Replacement part per hundred pph, % 1(T -2 The degree of dissociation is 1.5 %^ part per thousand, ppt, %o 10~ -3 An approximate preindustrial value of the mmol/mol permille 1 CO2 content of the Earth's atmosphere was 0.275 %o (0.275 ppt). ' The element Ti has a mass fraction 5.65 %o (5.65 xlO3 ppm) in the EarN^'g crust. mg/g part per million ppm io- -6 The volume fraction of helium is 20 ppm. |i.mol/mol part per hundred pphm io- -8 The mass fraction of impurity in the metal million was less than 5 pphm. part per billion ppb io- -9 The air quality standard for ozone is a volume fraction \Ltp^ 120 ppb. nmol/mol part per trillion ppt io- -12 The natural background volume fraction of NO in air was found to be ip = 140 ppt. pmol/mol part per quadrillion ppq io- -15 fmol/mol The permille is also spelled per mill, permill, per mil, permil, per mille, or promille. 3.10.3 Units for logarithmic quantities: neper, bel, and decibel In some fields, especially in acoustics and telecommunications, special names are given to the number 1 when expressing physical quantities defined in terms of the logarithm of a ratio [131]. For a damped linear oscillation the amplitude of a q^antity^as a function of time is given by F(t) =f^fst cosuot = A Re[e(-s+iuj)t] From this relation it is clear that the coherent SI unit for the decay coefficient 5 and the angular frequency lo is the second to the power of minus one, s_1. However, the special names neper, Np, and radian, rad (see Section/27l, n. 13, Section 3.4, p. 89, and Section 3.7, p. 92), are used for the units of the dimensionless products 5t and u;t, respectively. Thus the quantities 5 and uj may be expressed in the units Np/s and rad/s, respectively. Used in this way the neper, Np, and the radian, rad, may both be thought of as special names for the number 1. In the fields of acoustics and signal transmission, signal power levels and signal amplitude levels (or field level) are usually expressed as the decadic or the napierian logarithm of the ratio of the power P to a reference power Pq, or of the field F to a reference field Fq. Since power is often proportional to the square of the field or amplitude (when the field acts on equal impedances in linear systems) it is convenient to define the power level and the field level to be equal in such a case. This is done by defining the field level and the power level according to the relations LF = HF/F0), and LP = (1/2) ln(P/P0) 98 so that if (P/Pq) = (F/Fq)2 then Lp = Lp. The above equations may be written in the form LF = ln(F/F0) Np, and LP = (1/2) ln(P/P0) Np The bel, B, and its more frequently used submultiple the decibel, dB, are used when dj^jeTd and power levels are calculated using decadic logarithms according to the relations ^\ \ LP = lg(P/P0) B = 10 lg(P/P0) dB and Lp = 2 Ig(F/F0) B = 20 Ig(F/F0) dB The relation between the bel and the neper follows from comparing th^e equations with the preceeding equations. We obtain Lp = ln(F/F0) Np = 2 lg(F/F0) B = ln(10) Ig(F/i^N? giving 1 B = 10 dB = (1/2) ln(10) Np « 1.151 X&tfp In practice the bel is hardly ever used. Only the decibel is used, to represent the decadic logarithm, particularly in the context of acoustics, and in labeling the controls of power amplifiers. Thus the statement Lp = n dB implies that 10 lg(P/Po) = n- The general use of special units for logarithmic quantitfes is Hiscussed in [131]. The quantities power level and field level, and the units bel, decibel and neper, are given in the table and notes that follow. Name Quantity Numerical value multiplied by unit Notes field level Lp = ln(P/P0) = HF/Fo) Np = 2 Ig(F/F0) B = 20 lg(F/F0) dB 1-3 power level LP = (1/2) ln(P/P0) = (1/2) ln(P/P0) Np = lg(P/P0) B = 10 Ig(P/P0) dB 4-6 (1) Fo is a reference field quantity, which shopQctye specified. (2) In the context of acoustics the field level is called the sound pressure level and given the symbol Lp, and the reference pressure po = 20 (3) For example, when Lp = 1 Np, F^b =T« 2.718 281 8. (4) Po is a reference power, which should be specified. The factor 1/2 is included in the definition to make Lp=Lp. (5) In the context of acoustics the power level is called the sound power level and given the symbol Lw, and the reference power Po = 1 pW. (6) For example, when LP = 1 B = 10 dB, P/P0 = 10; and when LP = 2 B = 20 dB, P/P0 = 100; etc. o 99 100 RECOMMENDED MATHEMATICAL SYMBOLS 101 102 4.1 1. 2. 3. 4. 5. 6. PRINTING OF NUMBERS AND MATHEMATICAL SYMBOLS [5.a] Numbers in general shall be printed in Roman (upright) type. The decimal sigrkbetween digits in a number should be a point (e.g. 2.3) or a comma (e.g. 2,3). When the decimal sign is placed before the first significant digit of a number, a zero shall always precede the decimal sign. To facilitate the reading of long numbers the digits may be separated into groups of three about the decimal sign, using only a thin space (but never a point or a comma, nor any other symbol). However, when there are only four digits before or after the decimal marker we recommend that no space is required and no space should be used. Examples 2573.421 736 or 2573,421 736 or 0.257 342 173 6xl(^br 0,257 342 173 6xl04 32 573.4215 or 32 573,4215 Numerical values of physical quantities which have been experimentally determined are usually subject to some uncertainty. The experimental uncertainty should always be specified. The magnitude of the uncertainty may be represented as follows Examples I = [5.3478 - 0.0064, 5.3478 + 0.0064] cm I = 5.3478(32) cm In the first example the range of uncertainty is indicated directly as [a — b, a + b]. It is recommended that this notation should be used only with the meaning that the interval [a — b, a + b] contains the true value with a high deiree oj certainty, such that b ^ 2a, where a denotes the standard uncertainty or standard deV^arWri (see Chapter 8, p. 149). In the second example, a(c), the range of uncertainty c indicated in parentheses is assumed to apply to the least significant digits of a. It is recommended that this notation be reserved for the meaning that b represents la in the final digits of a. Letter symbols for mathematical constants e, k, i = a/^T ) shall be printed in Roman (upright) type, but letter symbols for numbers other than constants (e.g. quantum numbers) should be printed in italic (sloping) type^imilar to physical quantities. Symbols for specific mathematical functions and operators (e.g. lb, In, lg, exp, sin, cos, d, 8, A,V, ...) shall be printed in Ronrai^type, but symbols for a general function (e.g. f(x), F(x,y), ...) shall be printed in italic type. The operator p (as in paH+, pK^^^lgK etc., see Section 2.13.1 (viii), p. 75) shall be printed in Roman type. Symbols for symmetry species in group theory (e.g. S, P, D, s, p, d, S, n, A, Aig, B2', ...) shall be printed in Roman (upright) type when they represent the state symbol for an atom or a molecule, although they are often printed in italic type when they represent the symmetry species of a point group. 103 7. Vectors and matrices shall be printed in bold italic type. Examples force F, electric field E, position vector r Ordinary italic type is used to denote the magnitude of the corresponding vector. Example r = \r\ Tensor quantities may be printed in bold face italic sans-serif type. Examples S, T Vectors may alternatively be characterized by an arrow, A, a and second-rank tensors by a double arrow, S, T. 104 4.2 SYMBOLS, OPERATORS, AND FUNCTIONS [5.k] Description Symbol Nates signs and symbols equal to = not equal to 7^ def identically equal to equal by definition to approximately equal to « asymptotically equal to ~ corresponds to = proportional to ~, oc tends to, approaches —> infinity 00 less than < greater than > less than or equal to ^ greater than or equal to ^ much less than operations NT plus minus plus or minus minus or plus =F a multiplied by b a b, ab, a ■ b, a x b 1 a divided by b WV^> ab^1, — 2 magnitude of a \a\ a to the power n an square root of a, and of a2 + b2 y/a, a1/2, and Va2 + b2, (a2 + b2)1^2 nth root of a a1/™, y/a mean value of a (a), a sign of a (equal to aj \a\ if a 7^ 0, 0 if a = 0 ) sgn a n factorial n\ binominal coefficient, n\/p\(n — p)\ C™, (™) Y,Cii, Ej«i> E ai 11 product of ai fl ah Hi ah II a i=i n i=l functions sine of x sin x cosine of x cos x tangent of x tan x cotangent of x cot x (1) When multiplication is indicated by a dot, the dot shall be half high: a ■ b. (2) a : b is aarwerWsed for "divided by". However, this symbol is mainly used to express ratios such as length scales in maps. 105 Description arc sine of x arc cosine of x arc tangent of x arc cotangent of x hyperbolic sine of x hyperbolic cosine of x hyperbolic tangent of x hyperbolic cotangent of x area hyperbolic sine of x area hyperbolic cosine of x area hyperbolic tangent of x area hyperbolic cotangent of x base of natural logarithms exponential of x logarithm to the base a of x natural logarithm of x logarithm to the base 10 of x logarithm to the base 2 of x greatest integer ^ x integer part of x integer division remainder after integer division change in x infinitesimal variation of / limit of f(x) as x tends to a 1st derivative of / 2nd derivative of / nth derivative of / partial derivative of / total differential of / inexact differential of / first derivative of x with respect to time integral of f(x) Kronecker delta Levi-Civita symbol Dirac delta function (distribution) v Symbol Notes arcsin x 3 arccos x 13 arctan x 3 arccot x 3 sinh x cosh x tanh x coth x arsinh x 3 arcosh x 3 artanh x 3 arcoth x 3 e exp x, ex loga x 4 In x, loge x 4 lg x, logio x 4 lb x, log2 x 4 ent x int x int (n/m) n/m — int(n/m)V Ax = x (final) — x (initial) 8/ lim f(x) df/dx, /', (d/dx)/ d2//dx2, r dnf/dxnfQty df/dx, dxf, DJ 5 x, dr^dt f f(x) dx, f dxf(x) 0 if.^j 1 if ijk is a cyclic permutation of 123 £123 = £231 = £312 = 1 £ijk = < — 1 if ijk is an anticyclic permutation of 123 £132 = s321 = £213 = — 1 0 otherwise 5(x), J7(x)5(x) dx = /(0) (3) These are the inverse of the parent (4) For positiv^^^ (5) Notation/usCTkJii thermodynamics, function, i.e. arcsin x is the operator inverse of sinx. see Section 2.11, note 1, p. 56. 106 Description Symbol Notes unit step function, Heaviside function gamma function convolution of functions / and g complex numbers square root of —1, a/^I real part of z = a + i b imaginary part of z = a + i b modulus of z = a + i b, absolute value of z = a + i b argument of z = a + i b complex conjugate of z = a + i b e(x), H(x), h(x), e(x) = 1 for x > 0, e(x) = 0 for x < 0. oo r(x) = f t^-VMt 0 r(n + 1) = (n)! for positive integers n +oo f*9= J f(x ~ x')g(x') dx' —oo i Re z = a Im z = b \z\ = {a2+b2)1/2 arg z; tan(arg z) = b/a z* = a — lb vectors vector a cartesian components of a unit vectors in cartesian coordinate system scalar product vector or cross product nabla operator, del operator Laplacian operator gradient of a scalar field V divergence of a vector field A rotation of a vector field A a, &x i '-y i a ■ b ax b V V2, A = d2/dx2 + d2/dy2 + d2/dz2 grad V, W dm^ArV ■ A Vxi, (curl A) a Ab) exd/dx + eyd/dy + ezd/dz matrices matrix of element Aij product of matrices A and B unit matrix inverse of a square matrix A transpose of matrix A complex conjugate of matrix^A conjugate transpose (adjojit;) of n (hermitian conjugate cff A) \ trace of a square matrix A determinant of a square' msrtrix A AB, where (AB)ik = J2AijBjk E, I A-\ AT, A A* A", A\ where (A^ij=AJt* Y2 tr A i det A, \A\ sets and logical operators p and q (conjunction sign) p A q p or q or both (drejiuiction sign) p V q negation of p, not p ->p p implies q p q p is equivalent to q p <=> q 107 Description _Symbol_Notes A is contained in B Ac B union of A and B AUB intersection of A and B AnB x belongs to A x e A x does not belong to A x £ A the set A contains x A3 x A but not B A\B 108 FUNDAMENTAL PHYSICAL CONSTANTS 109 110 The data given in this table are from the CODATA recommended values of the fundamental physical constants 2006 [23] (online at http://physics.nist.gov/constants) and from the 2006 compilation of the Particle Data Group [133] (online at http://pdg.lbl.gov), see notes for derails. The standard deviation uncertainty in the least significant digits is given in parentheses. Quantity Symbol Value Notes magnetic constant Po 4kx10-7 H m-1 (defined) y l speed of light in vacuum co, c 299 792 458 m s"1 (defined! electric constant £0 = : iMico2 8.854 187 817... xlO~12 Bm\ 1, 2 characteristic impedance Z0 = = A^oco 376.730 313 461... fl 2 of vacuum Planck constant h h = hc0 h/2K 6.626 068 96(33) xlOjKj s 1.054 571 628(53)xlQ^y s 1.986 445 501(99) xl0~25 J m Fermi coupling constant GF/(hc0f 1.166 37(l)xlO~5@yN2 3 weak mixing angle #w sin 0w 0.222 55(56) 4, 5 elementary charge e 1.602 176 487(40) xl0~19 C electron mass me 9.109 382 15l4Vx^ut31 kg proton mass mp 1.672 621 637(83) xl0~27 kg neutron mass mn 1.674 927 i^4)xl0-27 kg atomic mass constant mu : = 1 u 1.660 538 782(83) xl0~27 kg 6 Avogadro constant L, NA 6.0aft4I*re(30) xlO23 mor1 7 Boltzmann constant k, k] b 1.38^50/4(24) xl0~23 J KT1 Faraday constant F = Le 9.648 533 99(24)xlO4 C mor1 molar gas constant R 8^^172(15) J K-1 mor1 zero of the Celsius scale ^V^/K (defined) molar volume of ideal gas, Vm *&lf10 981(40) dm3 mor1 p = 100 kPa, t = 0 °C molar volume of ideal gas, ^22.413 996(39) dm3 mor1 p = 101.325 kPa, t = 0 ' standard atmosphere 101 325 Pa (defined) fine-structure constant a = a'1 7.297 352 537 6(50) xl0~3 137.035 999 676(94) Bohr radius a0 = - 4^olrpfriee2 5.291 772 085 9(36) xl0~n m Hartree energy Eh = = h2/meaQ2 4.359 743 94(22) xl0~18 J Rydberg constant Roo j^*f2hc0 1.097 373 156 852 7(73)xlO7 m"1 Bohr magneton PB = = eh/2me 9.274 009 15(23) xl0~24 J T"1 electron magnetic moment -9.284 763 77(23) xl0~24 J T"1 Lande g-factor for 9e = ¥pe/PB -2.002 319 304 362 2(15) free electron = nuclear magneton = eh/2mp 5.050 783 24(13) xl0~27 J T"1 (1) H m-1 = n A~2 = nV >2;F m-1 = C2 J"1 m-1. (2) eo and Zq may be calculated exactly from the defined values of fiQ and cq. (3) The value of the Fenl^coupling constant is recommended by the Particle Data Group [133]. (4) With the weak mixing angle #w, sin2 #w is sometimes called Weinberg parameter. There are a number of schemes^drrMing in the masses used to determine sin2 #w (see Chapter 10 in [133]). The value given here for sin2 #w [23] is based on the on-shell scheme which uses sin2 6^ = 1 — (mw/mz)2, where the quantities mw and mz are the masses of the W±- and Z°-bosons, respectively. (5) The Particle Data Group [133] gives mw = 80.403(29) GeV/c02, mz = 91.1876(21) GeV/c02 and recomm%jd^3in2 #w = 0.231 22(15), based on the MS scheme. The corresponding value in 111 Quantity Symbol Value Notes proton magnetic moment Pp 1.410 606 662(37) xl(T26 J T"1 proton gyromagnetic ratio 7p = AKfip/h 2.675 222 099(70) x 108 s"1 T"1 shielded proton magnetic Pp'/PB 1.520 993 128(17) x 10~3 moment (H2O, sphere, 25 °C) shielded proton gyromagnetic 7p72ji 42.576 388 1(12) MHz T~i^ ratio (H20, sphere, 25 °C) Stefan-Boltzmann constant a = 2K5k4/15h3c02 c\ = 2iihco2 5.670 400(40) x 10"8 W m"2 K~4 first radiation constant 3.741 771 18(19) x 10-16 W m2 second radiation constant c2 = hco/k 1.438 775 2(25) x 10"2 n?V Newtonian constant G 6.674 28(67) x 10"11 m3 kg"1 s~2 of gravitation standard acceleration 9n 9.806 65 m s"2 (defined) of gravity (5) (continued) the on-shell scheme is sin2 #w = 0.223 06(33). The effective parameter also depends on the energy range or momentum transfer considered. (6) u is the (unified) atomic mass unit (see Section 3.9, p. 94). (7) See [134] and other papers in the same special issue of Metrológia on the precise measurement of the Avogadro constant. Values of common mathematical constants Mathematical constant Symbol Value Notes ratio of circumference to diameter of a circlft^^p base of natural logarithms y e natural logarithm of 10 In 10 3.141 592 653 59-•• 1 2.718 281 828 46-• • 2.302 585 092 99-• • (1) A mnemonic for k, based on the number of letters in words of the English language, is: "How I like a drink, alcoholic of course, after the heavy lectures involving quantum mechanics!" There are similar mnemonics in poem form in French: "Que j'aime á faire apprendre ce nombre utile aux sages! Immortel Archiměde, artiste ingénieur, Qui de ton jugement peut priser la valeur? Pour moi, ton probléme eut de pareils avantages." and German: "Wie? O! Dies n Macht ernstlich so vielen viele Müh'! Lernt immerhin, Jünglinge, leichte Verselein, Wie so zum Beispiel dies dürfte zu merken sein!" See the Japanese [2.e] and Russian [2.c] editions for further mnemonics. 112 6 PROPERTIES OF PARTICLES, ELEMENTS, AND NUCLIDES The symbols for particles, chemical elements, and nuclides have been discussed in Section 2.10.1 (ii), p. 50. The Particle Data Group [133] recommends the use of italic symbols for particles and this has been adopted by many physicists (see also Section 1.6, p. 7). 113 114 6.1 PROPERTIES OF SELECTED PARTICLES The data given in this table are from the CODATA recommended values of the fundamental physical constants 2006 [23] (online at http://physics.nist.gov/constants) and from the 2006 compilation of the Particle Data Group (PDG) [133] (online at http://pdg.lbl.gov), see notes for details. The standard deviation uncertainty in the least significant digits is given in parentheses/ Charge Symbol Spin number Mass Name I z m/u mco/JílgV Notes photon Y 1 0 0 0 neutrino ve 1/2 0 « 0 ps 0 1, 2 electron e~ 1/2 -1 5.485 799 094 3(23)xlO~4 0.51(tf^ 910(13) 3 muon 1/2 ±1 0.113 428 926 4(30) 105.658 369 2(94) 2 pion 0 ±1 0.149 834 76(37) 139.570 18(35) 2 pion *° 0 0 0.144 903 35(64) 2 proton P 1/2 1 1.007 276 466 77(10) 938.272 013(23) neutron n 1/2 0 1.008 664 915 97(43) 939.565 346(23) deuteron d 1 1 2.013 553 212 724(78) MAK). 612 793(47) triton t 1/2 1 3.015 500 713 4(25) %80#.920 906(70) 4 helion h 1/2 2 3.014 932 247 3(26) 2808.391 383(70) 4 oc-particle a 0 2 4.001 506 179 127(62) ) 3727.379 109(93) Z-boson Z° 1 0 l 91.1876(21) xlO3 2, 5 W-boson w± 1 ±1 v ^^^^ 80.403(29) xlO3 2, 5 (1) The neutrino and antineutrino may perhaps have a small mass, m^e < 2 eV/co2 [133]. In addition to the electron neutrino ve one finds also a taujTamJino, vT, and a myon neutrino, (and their antiparticles v). (2) These data are from the Particle Data Group [133]. (3) The electron is sometimes denoted by e or as a p-particle by p~. Its anti particle e+ (positron, also p+) has the same mass as the electron e# b<|^^posite charge and opposite magnetic moment. (4) Triton is the 3H+, and helion the 3He2+ particle. (5) Z° and W± are gauge bosons [133]. Symbol Magnetic moment Mean life 1 Name /i//iN r/s Notes photon Y 0 neutrino ve « n 2,6 electron e~ -1.001 159 652 181 11(74) 7,8 muon £.890 9Í6 98(23) 2.197 03(4) xl0~6 2, 8,9 pion ÚJ 2.6033(5) xl0~8 2 pion *° 8.4(6) xl0~17 2 proton P 2.792 847 356(23) 8, 10 neutron n ^913 042 73(45) 885.7(8) 8 deuteron d ^0.857 438 230 8(72) 8 triton t 2.978 962 448(38) 8, 11 helion h / -2.127 497 718(25) 8, 12 oc-particle a 0 1 The PDG [133] gives the mean life (r) values, see also Section 2.12, note 8, p. 64. (6) The Particle Data Group [133] gives /i//iB < 0.9 x 10~10. (7) The value of the magnetic moment is given in Bohr magnetons fi/fiB, Pb = eh/2me. 115 In nuclear physics and chemistry the masses of particles are often quoted as their energy equivalents (usually in megaelectronvolts). The unified atomic mass unit corresponds to 931.494 028(23) MeV [23]. Atom-like pairs of a positive particle and an electron are sometimes sufficiently stable to be treated as individual entities with special names. Examples positronium (e+e~; Ps) m(Ps) = 1.097 152 515 21(46)xl0~3 u muonium (ji+e-; Mu) m(Mu) = 0.113 977 490 9(29) u (Notes continued) (8) The sign of the magnetic moment is defined with respect to the direction of the spin angular momentum. (9) [jl and |i+ have the same mass but opposite charge and opposite magnetic moment. (10) The shielded proton magnetic moment, /xp', is given by /xp'//xn = 2.792 775 598(30) (H2O, sphere, 25 °C). / (11) The half life, t1/2, of the triton is about 12.3 a (see Section 2.12, p. 64) with a corresponding mean life, r, of 17.7 a. (12) This is the shielded helion magnetic moment, /V, given as /V/^n (gas, sphere, 25 °C). 116 6.2 STANDARD ATOMIC WEIGHTS OF THE ELEMENTS 2005 As agreed by the IUPAC Commission on Atomic Weights and Isotopic Abundances (CAWIA) in 1979 [135] the relative atomic mass (generally called atomic weight [136]) of an element, E, can be defined for any specified sample. It is the average mass of the atoms in the sample divided by the unified atomic mass unit1 or alternatively the molar mass of its atoms divided by the molar mass constant Mu = NAinu = 1 g mol-1: Ar(E) = ma(E)/u = M(E)/MU The variations in isotopic composition of many elements in samples of different origin limit the precision to which an atomic weight can be given. The standard atomic weights revised biennially by the CAWIA are meant to be applicable for normal materials. This means that to a high level of confidence the atomic weight of an element in any normal sample will be within the uncertainty limits of the tabulated value. By "normal" it is meant here that the material is a reasonably possible source of the element or its compounds in commerce for industry and science and that it has not been subject to significant modification of isotopic composition within a geologically brief period [137]. This, of course, excludes materials studied themselves for very anomalous isotopic composition. New statistical guidelines have been formulated and used to provide uncertainties on isotopic abundances in the isotopic composition of the elements 1997 [138]. Table 6.2 below lists the atomic weights of the elements'S^ife [139] and the term symbol 28+1Lj for the atomic ground state [140] in the order of the atomic number. The atomic weights have been recommended by the IUPAC Commission on Isotopic Abundances and Atomic Weights (CIAAW) in 2005 [139] and apply to elements as they exist naturally on earth. An electronic version of the CIAAW Table of Standard Atomic Weights 2005 c^n be found on the CIAAW web page at http://www.ciaaw.org/atomic_weights4.htm. The list includes the approved names of elements 110 and 111 (Ds and Rg) [141,142]. The symbol Rg has also been used for "rare gas". For a history of recommended atomic weight values from 1882 to 1997, see [143]. The atomic weights of many elements depend on the origin and treatment of the materials [138]. The notes to this table explain the types of variation to be expected for individual elements. When used with due regard to the notes the values are considered reliable to ± the figure given in parentheses being applicable to the last digit. For elements without a characteristic terrestrial isotopic composition no standard atomic weight is recommended. The atomic mass of its most stable isotope can be found in Section 6.3 below. Atomic Atomic weight Ground state Symbol number Name (Relative atomic mass) term symbol Note H 1 hydrogen 1.007 94(7) 2Sl/2 g, m, r He 2 helium 4.002 602(2) 1So g, r Li 3 lithium [6.941(2)]t 2Sl/2 g, m, r Be 4 beryllium 9.012 182(3) 1So B 5 boron 10.811(7) 2po rl/2 g, m, r C 6 carbon 12.0107(8) 3Po g, r N 7 nitrogen 14.0067(2) 4qo D3/2 g, r O 8 oxygen 15.9994(3) 3P2 g, r F 9 fluorine 18.998 403 2(5) 2po r3/2 Ne lOf neon 20.1797(6) 1So g, m Na 11 solium 22.989 769 28(2) 2Sl/2 Mg 12 magnesium 24.3050(6) 1Sq Note that the atomic mass constant mu is equal to the Dalton, Da, or the unified atomic mass unit, u, and is defined in terms of the mass of the carbon-12 atom: mu = 1 u = 1 Da = ma(12C)/12. 117 Atomic Atomic weight Ground state Symbol number Name (Relative atomic mass) term symbol Note Al 13 aluminium (aluminum) 26.981 538 6(8) 2po rl/2 Si 14 silicon 28.0855(3) 3Po r P 15 phosphorus 30.973 762(2) 4co D3/2 S 16 sulfur 32.065(5) 3P2 yg'r CI 17 chlorine 35.453(2) 2po r3/2 g, m, r Ar 18 argon 39.948(1) 1So g, r K 19 potassium 39.0983(1) Ca 20 calcium 40.078(4) 1So g Sc 21 scandium 44.955 912(6) 2D3/2 Ti 22 titanium 47.867(1) 4F2 V 23 vanadium 50.9415(1) F3/2 Cr 24 chromium 51.9961(6) 7s3 Mn 25 manganese 54.938 045(5) 6S5/2 Fe 26 iron 55.845(2) 4D' Co 27 cobalt 58.933 195(5) ' F9/2 Ni 28 nickel 58.6934(2) Cu 29 copper 63.546(3) Sl/2 r Zn 30 zinc 65.409(4) 1So Ga 31 gallium 69.723(1) 2po rl/2 Ge 32 germanium 72.64(1) 3Po As 33 arsenic 74.921 60^" 4co D3/2 Se 34 selenium 78.96(3^ 3P2 r Br 35 bromine 79.90#JV 2po r3/2 Kr 36 krypton 1So g, m Rb 37 rubidium 85.4678(3) 2Sl/2 g Sr 38 strontium 1So g, r Y 39 yttrium 88.905 85(2) 2D3/2 Zr 40 zirconium '^5^.224(2) 3F2 g Nb 41 niobium ^02.906 38(2) 6D1/2 Mo 42 molybdenum , 95.94(2) 7s3 g Tc 43 technetium 6S5/2 A Ru 44 ruthenium 101.07(2) 4F5 g Rh 45 rhodium 102.905 50(2) F9/2 Pd 46 palladium a 106.42(1) 1So g Ag 47 silver 107.8682(2) 2s1/2 g Cd 48 cadmium 112.411(8) So g In 49 indium( 114.818(3) 2po rl/2 Sn 50 tin 118.710(7) 3Po g Sb 51 antimony 121.760(1) 4co D3/2 g Te 52 tellurium 127.60(3) 3P2 g I 53 iodine 126.904 47(3) 2po r3/2 Xe 54* xenon 131.293(6) 1So g, m Cs 55 caesium (cesium) 132.905 451 9(2) 2Sl/2 Ba 56 barium 137.327(7) 1So La 57 lanthanum 138.905 47(7) 2D3/2 g Ce 58 cerium 140.116(1) g 118 Atomic Atomic weight Ground state Symbol number Name (Relative atomic mass) term symbol Note Pr 59 praseodymium 140.907 65(2) 4 to L9/2 Nd 60 neodymium 144.242(3) % g Pm 61 promethium ()TTO H5/2 A Sm 62 samarium 150.36(2) 7F0 g Eu 63 europium 151.964(1) 8oo °7/2 g Gd 64 gadolinium 157.25(3) 9n° u2 g Tb 65 terbium 158.925 35(2) 6tjo n15/2 Dy 66 dysprosium 162.500(1) 5I8 g Ho 67 holmium 164.930 32(2) 4 to i15/2 Er 68 erbium 167.259(3) 3H6 g Tm 69 thulium 168.934 21(2) ^7/2 Yb 70 ytterbium 173.04(3) % g Lu 71 lutetium 174.967(1) 2D3/2 g Hf 72 hafnium 178.49(2) Ta 73 tantalum 180.947 88(2) F3/2 W 74 tungsten 183.84(1) 5D0 Re 75 rhenium 186.207(1) 6S5/2 Os 76 osmium 190.23(3) 5D4 g Ir 77 iridium 192.217(3) ^9/2 Pt 78 platinum 195.084(9) 3D3 Au 79 gold 196.966 569(4) 2s1/2 Hg 80 mercury 200.59(2) % Tl 81 thallium 204.3833(2) 2po rl/2 Pb 82 lead 207.2(1) 3Po g. r Bi 83 bismuth 208.980 40(1) 4 go °3/2 Po 84 polonium 3P2 A At 85 astatine 2po r3/2 A Rn 86 radon % A Fr 87 francium 2s1/2 A Ra 88 radium % A Ac 89 actinium 2D3/2 A Th 90 thorium 232.038 06(2) 3F2 g,z Pa 91 protactinium 231.035 88(2) 4K11/2 Z U 92 uranium 238.028 91(3) 5L6 g> m> z Np 93 neptunium ^11/2 A Pu 94 plutonium 7F0 A Am 95 americium 8qo D7/2 A Cm 96 curium 9D§ A Bk 97 berkelium 6t_to nl5/2 A Cf 98 californium 5I8 A Es 99 einsteinium 4to i15/2 A Fm 100 fermiurn 3H6 A Md 101 mendelevium 2-po ^7/2 A No 102 nobelium lS0 A 119 Symbol Atomic number Name Atomic weight (Relative atomic mass) Ground state term symbol Note Lr 103 lawrencium A Rf 104 rutherfordium Db 105 dubnium , A Sg 106 seaborgium A Bh 107 bohrium A Hs 108 hassium A Mt 109 meitnerium A Ds 110 darmstadtium A Rg 111 roentgenium A t Commercially available Li materials have atomic weights that range beJwIwi 6.939 and 6.996; if a more accurate value is required, it must be determined for the specific material, (g) Geological specimens are known in which the element has an isotopic composition outside the limits for normal material. The difference between the atomic weight of the element in such specimens and that given in the table may exceed the stated uncertainty. (m) Modified isotopic compositions may be found in commercially available material because it has been subjected to an undisclosed or inadvertent isotopic fractionation. Substantial deviations in atomic weight of the element from that given in the table can occur. (r) Range in isotopic composition of normal terrestrial material prevents a more precise AT(E) being given; the tabulated AT(E) value and uncertainty should be applicable to normal material. (A) Radioactive element without stable nuclide that lacks a characteristic terrestrial isotopic composition. In the IUPAC Periodic Table of the Elements on the inside back cover, a value in brackets indicates the mass number of the longest-lived isotope of the element (see also the table of nuclide masses, Section 6.3). (Z) An element without stable nuclide(s), exhibiting a range of characteristic terrestrial compositions of long-lived radionuclide(s) such that a meaningful atomic weight can be given. O 120 6.3 PROPERTIES OF NUCLIDES The table contains the following properties of naturally occurring and some unstable nuclides Column 1 Z is the atomic number (number of protons) of the nuclide. 2 Symbol of the element. 3 A is the mass number of the nuclide. The asterisk * denotes an unstable nuclide (for elements without naturally occurring isotopes it is the most stable nuclide) and the # sign a nuclide of sufficiently long lifetime (greater than 105 years) [144] to enable the determination of its isotopic abundance. 4 The atomic mass is given in unified atomic mass units, 1 u = ma(12C)/12, together with the standard errors in parentheses and applicable to the last digits quoted. The data were extracted from a more extensive list of the Ame2003 atomic mass evaluation [145,146]. 5 Representative isotopic compositions are given as amount-of-substance fractions (mole fractions), x, of the corresponding atoms in percents. According to the opinion of CAWIA, they represent the isotopic composition of chemicals or materials most commonly encountered in the laboratory. They may not, therefore, correspond to the most abundant natural material [138]. It must be stressed that those values sCTMdbe used to determine the average properties of chemicals or materials of unspecified natural terrestrial origin, though no actual sample having the exact composition listed may be available. The values listed here are from the 2001 CAWIA review as given in column 9*fcfrefy[147] as representative isotopic composition. This reference uses the Atomic Mass Evaluation 1993 [148,149]. There is an inconsistency in this Table because column 4 uses the most recent masses Ame2003 [145,146], whereas column 5 is based on previous atomic maJkmJfit&, 149]. When precise work is to be undertaken, such as assessment of individual properties, samples with more precisely known isotopic abundances (such as listed in column 8 of ref. [147]) should be obtained or suitable measurements should be made. The uncertainties given in parentheses are applicable to the last digits quoted and cover the range of probable variations in the materials as well as experimental errors. For additional data and background information on ranges of isotope-abundance variations in natural and anthropogenic material, see [149,150]. 6 I is the nuclear spin quantum number. A plus sign indicates positive parity and a minus sign indicates negative parity. Parentheses denotes uncertain values; all values have been taken from the Nubase evaluation [144]. 7 Under magnetic moment the ma|rimam ^-component expectation value of the magnetic dipole moment, m, in nuclear magnetons is given. The positive or negative sign implies that the orientation of the magnetic dipole with respect to the angular momentum corresponds to the rotation of a positive or negative charge, respectively. The data were extracted from the compilation by N. J. Stone [151]. An asterisk * indicates that more than one value is given in the original compilation; ** indicates that an older value exists with a higher stated precision. The absence of a plus or minus sign means that the sign has not been determined by the experimenter. 8 Under quadrupole moment, the electric quadrupole moment area (see Section 2.5, notes 14 and 15 on p. 23 and 24) is given in units of square femtometres, 1 fm2 = 10~30 nfWalthough most of the tables quote them in barn (b), 1 b = 10~28 m2 = 100 fr*2. The positive sign implies a prolate nucleus, the negative sign an oblate nucleus. The data are taken from N. J. Stone [151]. An asterisk * indicates that more than one value is given in the original compilation; ** indicates that an older value exists with a higher stated precision. The absence of a plus or minus sign means that the sign has not been determined by the experimenter. 121 Isotopic Nuclear Magnetic Quadrupole Atomic mass, composition, spin, moment, moment, z Symbol A ma/u 100 x I m//xN 0/fm2 1 H 1 1.007 825 032 07(10) 99.9885(70) 1/2+ +2.792 847 34(3) (D) 2 2.014 101 777 8(4) 0.0115(70) 1+ +0.857 438 228(9) +0.286(2)* (T) 3* 3.016 049 277 7(25) 1/2+ +2.978 962 44(4) 2 He 3 3.016 029 319 1(26) 0.000 134(3) 1/2+ -2.127 497 72(3)/ 4 4.002 603 254 15(6) 99.999 866(3) 0+ 0 3 Li 6 6.015 122 795(16) 7.59(4) 1+ +0.822 047 3|6)* -0.082(2)* 7 7.016 004 55(8) 92.41(4) 3/2- +3.256 427(2)* -4.06(8)* 4 Be 9 9.012 182 2(4) 100 3/2- -1.177 432(3)* +5.29(4)* 5 B 10 10.012 937 0(4) 19.9(7) 3+ +1.800 6^8j6) +8.47(6) 11 11.009 305 4(4) 80.1(7) 3/2- +2.688 648 9(10) +4.07(3) 6 C 12 12 (by definition) 98.93(8) 0+ 0 13 13.003 354 837 8(10) 1.07(8) 1/2- +0.702 411 8(14) 14* 14.003 241 989(4) 0+ 0 7 N 14 14.003 074 004 8(6) 99.636(20) 1+ +JH ■*o 56 55.934 937 5(7) 91.754(36) 0+ 0 57 56.935 394 0(7) 2.119(10) 1/2- +0.090 623 00(9)* 58 57.933 275 6(8) 0.282(4) g£s 0 27 Co 59 58.933 195 0(7) 100 7/2- +4.627(9) +41.0(10)* 28 Ni 58 57.935 342 9(7) 68.0769(89) Ř 0+ 0 60 59.930 786 4(7) 26.2231(77) 0 61 60.931 056 0(7) 1.1399(6)% T3/2- -0.750 02(4) +16.2(15) 62 61.928 345 1(6) 3.6345(17) 0+ 0 64 63.927 966 0(7) 0.9256(9) 0+ 0 29 Cu 63 62.929 597 5(6) 69.15(15) 3/2- 2.227 345 6(14)* -21.1(4)* 65 64.927 789 5(7) 30.85(15) 3/2- 2.381 61(19)* -19.5(4) 30 Zn 64 63.929 142 2(7) 48.268(321) 0+ 0 66 65.926 033 4(10) t .27.975(77) 0+ 0 67 66.927 127 3(10) 4.102(21) 5/2- +0.875 204 9(11)* +15.0(15) 68 67.924 844 2(10) 19.024(123) 0+ 0 70 69.925 319 3(21)/ 0.631(9) 0+ 0 31 Ga 69 68.925 573 6(13) * 60.108(9) 3/2- +2.016 59(5) +16.50(8)* 71 70.924 701 3CH)y 39.892(9) 3/2- +2.562 27(2) +10.40(8)* 32 Ge 70 69.924 2/7*4(11) 20.38(18) 0+ 0 72 71.922 075 8(18) 27.31(26) 0+ 0 73 72.923 458 9(18) 7.76(8) 9/2+ -0.879 467 7(2) -17.0(30) 74 73.921 177 8(18) 36.72(15) 0+ 0 76# 75.921 402 6(18) 7.83(7) 0+ 0 33 As 75 74.921 596 5(20) 100 3/2- +1.439 48(7) +30.0(50)** O 123 Isotopic Nuclear Magnetic Quadrupole Atomic mass, composition, spin, moment, moment, z Symbol A ma/u 100 x I m//zN 34 Se 74 73.922 476 4(18) 0.89(4) 0+ 0 76 75.919 213 6(18) 9.37(29) 0+ 0 77 76.919 914 0(18) 7.63(16) 1/2- +0.535 042 2(6)* 78 77.917 309 1(18) 23.77(28) 0+ 0 \ \ 80 79.916 521 3(21) 49.61(41) 0+ 0 82# 81.916 699 4(22) 8.73(22) 0+ 0 35 Br 79 78.918 337 1(22) 50.69(7) 3/2- +2.106 400(4^ ) 31.8(5)** 81 80.916 290 6(21) 49.31(7) 3/2- +2.270 562(4) +26.6(4)** 36 Kr 78 77.920 364 8(12) 0.355(3) 0+ 0 80 79.916 379 0(16) 2.286(10) 0+ 0 82 81.913 483 6(19) 11.593(31) 0+ 0 83 82.914 136(3) 11.500(19) 9/2+ -0.970 669(3) +25.9(1)* 84 83.911 507(3) 56.987(15) 0+ 0 86 85.910 610 73(11) 17.279(41) 0+ 0 • 37 Rb 85 84.911 789 738(12) 72.17(2) 5/2- +1.352 98(10)** +27.7(1)** 87# 86.909 180 527(13) 27.83(2) 3/2- -V$7fy 31(12)** +13.4(1)* 38 Sr 84 83.913 425(3) 0.56(1) 0+ 0 86 85.909 260 2(12) 9.86(1) 0+ 0 87 86.908 877 1(12) 7.00(1) 9/2+ -1.093 603 0(13)* +33.0(20)* 88 87.905 612 1(12) 82.58(1) 0+ 39 Y 89 88.905 848 3(27) 100 1/2- -0.137 415 4(3)* 40 Zr 90 89.904 704 4(25) 51.45(40) 0+ 0 91 90.905 645 8(25) 11.22(5) 5/2+ -1.303 62(2) -17.6(3)* 92 91.905 040 8(25) 17.15(8) 0+ 0 94 93.906 315 2(26) 17.38(28) 0+ 0 96# 95.908 273 4(30) 2.80(9) 0+ 0 41 Nb 93 92.906 378 1(26) 100 f9/2+ +6.1705(3) -37.0(20)* 42 Mo 92 91.906 811(4) 14.77(31) 0+ 0 94 93.905 088 3(21) 9.23(10) 0+ 0 95 94.905 842 1(21) 15.90(9) 5/2+ -0.9142(1) -2.2(1)* 96 95.904 679 5(21) 16.68(1) 0+ 0 97 96.906 021 5(21) 9.56(5) 5/2+ -0.9335(1) +25.5(13)* 98 97.905 408 2(21) ^U9(26) 0+ 0 100# 99.907 477(6) 9.67(20) 0+ 0 43 Tc 98* 97.907 216(4) (6)+ 44 Ru 96 95.907 598(8) y5.54(14) 0+ 0 98 97.905 287(7) 1.87(3) 0+ 0 99 98.905 939 3[m J 12.76(14) 5/2+ -0.641(5) +7.9(4) 100 99.904 219 5(22)/ 12.60(7) 0+ 0 101 100.905 582 1(22) 17.06(2) 5/2+ -0.719(6)* +46.0(20) 102 101.904 349 3(22) 31.55(14) 0+ 0 104 103.905 433(3) 18.62(27) 0+ 0 45 Rh 103 102.9^^1(3) 100 1/2- -0.8840(2) 46 Pd 102 101.905 609(3) 1.02(1) 0+ 0 104 103.904 036(4) 11.14(8) 0+ 0 105 104.905 085(4) 22.33(8) 5/2+ -0.642(3) +65.0(30)** 106 105.903 486(4) 27.33(3) 0+ 0 108_ 107.903 892(4) 26.46(9) 0+ 0 110 109.905 153(12) 11.72(9) 0+ 0 o 124 Isotopic Nuclear Magnetic Quadrupole Atomic mass, composition, spin, moment, moment, z Symbol : A ma/u 100 x I m//xN 47 Ag 107 106.905 097(5) 51.839(8) 1/2- -0.113 679 65(15)* 109 108.904 752(3) 48.161(8) 1/2- -0.130 690 6(2)* 48 Cd 106 105.906 459(6) 1.25(6) 0+ 0 108 107.904 184(6) 0.89(3) 0+ 0 » T 110 109.903 002 1(29) 12.49(18) 0+ 0 111 110.904 178 1(29) 12.80(12) 1/2+ -0.594 886 1(8)^ 112 111.902 757 8(29) 24.13(21) 0+ 0 113# 112.904 401 7(29) 12.22(12) 1/2+ -0.622 300 9(9) 114 113.903 358 5(29) 28.73(42) 0+ 0 116# 115.904 756(3) 7.49(18) 0+ 0 49 In 113 112.904 058(3) 4.29(5) 9/2+ +5.5282(2^ +80.0(40) 115# 114.903 878(5) 95.71(5) 9/2+ +5.5408(2) +81.0(50)* 50 Sn 112 111.904 818(5) 0.97(1) 0+ 0 114 113.902 779(3) 0.66(1) 0+ 0 115 114.903 342(3) 0.34(1) 1/2+ -0.918 83(7) 116 115.901 741(3) 14.54(9) 0+ oa! 117 116.902 952(3) 7.68(7) 1/2+ -1.001 04(7) 118 117.901 603(3) 24.22(9) 0+ 0 119 118.903 308(3) 8.59(4) 1/2+ -1.047 28(7) 120 119.902 194 7(27) 32.58(9) 0+ 122 121.903 439 0(29) 4.63(3) 0+> 0 124 123.905 273 9(15) 5.79(5) 0+ 0 51 Sb 121 120.903 815 7(24) 57.21(5) 5/2+ +3.3634(3) -36.0(40)** 123 122.904 214 0(22) 42.79(5) 7/2+ +2.5498(2) -49.0(50) 52 Te 120 119.904 020(10) 0.09(1) 0+ 0 122 121.903 043 9(16) 2.55(12) 0+ 0 123# 122.904 270 0(16) 0.89(3) « -0.736 947 8(8) 124 123.902 817 9(16) 4.74(14)^ Jo+ 0 125 124.904 430 7(16) 7.07(15)* 7 1/2+ -0.888 450 9(10)* 126 125.903 311 7(16) 18.84(25) 0+ 0 128# 127.904 463 1(19) 31.74(8) 0+ 0 130# 129.906 224 4(21) 34.08(62) 0+ 0 53 I 127 126.904 473(4) 100 5/2+ +2.813 27(8) 72.0(20)** 54 Xe 124 123.905 893 0(20) 0.0952(3) 0+ 0 126 125.904 274(7) . 0.0890(2) 0+ 0 128 127.903 531 3(15) 1.9102(8) 0+ 0 129 128.904 779 4(8) *«6.4006(82) 1/2+ -0.777 976(8) 130 129.903 508 0(8)^ Y 4.0710(13) 0+ 0 131 130.905 082 4(10) 21.2324(30) 3/2+ +0.6915(2)** -11.4(1)* 132 131.904 153 5(10|> 26.9086(33) 0+ 0 134 133.905 394 5(9) 10.4357(21) 0+ 0 136 135.907 219(8) 8.8573(44) 0+ 0 55 Cs 133 132.905 451 933(24) 100 7/2+ +2.582 025(3)** -0.355(4)* 56 Ba 130 129.906 320(8) 0.106(1) 0+ 0 132 131.905 061(3) 0.101(1) 0+ 0 134 133.904 508 4(4) 2.417(18) 0+ 0 135 134.905 688 6(4) 6.592(12) 3/2+ 0.838 627(2)* +16.0(3)* 136 J35.904 575 9(4) 7.854(24) 0+ 0 137 WK905 827 4(5) 11.232(24) 3/2+ 0.937 34(2)* +24.5(4)* 138/i 13*905 247 2(5) 71.698(42) 0+ 0 o 125 Isotopic Nuclear Magnetic Quadrupole Atomic mass, composition, spin, moment, moment, z Symbol A ma/u 100 x I m//iN Q/frf* 57 La 138# 137.907 112(4) 0.090(1) 5+ +3.713 646(7) +45Nöflf 139 138.906 353 3(26) 99.910(1) 7/2+ +2.783 045 5(9) +20.0(10) 58 Ce 136 135.907 172(14) 0.185(2) 0+ 0 138 137.905 991(11) 0.251(2) 0+ 0 ^ T 140 139.905 438 7(26) 88.450(51) 0+ o 142 141.909 244(3) 11.114(51) 0+ 0 59 Pr 141 140.907 652 8(26) 100 5/2+ +4.2754(5) '-7.7(6)* 60 Nd 142 141.907 723 3(25) 27.2(5) 0+ 0 143 142.909 814 3(25) 12.2(2) 7/2- -1.065(5) -61.0(20)* 144# 143.910 087 3(25) 23.8(3) 0+ 0 145 144.912 573 6(25) 8.3(1) 7/2- -0.656(4) -31.4(12)** 146 145.913 116 9(25) 17.2(3) 0+ 0 148 147.916 893(3) 5.7(1) 0+ o. 150# 149.920 891(3) 5.6(2) 0+ 0* 61 Pm 145* 144.912 749(3) 5/2+ 62 Sm 144 143.911 999(3) 3.07(7) 0+ 0 147# 146.914 897 9(26) 14.99(18) 7/2- -0.812(2)** -26.1(7)* 148# 147.914 822 7(26) 11.24(10) 0+ 149 148.917 184 7(26) 13.82(7) 7/2- -0.6677(11)** +7.5(2)* 150 149.917 275 5(26) 7.38(1) 0 152 151.919 732 4(27) 26.75(16) 0 154 153.922 209 3(27) 22.75(29) 0+ 0 63 Eu 151 150.919 850 2(26) 47.81(6) 5/2+ +3.4717(6) 83.0** 153 152.921 230 3(26) 52.19(6) 5/2+ +1.5324(3)* +222.0* 64 Gd 152# 151.919 791 0(27) 0.20(1) 0+ 0 154 153.920 865 6(27) 2.18(3) y)+ 0 155 154.922 622 0(27) 14.80(123/' f 3/2- -0.2572(4)* +127.0(50)* 156 155.922 122 7(27) 20.47(9) 0+ 0 157 156.923 960 1(27) 15.65(2) 3/2- -0.3373(6)* +136.0(60)** 158 157.924 103 9(27) 24.842tK 0+ 0 160 159.927 054 1(27) 21.86(19) 0+ 0 65 Tb 159 158.925 346 8(27) 100 3/2+ +2.014(4) +143.2(8) 66 Dy 156 155.924 283(7) 0.056(3) 0+ 0 158 157.924 409(4) 0.095(3) 0+ 0 160 159.925 197 5(27) ( 2.329(18) 0+ 0 161 160.926 933 4(2^ 18.889(42) 5/2+ -0.480(3)* 247.7(30)** 162 161.926 798 4(27) 25.475(36) 0+ 0 163 162.928 731 2(27) ' 24.896(42) 5/2- +0.673(4) +265.0(20)** 164 163.929 174^8(2dr 28.260(54) 0+ 0 67 Ho 165 164.930^2 1(27) 100 7/2- +4.17(3) 358.0(20)** 68 Er 162 161.928 778(4) 0.139(5) 0+ 0 164 163.929 200(3) 1.601(3) 0+ 0 166 165.930 293 1(27) 33.503(36) 0+ 0 167 166.932 048 2(27) 22.869(9) 7/2+ -0.563 85(12) +357.0(3)** 168 167.932 370 2(27) 26.978(18) 0+ 0 170 169.935 464 3(30) 14.910(36) 0+ 0 69 Tm 169 1 168.934 213 3(27) 100 1/2+ -0.2310(15)* O 126 Isotonic Nuclear Magnetic Quadrupole Atomic mass, composition, spin, moment, moment, z Symbol A ma/u 100 x I m//xN 0/fm2 70 Yb 168 167.933 897(5) 0.13(1) 0+ 0 170 169.934 761 8(26) 3.04(15) 0+ 0 171 170.936 325 8(26) 14.28(57) 1/2- +0.493 67(1)* 172 171.936 381 5(26) 21.83(67) 0+ 0 173 172.938 210 8(26) 16.13(27) 5/2- -0.648(3)** Jf280.0(40) 174 173.938 862 1(26) 31.83(92) 0+ 0 176 175.942 571 7(28) 12.76(41) 0+ 0 71 Lu 175 174.940 771 8(23) 97.41(2) 7/2+ +2.2323(11)^L +349.0(20)* 176# 175.942 686 3(23) 2.59(2) 7- +3.162(12)** +492.0(50)* 72 Hf 174# 173.940 046(3) 0.16(1) 0+ 0 176 175.941 408 6(24) 5.26(7) 0+ 0 177 176.943 220 7(23) 18.60(9) 7/2- +0.7935(6) +337.0(30)* 178 177.943 698 8(23) 27.28(7) 0+ 0 179 178.945 816 1(23) 13.62(2) 9/2+ -0.6409(13) +379.0(30)* 180 179.946 550 0(23) 35.08(16) 0+ 0 73 Ta 180 179.947 464 8(24) 0.012(2) 9 181 180.947 995 8(19) 99.988(2) 7/2+ +2.3705(7) +317.0(20)* 74 W 180 179.946 704(4) 0.12(1) 0+ 182 181.948 204 2(9) 26.50(16) 0+ 0 183 182.950 223 0(9) 14.31(4) V2> yiTl7 784 76(9) 184 183.950 931 2(9) 30.64(2) 0+/ 0 186 185.954 364 1(19) 28.43(19) 0-4 0 75 Re 185 184.952 955 0(13) 37.40(2) 5/2+ +3.1871(3) +218.0(20)* 187# 186.955 753 1(15) 62.60(2) 5/2+ +3.2197(3) +207.0(20)* 76 Os 184 183.952 489 1(14) 0.02(1) 0+ 0 186# 185.953 838 2(15) 1.59(3) 0+ 0 187 186.955 750 5(15) 1.96(2) 1/2- +0.064 651 89(6)* 188 187.955 838 2(15) 13.24(8) ^ 70+ 0 189 188.958 147 5(16) 16.15(5)- ' 3/2- +0.659 933(4) +98.0(60)** 190 189.958 447 0(16) 26.26,(2^ 0+ 0 192 191.961 480 7(27) 40.78^««^ 0+ 0 77 Ir 191 190.960 594 0(18) 37.3(2) 3/2+ +0.1507(6)* +81.6(9)* 193 192.962 926 4(18) 3/2+ +0.1637(6)* +75.1(9)* 78 Pt 190# 189.959 932(6) O.Ollfi) 0+ 0 192 191.961 038 0(27) 0.782(7) 0+ 0 194 193.962 680 3(9) 32.967(99) 0+ 0 195 194.964 791 1(9) r 33.832(10) 1/2- +0.609 52(6) 196 195.964 951 5(9) J^5.242(41) 0+ 0 198 197.967 893(3) 7.163(55) 0+ 0 79 Au 197 196.966 5fc8 7(6)^ 100 3/2+ +0.145 746(9)** +54.7(16)** 80 Hg 196 195.965 833(3) 0.15(1) 0+ 0 198 197.966 769 0(4) 9.97(20) 0+ 0 199 198.968 279 9(4) 16.87(22) 1/2- +0.505 885 5(9) 200 199.968 326 0(4) 23.10(19) 0+ 0 201 200.970 302 3(6) 13.18(9) 3/2- -0.560 225 7(14)* +38.0(40)* 202 201.970 643 0(6) 29.86(26) 0+ 0 204 203.9TO 493 9(4) 6.87(15) 0+ 0 81 Tl 203 202.972 344 2(14) 29.52(1) 1/2+ +1.622 257 87(12)* 205/* 204.974 427 5(14) 70.48(1) 1/2+ +1.638 214 61(12) 82 Pb 204 203.973 043 6(13) 1.4(1) 0+ 0 206 205.974 465 3(13) 24.1(1) 0+ 0 207 206.975 896 9(13) 22.1(1) 1/2- +0.592 583(9)* 208 207.976 652 1(13) 52.4(1) 0+ 0 127 Isotopic Nuclear Magnetic Quadrupole z Symbol A Atomic mass, ma/u composition, spin, 100 x I moment, m//xN moment, Q/fm2 83 Bi 209# 208.980 398 7(16) 100 9/2- +4.1103(5)* -5flntt*r 84 Po 209* 208.982 430 4(20) 1/2- 85 At 210* 209.987 148(8) (5)+ ^ \ 86 Rn 222* 222.017 577 7(25) 0+ o 87 Fr 223* 223.019 735 9(26) 3/2R +1.17(2W +117.0(10) 88 Ra 226* 226.025 409 8(25) 0+ 0 89 Ac 227* 227.027 752 1(26) 3/2- +1-1^) +170.0(200) 90 Th 232# 232.038 055 3(21) 100 0+ 0 91 Pa 231* 231.035 884 0(24) 100 3/2- 2.01(2) 92 U 233* 234# 235# 238# 233.039 635 2(29) 234.040 952 2(20) 235.043 929 9(20) 238.050 788 2(20) 5/2+ 0.0054(5) 0+ 0.7204(6) 7/2-99.2742(10) 0+ 0.59(5) 0 ^38(3)* 0 366.3(8)* 493.6(6)* 93 Np 237* 237.048 173 4(20) 5/2+J +3.14(4)* +386.6(6) 94 Pu 244* 244.064 204(5) 0+ 95 Am 243* 243.061 381 1(25) 5/2- +1.503(14) +286.0(30)* 96 Cm 247* 247.070 354(5) S 0.36(7) 97 98 Bk Cf 247* 251* 247.070 307(6) 251.079 587(5) 99 Es 252* 253* 252.082 980(50) 253.084 824 7(28) 7/2+ +4.10(7) 670.0(800) 100 Fm 257* 257.095 105(7) 101 Md 258* 258.098 431(5) 102 No 259* 259.101 03(11) 103 Lr 262* 262.109 63(22) 104 Rf 261* 261.108 77(3) y 105 Db 262* 262.114 08(20) 106 Sg 263* 263.118 32(13) 107 Bh 264* 264.1246(3) 108 Hs 265* 265.130 09(lW 109 Mt 268* 268.138 73(34) 110 Ds 271* 271.1^606/11) 111 Rg 272* 272.153 62(36) o 128 7 CONVERSION OF UNITS Units of the SI are recommended for use throughout science and technology. However, some non-rationalized units are in use, and in a few cases they are likely to remain so for many years. Sltore^er, the published literature of science makes widespread use of non-SI units. It is thus often necessary to convert the values of physical quantities between SI units and other units. This chapter is concerned with facilitating this process, as well as the conversion of units in general. Section 7.1, p. 131 gives examples illustrating the use of quantity calculus for converting the numerical values of physical quantities expressed in different units. The table in Section^, p. 135 lists a variety of non-rationalized units used in chemistry, with the conversion factors to the corresponding SI units. Transformation factors for energy and energy-related units (repetency, wavenumber, frequency, temperature and molar energy), and for pressure units, are also*£)resented in tables in the back of this manual. Many of the difficulties in converting units between different systems are associated either with the electromagnetic units, or with atomic units and their relation to the electromagnetic units. In Sections 7.3 and 7.4, p. 143 and 146 the relations involving electromagnetic and atomic units are developed in greater detail to provide a background for the conversion factors presented in the table in Section 7.2, p. 135. o 129 130 7.1 THE USE OF QUANTITY CALCULUS Quantity calculus is a system of algebra in which symbols are consistently used to represent physical quantities and not their numerical values expressed in certain units. Thus we always take the values of physical quantities to be the product of a numerical value and a unit (see Section 1/^Hj. 3), and we manipulate the symbols for physical quantities, numerical values, and units by the ordinary rules of algebra (see footnote 1, below). This system is recommended for general use in science and technology. Quantity calculus has particular advantages in facilitating the problems of converting between different units and different systems of units. Another important advantage of quantity calculus is that equations between quantities are independent of the choice of units, and must always satisfy the rule that the dimensions must be the same for each term on either side of the equal sign. These advantages are illustrated in the examples below, where the numerical values are approximate. Example 1. The wavelength A of one of the yellow lines of sodium is given by A ps 5.896 x 1(T7 m, or A/m ps 5.896 x iO~W{ The angstrom is defined by the equation (see Section 7.2, "length", p. 135) 1 A = A := 1(T10 m, or m/A := 1010 Substituting in the first equation gives the value of A in angstrom A/A = (A/m) (m/A) ps (5.896 x 1(T7) (1010) = 5896 or A ps 5896 A Example 2. The vapor pressure of water at 20 °C is recorded to be p(H20, 20 °C) ps 17.5 Torr The torr, the bar, and the atmosphere are given by the equations (see Section 7.2, "pressure", p. 138) 1 Torr^^y 133.3 Pa 1 bar := 105 Pa latolL,:= 101 325 Pa Thus p(H20, 20 °C) ps l^yT 133.3 Pa ps 2.33 kPa = (2.33 x 103/105) bar = 23.3 mbar = S(Z83 x 103/101 325) atm ps 2.30 x 10~2 atm Example 3. Spectroscopic nfeasurements show that for the methylene radical, CH2, the a xAi excited state lies at a repetency (wavenumber) 3156 cm-1 above the X 3Bi ground state u(a - X) = T0 (a) - T0(X) ps 3156 cm"1 The excitation energy from the ground triplet state to the excited singlet state is thus AE =^c0?ps (6.626 x 10~34 J s) (2.998 x 108 m s"1) (3156 cm"1) ps I 6.269 x 10~22 J m cm"1 = _/\^ * 6.269 x 10~20 J = 6.269 x 10~2 aJ_ 1 A more appropriate name for "quantity calculus" might be "algebra of quantities", because the principles of algebra rather than calculus (in the sense of differential and integral calculus) are involved. 131 where the values of h and cq are taken from the fundamental physical constants in Chapter 5, p. Ill and we have used the relation 1 m = 100 cm, or 1 m 1 cm-1 = 100. Since the electronvoltjis given by the equation (Section 7.2, "energy", p. 137) 1 eV ps 1.6022xl0~19 J, or 1 aJ ps (l/0.1ffi22) leV, AE ps (6.269 x 10_2/0.160 22) eV ps 0.3913 eV Similarly the hartree is given by Eh = h2/mea02 ps 4.3597 aJ, or 1 aJ ps (l/4.3597)-Eh (Section 3.9.1, p. 94), and thus the excitation energy is given in atomic units by AE ps (6.269 x 10~2/4.3597) Eh ps 1.4379 x 10~2£h Finally the molar excitation energy is given by AEm = LAE ps (6.022 x 1023 mor1)(6.269 x 10~2 aJ) ps 3^FW,mol-1 Also, since 1 kcal := 4.184 kJ, or 1 kJ := (1/4.184) kcal, AEm ps (37.75/4.184) kcal mol-1 ps 9.023 kcatfmoF1 In the transformation from AE to AEm the coefficient L is not a number, but has a dimension different from one. Also in this example the necessary transformation coefficient could have been taken directly from the table in the back of this manual. Example 4. The molar conductivity, A, of an electrolyte is defined by the equation A = k/c where re is the conductivity of the electrolyte solution minus the conductivity of the pure solvent and c is the electrolyte concentration. Conductivities of electrolytes are usually expressed in S cm-1 and concentrations in mol dm~3; f»^je^mple re(KCl) ps 7.39 x 10~5 S cm-1 for c(KCl) ps 0.000 500 mol dm~3. The molar conductivity can then be calculated as follows A ps (7.39 x 10~5 S cm^wfap 500 mol dm"3) ps 0.1478 S mol-1 cm"1 dm3 = 147.8 S mor1 cm2 since 1 dm3 = 1000 cm3. The abov^elatioii has previously often been, and sometimes still is, written in the not-recommended form r A = 1000 k/c However, in this form the symbols do not represent physical quantities, but the numerical values of physical quantities expressed in J^tain units. Specifically, the last equation is true only if A is the numerical value of the molar conductivity in S mol-1 cm2, re is the numerical value of the conductivity in S cm-1, and c is the numerical value of the concentration in mol dm~3. This form does not follow the rules of quantity calculus, and should be avoided. The equation A = k/c, in which the symbols represent physical quantities, is true in any units. If it is desired to write the relation between numerical values it should be written in the form . ..c 2. 1000 re/(S cnT A/(S mol cmz) = ———-^ c/(mol dm o 132 Example 5. A solution of 0.125 mol of solute B in mg pa 953 g of solvent S has a molality 6b given by (see also footnote 2, below) 6B = nB/ms Pa (0.125/953) mol g"1 pa 0.131 mol kg"1 The amount-of-substance fraction of solute is approximately given by xB = nB/ (ns + raB) ~ raB/ras = 6BMS where it is assumed that nB Eh/e « 27.211 383 86(68) V 1.000 34 V 1.000 330 V 10 11 electric resistance, R ohm (SI unit) mean international ohm US international ohm Gaussian conductivity, k, a SI Gaussian ft s cm S m o-l 1 V A"1 = 1 m2iig*. 3 A"2 1.000 49 n 1.000 495 n C2 x io~9 n 8.987 551 787X1011 Ü 1 kg_1/ri*S?s3 A2 (10n/C2) S m-1 « 1.112 650 056xl0~10 S m"1 10 10 capacitance, C farad (SI unit) Gaussian F cm 1 kg-1 m-2 s4 A2 N^09/C2) F « 1.112 650 056xl0~12 F 10 electric field strength, E SI unit esu, Gaussian "cm-2" au V m-1 Fr-1 cm-2 e cm~2/4rc£o e/4^£oao2 c 1 J C-1 m-1 = 1 kg m s~3 A = C x 10~6 V m"1 = 2.997 924 58xl04 V m"1 : 1.439 964 41(36) xl0~5 V m"1 Eh/ea0 « 5.142 206 32(13)xlO11 V m"1 10 11 (11) The units in quotation marks for electric potential through polarizability may be found in the literature, although they are strictly incorrect; the entry suggested in the column Symbol defines the unit in terms of physical quantities and other units, so that, for a conversion into the SI, the physical quantities only need to be replaced by their values in the SI and the units need to be interpreted as units in the SI. 140 Name Symbol Expressed in SI units Notes electric field gradient, E^, qap SI unit V m~2 esu, Gaussian Fr-1 cm~3 "cm-3" au e cm 3/4tc£o e/4rc£oao3 1 J C"1 m-2 1 kg s~3 A-1 C x 1(T4 V m"2 = 2.997 924 58xl06 V m~2 1.439 964 41(36) xl(T3 V m"2 Eh/ea02 sa 9.717 361 66(24) xlO21 V m"2 10 11 electric dipole moment, p, fi SI unit C m esu, Gaussian Fr cm debye D "cm", dipole length e cm au eciQ 1 A s m = (IO-7O Cm« 3.335 640 952xl0~12 C m 10-i8 Fr cm sa 3.335 640 952 x 10-í0 1.602 176 487(40) x 10"21 C m 8.478 352 81(21) x ÍO"30 C m 10 11 electric quadrupole moment, Qaß, @aß, eQ SI unit esu, Gaussian "cm2", quadrupole area au Cm2 Fr cm2 e cm 2 ecio 1 Asm2 = (l0-3/C>£^« 3.335 640 952k 10~14 C m2 1.602 176 487(40) x ÍO"23 C m2 4.486 551 07(11) x 10~40 C m2 10 11 polarizability, a SI unit esu, Gaussian ,3» J-1 C2 m2 Fr2 cm3 "cm3", polarizability volume 4tc£q cm :íj^3» au electric displacement, D (volume) polarization, P SI unit esu, Gaussian magnetic flux density, BJ (magnetic field) tesla (SI unit) gauss (emu, Gaussian) au 4tc£0 A3 4tc£q ao3 -1 s4 A2 m 1 C2 m2 C m"2 Fr cm~2 •l^nlr = 1 kg *£w/C2) J"1 C2 m2 sa 1.112 650 056xl0~16 J"1 C2 1.112 650 056xl0~16 J f1112 650 056 x 10~40 J"1 C2 m2 e2a02/Eh sa 1.648 777 253 6(34) x 10~41 J"1 C2 1 A s m~2 = (107C) C m"2 sa 3.335 640 952xl0~6 C m"2 m T G 1 J A m 1 V s m- 1 Wb nT 10~4 T 2.350 517 382(59) x 105 T 10 11 11 12 10 (12) The usafoi^fce esu or Gaussian unit for electric displacement usually implies that the non-rationalized displacement is being quoted, = 4kD (see Section 7.3, p. 143). 141 Name Symbol Expressed in SI units Notes magnetic flux, <2> weber (SI unit) Wb = 1 J A-1 = 1 V s = 1 kg m2 s~2 A"1 maxwell (emu, Gaussian) Mx = 10~8 Wb [= 1 G cm2] magnetic field strength, H SI unit A m_1 oersted (emu, Gaussian) Oe = (103/4rc) A m_1 13 (volume) magnetization, M SI unit A m_1 gauss (emu, Gaussian) G = 103 A m_1 13 magnetic dipole moment, m, fi SI unit jr1 = 1 A m2 emu, Gaussian erg G 1 = 10 A cm2 = 10~3 J T_1 Bohr magneton /xb := eh/2me ~ 14 9.274 009 15(23) xl0~24 J T"1 au eh/me := 2/iB « 1-854 SjJ^3»(46) x 10~23 J T"1 nuclear magneton fi^ := (me/mp)fiB ~ 5.050 783 24(13) xl0~27 J T"1 magnetizability, £ SI unit jr2 =1 A2 f£^2 kg"1 Gaussian erg G"2 = 10 jQ^T) au e2a02/me « 7.891 036 433(27) x 10~29 J T~2 magnetic susceptibility, \t K SI unit 1 emu, Gaussian 1 15 molar magnetic susceptibility, Xm SI unit m3 mol^^^ emu, Gaussian cm3 mol-1 = 10~6 m3 mol-1 16 inductance, self-inductance, L henry (SI unit) H = 1 V s A"1 = 1 kg m2 s~2 A~2 Gaussian s2 cm"1 = C2 x 10~9 H « 8.987 551 787x 1011 H 10 emu cm = 10~9 H (13) In practice the oersted, Oe, is only used as a unit for = AiiH, thus when = 1 Oe, H = (103/4rc) A m_1 (see Section 7.3, p. 143). In the Gaussian or emu system, gauss and oersted are equivalent units. ^ (14) The Bohr magneton /Tb is sometimes denoted BM (or B.M.), but this is not recommended. (15) In practice suscllpNbilities quoted in the context of emu or Gaussian units are always values for x(ir) = x/4rc/S|y&\*nen x(ir) = lO"6, X = 4n:xl0-6 (see Section 7.3, p. 143). (16) In practice the units cm3 mol-1 usually imply that the non-rationalized molar susceptibility is being quoted Xm" = Xm/4rc ; thus, for example if Xm^ = — 15xl0~6 cm3 mol-1, which is often written as "-15 cgs ppm", then Xm = -1.88xl0~10 m3 mol-1 (see Section 7.3, p. 143). 142 7.3 THE ESU, EMU, GAUSSIAN, AND ATOMIC UNIT SYSTEMS IN RELATION TO THE SI The ISQ (see Section 1.2, p. 4) equations of electromagnetic theory are usually used with physical quantities in SI units, in particular the four base units metre (m), kilogram (kg), second (s), and ampere (A) for length, mass, time, and electric current, respectively. The basic equations for the electrostatic force Fes between particles with charges Qi and Q2 in vacuum, and for the infinitesimal electromagnetic force d2-Fem between conductor elements of length dZi and dl2 and corresponding currents I\ and I2 in vacuum, are Fes = QiQ2r/4K£0r6 d2Fem = (/i0/4n) h d/i x (I2 dl2 x r)/r3 where particles and conductor elements are separated by the vector r (^jUn^l = r). The physical quantity £o, the electric constant (formerly called permittivity of ^aciWmfin is defined in the SI system to have the value £0 = (107/4tcc02) kg"1 nT1 C2 ps 8.854 187 817 x 1CT12 C2 nT1 J"1 Similarly, the physical quantity fiQ, the magnetic constant (formerly called the permeability of vacuum), has the value Po 4k x 1(T7 N A"2 ps 1.256 637 061 4 x 10~6 N A"2 in the SI system. In different unit systems £o and fiQ may have different numerical values and units. In this book we use £o and fiQ as shorthands for the SI values as given above and prefer to use more general symbols kes and kem to describe these quanfid^in other systems of units and equations as discussed below. The SI value of fiQ results from the definition of the ampere (Section 3.3, p. 87). The value of £o then results from the Maxwell relation £0/i0 = l/c02 where cq is the speed of light in vacuum (seSyChapter 5, p. 111). More generally, following ideas outlined in [153] and [154], the basic equations for the electrostatic and electromagnetic forces may be wrifttar as QiQ2r Fes — ^es ^3 ,2„ kem hdh x (I2dl2 x r) o -Tern k2 r3 where the three constants kes, kem and k satisfy a more general Maxwell relation k * ^es/^em — The introduction oi trnrje constants is necessary in order to describe coherently the system of units and equations that are in common use in electromagnetic theory. These are, in addition to the SI, the esu (electmsrftto'unit) system, the emu (electromagnetic unit) system, the Gaussian system, and the system of atomic units. The constants k, kes, and kem have conventionally the values given in the following table. 143 SI esu emu Gaussian atomic units Notes k 1 1 1 ( cm s"1 1 J fces l/4rce0 1 C2 cm2 s~2 1 1 [2 fcem /i0/4rc (1/C2) s2 cm'2_1_1_a2^_V^/ (1) C is the exact number C = c0/(cm s"1) = 29 979 245 800, see p. 135. (2) 1/4k£0 = C2 x 10-11 N A"2 m2 s~2 (3) /i0/4n = 10~7 N A"2 (4) a is the fine-structure constant with or1 = 137.035 999 679(94) (see ChaptefS^^fll). This table can be used together with Section 7.4, p. 146 to transform expressions involving electromagnetic quantities between several systems. One sees that, particularly in the^Kaussian system, k = co- This guarantees, together with the general relation of Maxwell equ^ons given in Section 7.4, that the speed of light in vacuum comes out to be cq in the Gaussian system. Examples of transformations between the SI and the Gaussian system are also given in Section 7.4. Additional remarks (i) The esu system In the esu system, the base units are the centimetre (cm), gram (g), and second (s) for length, mass and time, respectively. The franklin, symbol Fr (see footnote 1, below) for the electrostatic unit of charge may be introduced alternatively as a fourth base unit. iTwo particles with electric charges of one franklin, one centimetre apart in a vacuum, repel each other with a force of 1 dyn = 1 cm g s~2. From this definition, one obtains the relation 1 BL^(10/C) C (see Example 7, Section 7.1, p. 134). Since kes = 1, from the general definition of the electrostatic force given above, the equation Fr = erg1/2 cm1/2 is true in the esu system, where erg1/2 cm1/2 is the esu of charge. In this book the franklin is thus used for convenience as a name for the esu of charge. (ii) The emu system In the emu system, the base units are the centijraitpe (cm), gram (g), and second (s) for length, mass and time. The biot (symbol Bi) for the electromagnetic unit of electric current may be introduced alternatively as a fourth base unit. Two Ifl^^vires separated by one centimetre with electric currents of one biot that flow in the same direction, repel each other in a vacuum with a lineic force (force per length) of 1 dyn/cm. From this definition one obtains the relation 1 Bi = 10 A. Since kem = 1, from the general definition of the electromagnetic force given above, the equation Bi = dyn1/2 is true in the emu system, where dyn1/2 is the emu of electric current. The biot has generally been used as a compact expression for the emu of electric current. (iii) The Gaussian systeni In the Gaussian system, the esu and emu systems are mixed. From the relation of the franklin and the biot to the SI units one readily obtains 1 Bi = C Fr s_1 Since kes = 1 and ii|k^= 1, the value of the constant k is determined to be cq by the more general Maxwell relation given above. In treatises about relativity theory k is sometimes set to 1, which corresponds to a transformation of the time axis from a coordinate t to a coordinate x = cq£. 1 The name "franklin", symbol Fr, for the esu of charge was suggested by Guggenheim [155], although it has not been widely adopted. The name "statcoulomb" has also been used for the esu of charge. 144 (iv) The atomic units system In the system of atomic units [22] (see also Section 3.9.1, p. 94), charges are given iry^tnits of the elementary charge e, currents in units of eE\Jh, where E^ is the hartree and h is the Planck constant divided by 2k, lengths are given in units of the bohr, ao, the electric field strength is given in units of E^/ea^, the magnetic flux density in units of h/ea^2. Conversion factors from these and other atomic units to the SI are included in Sections 3.9.1 and 7.2, p. 94 and 135. Thus, since conventionally k = 1, it follows that kes = 1 in this system (see footnote 2) and kem = a2 (see footnote 3). (v) Non-rationalized quantities The numerical constant 4k is introduced into the definitions of £o and po because of the spherical symmetry involved in the equations defining Fes and d2Fem above; in this way its appearance is avoided in later equations, i.e., in the Maxwell equations. When factors of 4k are introduced in this way, as in the SI, the equations are also called "rationalized". Furthermore, it is usual to include the factor 4k in the following quantities, when converting the electromagnetic equation from the SI system to the esu, emu, Gaussian and atomic units system: = 4kD = 4kH Xe(ir) = Xe/4l/ X(ir) = X/4K where the superscript (ir), for irrational, meaning non-rationalized, denotes the value of the corresponding quantity in a "non-rationalized" unit system as opposed to a system like the SI, which is "rationalized" in the sense described above. The magnetic permeability ii is given as ii <^^em 4k = /xr po in the SI, and in the non-rationalized unit system. fiT is the dimensionless relative permeability and is defined in terms of the magnetic susceptibility in Section 7.4, p. 147. The electric permittivity e is given as e = er 4K/kes = £r • £o in the SI, and as £ = £T/kes in non-rationalized unit systems. £r is the dimensionless relative permittivity and is defined in terms of the electric susceptibility in Section 7.4, p. 146. V Since Eh = JLT^E =---^ = —2--2 =-7^^2" 4Ke0a0 4Ke0rz ez rz ea0(r/a0) 3 Since the value of the speed of light in atomic units is the reciprocal of the fine-structure constant, oT1, the condition k2 kes/kem = cq2 yields kem = a2 in atomic units for k = 1. 145 7.4 TRANSFORMATION OF EQUATIONS OF ELECTROMAGNETIC THEORY BETWEEN THE ISQ(SI) AND GAUSSIAN FORMS General relation ISQ(SI) Gaussian Force between two localized charged particles in vacuum (Coulomb law): F = KsQ^r/r3 F = Q1Q2r/4K£0r3 F = Q^r/r3 Electrostatic potential around a localized charged particle in vacuum: 4> = kesQ/r 4> = Q/4K£0r Relation between electric field strength and electrostatic potential: E = -V4> E = -V4> Field due to a charge distribution p in vacuum (Gauss law): V • E = 4Kkesp V • E = p/eq Electric dipole moment of a charge distribution: p = f prdV p = f prdV Potential around a dipole in vacuum: 4> = kesp ■ r/r3 4> = p ■ v/Ane^r3 Energy of a charge distribution in an electric field: E = Q4>- p E+ ■■■ E = Q4> - p ■ E -\---- Electric dipole moment induced in an electric field: p = aE-\---- p = aE-\---- Dielectric polarization: P = XeE/4Kkes P = Xes0E Electric susceptibility and relative permittivity^ £r = 1 + Xe £r = 1 + ^ft J Electric displacement1: D = E/4Kkes + P D = e0E + P D = eTE/4Kkes D =^o£tE7 Capacitance of a parallel plate condenser, area A, separation d: C = eTA/4-nkesd C/^StErA/d Force between two current elements in vacuum: d2F- fcem iidli x (I2dZ2 x k2 r3 d2F- Po hdh x (hdh x r) 4k r3 Magnetic vector potential due to a current element in vacuum: dA = ^Idl/r dA = ^Idl/r 4> = Q/r E^^vly V • E = 4kp ja = j prdV 4> = P • r/r3 E = Qcf) - p ■ E + p = aE-\---- Xe & £r = 1 + 4-KXe (ir) D(ir) E + 4kP etE C = eTA/4-nd hdh x {hdh x r) d2F. Cn r 2 q, 3 dA = Idl/cor k ""' ' ~~~ 4k Relation between magnetic flux density and magnetic vector potential: B = VxA B = VxA B = VxA Magnetic flux density due to a current element in vacuum (Biot-Savart law): —IdH^/3 dB = f^Idl x r/r3 dB = Idl x r/c0r3 k 4k dB (1) The second equation holds in isotropic media. 146 General relation ISQ(SI) Gaussian Magnetic flux density due to a current density j in vacuum (Ampere law): k V x B = 4x-^j V x B = p0j V x B = Wc0^ k Force on a current element in a magnetic flux density: dF = Idl x B/k dF = Idl x B dF = Idl/Q/cJ Magnetic dipole of a current loop of area A: m = IA/k m = IA m = IA/co ^^^^^ Magnetic vector potential around a magnetic dipole in vacuum: A = kemm x r/r3 A = —m x r/r3 A ^trax r/r3 em / 4K ' /^T\J Energy of a magnetic dipole in a magnetic flux density: E = -m ■ B E = -m ■ B ^j£\-m-B Magnetic dipole induced by a magnetic flux density: m = £B H---- m = ^B^---- m = £B -\---- Magnetization: M= \H M= \H M= x(ir) H^T) Magnetic susceptibility and relative permeability: Pr = l + X Pr = l + X /xr = 1 + 4ji:x(ir) Magnetic field strength1: // = B/AKkem - M II B/p0 - BS^ = B - AkM H=B/4Kfirkem H=B/p0pr H^ir) = B/pr Conductivity: j = kE j = kE j = kE Self-inductance of a solenoid of volume O^tft windings per length: k ' L = 4k-^- pTn2V L^^/(^Tn2V L = 4KpTn2V/c02 Faraday induction law: „ „ 1 c)B ,s U„ n „ „ I dB n VxE+-—=0 VxE + dBdt = 0 VxE+— — =0 k dt Co at Relation between the electric field strength and electromagnetic potentials: E=-V4>-\d^- E=-X?4>-dA/dt E= -V0- k dt cq at Maxwell equations: V D = p V D = p V • Z>(ir) = 4np , 1 dD^ 4k kV x H-dD/dt = j V x H-dD/dt = j V x fl and A (in Lorentz gauge): . , kem a26 A , . , d2(j) . AJ1 a2 kem a2 A , . , a2 A . . 1 a2 A 4k . V^ + ^=° VA + e0,0d4 = 0 ^ + i| = 0 k kes at at cq at Energy density of radiation: U/V = (E-D + B H)/2 U/V = (E■ D + B ■ H)/2 U/V = (e-£>(ir) + B fl2ipmic Mass Evaluation. (II). Tables, Graphs and References. Nucl. Phys. A, 729:337-676, 2003. [147] J. K. Böhlke, J. R. de Laeter, P. de Bievre, H. Hidaka,/fl^S^eiser, K. J. R. Rosman, and P. D. P. Taylor. Inorganic Chemistry Division, Commission on Atomic Weights and Isotopic Abundances. Isotopic Compositions of the Elements, 2001. J. Phys. Chem. Ref. Data, 34:57-67, 2005. [148] G. Audi and A. H. Wapstra. The 1993 Atomic MÖBftaluation. I. Atomic Mass Table. Nucl. Phys. A, 565:1-65, 1993. [149] J. R. de Laeter, J. K. Böhlke, P. de Bievre, H. Hidaka, H. S. Peiser, K. J. R. Rosman, and P. D. P. Taylor. Inorganic Chemistry Diviai^^Commission on Atomic Weights and Isotopic Abundances. Atomic Weights of the Elements: Review 2000. Pure Appl. Chem., 75:683-800, 2003. [150] T. B. Coplen, J. K. Böhlke, P. De Bievre, T. Ding, N. E. Holden, J. A. Hopple, H. R. Krouse, A. Lamberty, H. S. Peiser, I^Revesz, S. E. Rieder, K. J. R. Rosman, E. Roth, P. D. P. Taylor, R. D. Vocke Jr., and Y. K. Xiao. Inorganic Chemistry Division, Commission on Atomic Weights and Isotopic Abundances, Subcommittee on Natural Isotopic Fractionation. Isotope-Abundance Variations of Selected Elements. Pure Appl. Chem., 74:1987-2017, 2002. [151] N. J. Stone. Table of Nuclear Magnetic Dipole and Electric Quadrupole Moments. Atomic Data and Nuclear Data*¥ables, 90:75-176, 2005. [152] B. J. Jaramillo and R.Xi^aaway The Astronomical Almanac for the Year 2003. U.S. Government Printing Office, Washington, 2003. [153] R. T. Birge. On Electric and Magnetic Units and Dimensions. The American Physics Teacher, 2:41-48, 1934. ' [154] J. D. Jackson. Classical Electrodynamics, 2nd edition. John Wiley, New York, 1975. [155] E. A. Guggenheim. Names of Electrical Units. Nature, 148:751, 1941. [156] D. R. Lide Jr. Use of Abbreviations in the Chemical Literature. Pure Appl. Chem., 52:2229-2232, 1980. 177 [157] H. Q. Porter and D. W. Turner. A Descriptive Classification of the Electron Spectroscopies. Pure Appl. Chem., 59:1343-1406, 1987. [158] N. Sheppard. English-Derived Abbreviations for Experimental Techniques in Surface Science and Chemical Spectroscopy. Pure Appl. Chem., 63:887-893, 1991. [159] D. A. W. Wendisch. Acronyms and Abbreviations in Molecular Spectroscopy. Springer, Heidelberg, 1990. 178 11 GREEK ALPHABET Pronounciation and Roman Italics Name Latin Equivalent Notes A, a A, a alpha A beta B | \ r, y A 7 gamma G A, 5 A, 5 delta D E, e E, e, e epsilon E z, C z,C zeta Z H,T1 eta Ae, A, Ee 1 e, ů, e &, e theta Th 2 i, i I, l iota I K, X, K K, x, n kappa K(f 2 A, X A, X lambda L M, [i M, ii mu, (my) N, v N, v nu, (ny) N s,5 xi X 0, o 0, o omikron 0 n, k n, it Pi yp p.p p,p rho R £, (7, q S, a, s sigma C s 2, 3 t,r tau T Y, u Y, u upsilon, ypsilon U, Y <2>, psi Ps $7, (0 fí, oj omega Oo 4 (1) For the Latin equivalent Ae is to be pronounced as the German A. The modern Greek pronounciation of the letter r) is like E, long ee as in^helfse, or short i as in lips. The Latin equivalent is also often called "long E". (2) For the lower case letters theta, kappa, sigma and phi there are two variants in each case. For instance, the second variant of the lower case theta is sometimes called "vartheta" in printing. (3) The second variant for lower case sigma is used in Greek only at the end of the word. (4) In contrast to omikron (short o) the letter omega is pronounced like a long o. O 179 180 INDEX OF SYMBOLS 181 182 This index lists symbols for physical quantities, units, some selected mathematical operators, states of aggregation, processes, and particles. Symbols for chemical elements are given in Section 6.2, p. 117. Qualifying subscripts and superscripts, etc., are generally omitted from this index, so that for example Ep foy^fotential energy, and Eea for electron affinity are both indexed under E for energy. The Latin alphabet is indexed ahead of the Greek alphabet, lower case letters ahead of upper case, bold ahead of italic, ahead or^Hmian, and single letter symbols ahead of multiletter ones. When more than one page reference is given, bold print is used to indicate the general (defining) reference. Numerical entries for the corresponding quantities are underlined. a acceleration, 13, 90, 137 AH Hall coefficient, 43 a fundamental translation vector, 42, 79, 80 Al Alfven number, 82 / * a reciprocal lattice vector, 42, 80 A, relative atomic mass, 47, 117 a absorption coefficient, 6, 36, 37, 40, 41 A ampere (SI unit), 4, 86, 87, 139, 143 a activity, 57-59, 70, 71, 72, 74, 75 A base-centred (crystal lattice), 44 a area per molecule, 77, 78 A symmetry label, 31-33, 51 a Hermann-Mauguin symbol, 44 A angstrom (unjVpNpngth), 24, 27, 131, 135 a hyperfme coupling constant, 27 a specific surface area, 77 b Burgers vector, 42 a thermal diffusivity, 81 b fundamental translation vector, 42, 79, 80 a unit cell length, 42 b* reciproc^arZice vector, 42, 80 a van der Waals coefficient, 57 b breadth, 13 «0 bohr (unit of length), 135 b Hermann-Mauguin symbol, 44 «0 Bohr radius, 9, 18, 20, 22, 24, 26, 95, 96, b impact parameter, 65 111, 132. 135, 145 b mobility ratio, 43 a adsorbed (subscript), 59 b ^olaKty, 48, 49, 59, 62, 70, 75, 133 a are (unit of area), 136 b Tafel slope, 72 a atom, 22, 24 b unit cell length, 42 a atto (SI prefix), 91 b van der Waals coefficient, 57 a symmetry label, 32, 33 b Wang asymmetry parameter, 25 a year (unit of time), 24, 137 b J barn (unit of area), 121, 136 am amorphous solid, 54, 55 symmetry label, 32, 33 aq aqueous solution, 54, 55, 57, 73, 74, 97 bar bar (unit of pressure), 48, 62, 131, 138 at atomization (subscript), 59, 61 abs absorbed (subscript), 34 B magnetic flux density (magnetic induction), ads, a adsorption (subscript), 54, 59 17, 27, 81, 82, 141, 146-148 app apparent (superscript), 60 B Debye-Waller factor, 42 atm atmosphere, 40, 62, 131, 138 B Einstein coefficient, Einstein transition amagat amagat unit, 139 probability, 35, 37, 39 B napierian absorbance, 36, 37 A conservation matrix, formula matrix, 53 B retarded van der Waals constant, 77 A magnetic vector potential, 17, 146-148 B rotational constant, 25-27 A absorbance, 5, 36, 37, 39, 40 B second virial coefficient, 57 A electron affinity, 22 B susceptance, 17 A integrated absorption coefficient, 37-41 B base-centred (crystal lattice), 44 A activity (radioactive), 24, 89, 139 B bel (unit of power level), 92, 99 A, A affinity of reaction, q8_6lV Bi biot (unit of electric current), 135, 139, 144 A area, 13, 14-17, 24, 34-36, 48, 49, 56, 71, Bq becquerel (SI unit), 89, 139 72, 77, 78, 81, 90, 136, 146 Btu British thermal unit (unit of energy), 137 A Einstein coeffiaWht, Efiistein transition probability, 35, 37, 39 c fundamental translation vector, 42 A Helmholtz energy, 56, 57, 78 c velocity, 13, 45, 90, 148 A hyperfme coupling constant, 27 c* reciprocal lattice vector, 42, 80 A nucleon number, mass number, 22, 49, 121 c amount concentration, 4, 37, 38, 48, 54, 58, A pre-exponential factor, 63, 64 59, 62-67, 70, 72, 81, 82, 90, 131 A rotational constant, 25-27 c Hermann-Mauguin symbol, 44 A spin-orbit coupling constant, 25 c specific heat capacity (constant pressure), A van der Waals-Hamaker constant, 77 6, 81, 82 183 c speed, 13, 64, 81, 90, 93, 95 c unit cell length, 42 c speed of light in medium, 34, Co speed of light in vacuum, 9, 13, 16, 22, 23, 25, 34-39, 41, 43, 95, 111, 112, 131, 134, 143, 145-148 ci first radiation constant, 36, 112 C2 second radiation constant, 36, 112 c centered superlattice, 79 c centi (SI prefix), 91 c combustion (subscript), 59, 61 c symmetry label, 32 cd candela (SI unit), 34, 86, 88 cd condensed phase, 40, 41, 48, 54, 58 cr crystalline, 54, 55, 61 cal calorie (unit of energy), 58, 137 ccc critical coagulation concentration, 78 cmc critical miscellisation concentration, 78 C spin-rotation interaction tensor, 28 C capacitance, 16, 140, 146 C heat capacity, 5, 6, 8, 43, 55-57, 90, 138 C integrated absorption coefficient (condensed phase), 41 C number concentration, number density, 43, 45, 48, 63-65, 68, 69, 81 C rotational constant, 25-27 C third virial coefficient, 57 C vibrational force constant, 27 Cn n-fold rotation symmetry operator, 31, 32 Co Cowling number, 82 C base-centred (crystal lattice), 44 C coulomb (SI unit), 89, 134, 139 Ci curie (unit of radioactivity), 139 CI clausius (unit of entropy), 138 °C degree Celsius (SI unit), 56, 89, 138 d centrifugal distortion constant, 25 d collision diameter, 64 d degeneracy, statistical weight, 26, 39, 45 d diameter, distance, thickness, 13, 15, 146 d lattice plane spacing, 42 r d relative density, 14 d day (unit of time), 92^136, 137 d deci (SI prefix), 91 d deuteron, 50, 115 da deca (SI prefix), 91 dB decibel, see bel (unit of power level), 92, 99 deg degree (unit of plane angle), 92, 136 dil dilution (subscript), 59 dpi displacement (subscript), 59 dyn dyne (unit of force), 27, 134, 137, 144 D dipolar interaction tensor, 28 D electric displacement, 16, 141, 145, 146, 148 D absorbed dose of radiation, 139 D centrifugal dstortion constant, 25w D Debye-Waller factor, 42 D diffusion coefficient, 43, 44, 72,\^9o} D dissociation energy, 22, 96 , DT thermal diffusion coefficient, 81 Dab dipolar coupling constant^! D debye (unit of electric dipole moment), 26, 39, 141 Da dalton (unit of mass^, 22} 47, 92, 94, 136 ^^^^^ e unit vector, 13, 81 e elementary charge, 9, 18-20, 22-24, 26, 29, 37, 43, 70, 92, 94-96, 111, 139, 145 e etendue, 35, 34t e linear strain, 15 e base of natural logarithm, 8, 99, 103, 106, 112 e electron,.$^Lp, 115 e symmetry label, 33 eV electronvolt (unit of energy) 9, 92, 94, 132, 137 erg erff^lS^f energy), 134, 135, 137, 138, * ^3/144 e.u. entropy unit, 139 E Wectric field strength, 7, 16, 24, 26, 40, 43, 73, 90, 104, 134, 140, 146, 148 E a activation energy, threshold energy, 64, 66, 67 EfS cell potential, 71, 74, 75 electric potential difference, 16, 44, 74 E electromotive force, cell potential, 16, 89 E energy, 5, 14, 15, 17, 19-23, 32, 33, 39, 43, 45, 56, 65, 67, 95, 132, 137, 140, 146, 147, 237 E etendue, 35, 36 E identity symmetry operator, 31 E irradiance, 35, 90 E modulus of elasticity, 15 E potential (electrochemistry), 71, 72, 76 E scattering amplitude, 42 E thermoelectric force, 43, 44 E* space-fixed inversion symmetry operator, 31 Eh Hartree energy, 9, 18, 20, 22, 95, 96, 111, 132, 137, 139, 145, 147 Eu Euler number, 82 E[x] expectation value of x, 151 E exa (SI prefix), 91 E excess quantity (superscript), 60 E symmetry label, 31, 32 Ei exabinary (prefix for binary), 91 / activity coefficient, 59, 61, 70, 75 / atomic scattering factor, 42 / finesse, 36 / frequency, 13, 28, 29, 34-36, 41, 68, 81, 184 82, 88,89, 129 G shear modulus, 15 / friction factor, 15 G thermal conductance, 81 / fugacity, 58, 74 G vibrational term, 25 / oscillator strength, 37 G weight, 14 / vibrational force constant, 27 Gf Fermi coupling constant, 111 f(cx) velocity distribution function, 45 Gr Grashof number (mass transtftr / SE f femto (SI prefix), 91 G gauss (unit of magnetic flux density), f fermi (unit of length), 135 27, 141, 142 f formation reaction (subscript), 60, 61 G giga (SI prefix), 91 f fluid phase, 54 Gi gigabinary (prefix foj^nnarj), 91 ft foot (unit of length), 135 Gy gray (SI unit), 89, i^N fus fusion, melting (subscript), 60 Gal gal (unit of acceleration), 137 F Fock operator, 20, 21 h coefficient of heat transfer, 81, 82 F force, 7, 14-17, 87, 89, 95, 104, 137, 138, h film thickness, 77 143, 144, 146-148 h height, 13 F angular momentum, 30 h Miller index, 44, 79, 80 F Faraday constant, 70-75, HI h, h Planck constant (h — h/2K), 7, 9, 18, 20, F fluence, 35 22, 23, 25, 28-30, 34-36,38, 39, 41-43, F frequency, 28 45, 65-68,95, 96, 111, 112, 131, 132, 145 F Helmholtz energy, 56 h hecto (SI prefix), 91 F rotational term, 25 h helion, 50, 115 F structure factor, 42 h hour (unit of time), 92, 137 F vibrational force constant, 27 ha hectare (unit of area), 136 F(c) speed distribution function, 45 hp horse power (unit of power), 138 Fo Fourier number (mass transfer), 82 hk \gprric horse power (unit of power), 138 Fr Froude number, 82 F face-centred (crystal lattice), 44 H magnetic field strength, 17, 90, 142, 145, F farad (SI unit), 89, 140 ) 147, 148 F symmetry label, 31 coulomb integral, resonance integral, 19 °F degree Fahrenheit (unit of temperature), H dose equivalent, 139 138 H enthalpy, 5, 6, 8, 55-58,60, 65, 66 Fr franklin (unit of electric charge), 134, 135, H fluence, 35 139, 144 H Hamilton function, hamiltonian, 14, 18, 20, 21 g reciprocal lattice vector, 80 ' Ha Hartmann number, 82 g acceleration of free fall, 13, 78, 81, 82, H Heaviside function, 45, 66, 67, 107 112, 137 H henry (SI unit), 89, 142 g degeneracy, statistical weight, 26, 39, 45 Hz hertz (SI unit), 6, 13, 29, 88, 89 g (spectral) density of vibrational modes, 43 g,ge q-factor, 23, 27, 28, 111 i unit vector, 13 g vibrational anharmonicity constant, 25 i electric current, 16, 72, 85, 86, 95, 139, g gas, 54, 61, 73 143, 144 g gram (unit of mass), 91, 136 i inversion symmetry operator, 31, 32 g gerade symmetry label, 32, 33 i square root of -1, 18, 32, 42, 43, 98, 103, 107 gr grain (unit of mass), 136 id ideal (superscript), 60 gal gallon (unit of volume), 136 in inch (unit of length), 135, 138 gon gon (unit of angle), 136 ir irrational (superscript), 145, see non-rationalized imm immersion (subscript), 60 G reciprocal lattice vector, 42, 80 iep isoelectric point, 78 G (electric) conductance, 17, 89 G Gibbs energy, 5, 6, 56-58, 60, 61, 65, 66, J nuclear spin angular momentum, 23, 27-30, 121 71, 74, 77 I differential cross section, 65 G gravitational constant, 14, 112 I electric current, 4, 16, 17, 72, 85, 86, 95, G intej^Kitecninet) absorption cross section, 139, 143, 146, 147 ^Mfl^ I intensity, 4, 34-37, 85, 86, 88 185 I ionic strength, 59, 70, 76 K I ionization energy, 22 KM I moment of inertia, 14, 25 Kab I body-centred (crystal lattice), 44 Kn 3 internal vibrational angular momentum, 30 K 3 electric current density, 16-18, 43, 72, 73, 90, 146-148 Ki 3 particle flux density, 43, 44 3 unit vector, 13 I J angular momentum, 30, 138 \ J coulomb operator, 20, 21 l J electric current density, 16, 18, 72, 90 J heat flux vector, 43, 44 l J angular momentum quantum number (component), 31-33, 51, 66, 67, 95 l J coulomb integral, 20 J flux density, 81 J Massieu function, 56 lb J moment of inertia, 14 lc Jab spin-spin coupling constant, 28 lm J joule (SI unit), 6, 9, 29, 89, 93, 137 lx k unit vector, 13 i-y-1 citin k angular wave vector, 43 k absorption index, 37, 40, 41 L k angular momentum quantum number L (component), 30 k, kB Boltzmann constant, 36, 38, 39, 45, 46, L J 64-67, 81, 111, 112, 133 Lu k coefficient of heat transfer, 81 L k coverage factor, 151-154 L kn Debye angular wavenumber, 43 L k decay constant, 24 L k Miller index, 44, 79, 80 L k rate coefficient, rate constant, 63jqQj72 k thermal conductivity, 81, 82, 90 kd mass transfer coefficient, 72, 81, 82 L kn Henry's law constant, 58, 59 L krst vibrational force constant, 27^ L k kilo (SI prefix), 91 Le kg kilogram (Si unit), 4, 86, 87, 91, 136, L 143 kp kilopond (unit of forte), 137 L kat katal, 89 kgf kilogram-force (unit of force), 137 m K exchange operator, 20, 21 m K rate (coefficient) matrix, 66, 67 K absorption coefficient, 36 m K angular mJrSWrtum quantum number m (component), 30, 31, 33 m K bulk modulus, 15 K coefficient of heat transfer, 81 K equilibrium constant, 58, 59, 61 m K exchange integral, 20 m kinetic energy, 14, 93, 94 Michaelis constant, 66 reduced nuclear spin-spin coupling constant, 28 Knudsen number, 82 kelvin (SI unit), 3, 5, 40, 56, 86, 87, 89, 138 kilobinary (prefix for binary), 91 electron orbital angular momentum, 30, 31 internal vibrational angular momentum, 30 cavity spacing, path length, 36, 37 length, 4, 13, 20,w42, 45; 53, 81, 82, 85-87, 103, 131, 135,^^43-145, 147 mean free path, 43, 44 Miller index, 44, 79, 80 vibrational quantum number, 26, 30 liquid, 54, 55, 61 litre (unit of volume), 6, 92, 136 pound ^unit of mass), 136 liquid crystal, 54 lumen (SI unit), 89 lux (SI unit), 89 light year (unit of length), 136 litre atmosphere (unit of energy), 137 angular momentum, 14, 25, 30, 32, 138 Avogadro constant, 9, 45, 47, 53, 65, 68, 78, } HI, 112, 132 angular momentum quantum number, 30, 32 Debye length, 77 diffusion length, 49 field level, power level, 98, 99 inductance, 17, 89, 142, 147 Lagrangian function, Lagrangian, 14, 45 length, 3, 4, 6, 13, 20, 42, 45, 53, 81, 85-87, 92, 94, 95, 131, 135, 136, 143-145, 147 Lorenz coefficient, 43 radiance, 35 term symbol, 117-120 Lewis number, 82 langmuir (unit of pressure-time product), 81 litre (unit of volume), 6, 92, 136 magnetic dipole moment, 17, 23, 95, 121, 133, 142, 147 angular momentum quantum number (component), 28, 30 effective mass, 43 electric mobility, 73 mass, 4, 6, 7, 9, 14, 18, 22, 28, 45, 47, 48, 64, 77, 81, 92, 93, 111, 115, 116, 133, 136, 143, 144 Hermann-Mauguin symbol, 44 molality, 48, 49, 58, 59, 62, 70, 75, 133 186 to order of reaction, 63, 66, 72 N mu atomic mass constant, 22, 47, 94, 111, 117 me electron mass, 9, 18, 20, 22, 23, 29, 37, N 94, 95, 96, 111, 132, 136 N mn neutron mass, 111 N mp proton mass, 29, 111 NA tow W-boson mass, 111 toz Z-boson mass, 111 NE m metre (SI unit), 3, 6, 26, 29, 40, 53, 86, Nu 87, 135, 143 Nu m milli (SI prefix), 91 mi mile (unit of length), 135 N min minute (unit of time), 92, 137 N mix mixing (subscript), 60 Np mol mole (SI unit), 4, 6, 40, 43, 47, 53, 59, 86, 88, 98 oz mon monomeric form, 54 mmHg millimetre of mercury (unit of pressure), Oe 138 M magnetization, 17, 28, 142, 147 V M torque, 14 M transition dipole moment, 26 V M angular momentum quantum number P (component), 30 P M matrix for superlattice notation, 79, 80 P M molar mass, 47, 54, 59, 117, 133 Mu molar mass constant, 47, 117 P M mutual inductance, 17 v i M radiant excitance, 34, 35 M Madelung constant, 43 P MT relative molecular mass, 47 P Ma Mach number, 82 M mega (SI prefix), 91 P M nautical mile (unit of length), 92 P m molar (unit of concentration), 48, 49 P Mi megabinary (prefix for binary), 91 pc Mx maxwell (unit of magnetic flux), 142 pH pol n amount of substance, chemical amount, ppb 4-7, 45, 47, 48, 51, 53, 56,^5^1, 77, pph 78, 85, 86, 88, 89 ppm n charge number of cell reaction, 71, 72 ppq n number density, number concentration, ppt 43-45, 48, 49, 53,C4, 67) 71, 81, 133 psi n number of electrons, 20 n order of (Bragg) /eflapton, 42 pzc n order of reaction, 63, 66, 72 pphm n principal quantum number, 23 n refractive index, 34, 37, 40, 41 P n nano (SI prefix), 91 P n nematic phase, 54 P n neutron, 8, 22, 50, 115 P N angular momentum, 30 / P N neutron number, 22, 24 P number of entities, 6, 24, 42, 45, 47, 77, 78, 89 number of events, 13, 35 number of measurements, 151 number of states, 44, 45, 66, 67 Avogadro constant, 9, 17, 37,^*3ö> 45, 47, 53, 92, 111, 112, 117, 133 density of states, 43, 45, 66, 67 Nusselt number (mass'fNmsiey), 82 (spectral) density of vibrational modes, 43, 44 newton (SI unit), 89Si^ nucleus, 22 neper, 8, 92, 98, 99 ounce (unit of mass), 136 oerstedt (unit of magnetic field strength), 142 electric dipole moment, 17, 23, 26, 39, 41, 95, 134, 141, 146 momentum, 14, 18, 45 bond order, 19, 21 fractional population, 38, 39, 46 number density, number concentration, 43, 44 permutation symmetry operator, 31 jj/ressure, 3, 5-8, 14, 15, 37-39, 48, 54, 56-58, 61-63, 72, 77, 81, 82, 89, 99, 131, 138 probability, 45 permutation-inversion symmetry operator, 31 pico (SI prefix), 91 primitive superlattice, 79 proton, 8, 22, 50, 115 parsec (unit of length), 136 pH, 5, 70, 75, 76 polymeric form, 54 part per billion, 98 part per hundred, 98 part per million, 98 part per quadrillion, 98 part per thousand, part per trillion, 98 pounds per square inch (unit of pressure), 138 point of zero charge, 78 part per hundred million, 98 density matrix, 21, 46 dielectric polarization, 16, 40, 146 heat flux, thermal power, 57, 81 permutation symmetry operator, 31 power, 14, 22, 98, 99, 138 pressure, 14, 39, 48, 58, 81, 89, 131, 138 probability, 45, 46, 65, 153, 154 187 P probability density, 18, 151 R P radiant power, 34-36 R P sound energy flux, 15 R P transition probability, 65 R P (volume) polarization, 141 P weight, 14 i?H P* permutation-inversion symmetry R operator, 31 R Pe Peclet number (mass transfer), 82 R Pr Prandtl number, 82 R P peta (SI prefix), 91 R P poise (unit of dynamic viscosity), 138 i?oo P primitive (crystal lattice), 44 R P symmetry label, 51 Ra Pa pascal (SI unit), 40, 48, 62, 89, 138 Re Pi petabinary (prefix for binary), 91 Rm,R, R q electric field gradient, 24, 29, 141 R q angular wave vector, 42, 43 Ry <7d Debye angular wavenumber, 43 °R q charge density, 16, 18, 23, 43, 44, 140 q charge order, 19 q flux of mass, 81 s q generalized coordinate, 13, 14, 45 s q heat, 56, 73, 89 s q partition function, 39, 45, 46, 66, 67 s q vibrationa normal coordinate, 27 s s Q quadrupole moment, 23, 24, 28, 29, 141 s Q disintegration energy, 24 m3"2^ Q electric charge, 4, 16, 17, 50, 70-72, 89, sN 95, 134, 139, 143, 145, 146, 148 Qw electroweak charge, 24 s Q heat, 56, 57, 73, 81, 89 sr Q partition function, 39, 45, 46, 67 ^sln Q quality factor, 36 ' sol Q radiant energy, 34 sub Q reaction quotient, 58, 61 Q vibrational normal coordinate, 27 S Q Q-branch label, 33 S S r position vector, 13, 16, 23, 42, 45, 95, S 104, 143, 145, 146 r interatomic distance, 27, 28, 96 S r internal vibrational /oordiilate, 27 S r radius, 13, 64, 70 r rate of concentration change, 63 S r spherical coordinate^!*, 18, 20, 21, 96 S r reaction (subsenwt^iO S rad rad (unit of radiation dose), 139 rad radian (Sljmit), 8(13, 29, 89, 98, 136 S rem rem (unit of dose equivalent), 139 S refl reflected (subscript), 34 S Sc R latti^vecTlir, 42 Sh R nuclear orbital angular momentum, 30 Sn particle position vector, 42 transition dipole moment, 26 electric resistance, 17, 89, 140 i gas constant, 7, 38, 45, 46, 54, 57-59, 61, 62, 64, 65, 71, 72, 74, 75, 111 Hall coefficient, 43 internal vibrational coordinate, 27 molar refraction, 37 position vector, 42, 43^ reflectance, 36 resolving power, 36 Rydberg constant, 22^|& 111 thermal resistance* 81 Rayleigh number, 82 Reynolds number, 5, 82 magnetic Rev^ds^number, 82 rhombohedral (crystal lattice), 44 röntgen (unit of exposure), 139 rydberg (unit of energy), 137 degree Rankine (unit of thermodynamic temperature), 138 electron spin angular momentum, 30 length of path, length of arc, 13 order parameter, 42 sedimentation coefficient, 13, 77 solubility, 48, 49 standard deviation, 151, 152 symmetry number, 46 variance, 151, 152 second (SI unit), 6, 29, 30, 34, 86, 87, 92, 98, 137, 143, 144 solid, 54, 73 steradian (SI unit), 13, 89 solution, 54 solution (subscript), 60 sublimation (subscript), 60 Poynting vector, 17, 148 probability current density, 18 scattering matrix, 65 electron spin angular momentum, 25, 27, 30, 32 nuclear spin operator, 28, 29 integrated absorption coefficient, 37, 39, 40 action, 14 area, 13, 24, 34, 36, 78, 81, 90, 136 entropy, 5, 55-58, 60, 61, 63, 65, 66, 78, 90, 138, 139 overlap integral, 19, 21 statistical entropy, 46 vibrational symmetry coordinate, 27 Schmidt number, 82 Sherwood number, 82 rotation-reflection symmetry operator, 31 188 Sr Strouhal number, 82 St Stanton number (mass transfer), 82 S Siemens (SI unit), 89, 132 St stokes (unit of kinematic viscosity), 138 Sv sievert (SI unit), 89, 139 Sv svedberg (unit of time), 137 t Celsius temperature, 38, 46, 56, 89, 111, 138 t film thickness, thickness of layer, 77 t time, 4, 8, 13, 15, 16, 24, 29, 34-36, 57, 63, 66-69, 72, 81, 82, 85-87, 92, 95, 98, 137, 143, 144 t transport number, 73 t1/2 half life, 24, 64, 116 t tonne (unit of mass), 92, 136 t triton, 50, 115, 116 tr transmitted (subscript), 34 tr triple point (subscript), 60 trs transition (subscript), 60 T hyperfine coupling constant (tensor), 27 T torque, 14 Tc Curie temperature, 43 T kinetic energy, 14, 18, 93, 94 Tn Neel temperature, 43 T period, characteristic time interval, 13 T relaxation time, 13, 23, 29, 43 T thermodynamic temperature, 3, 4, 6, 7, 35, 37, 39, 40, 43, 46, 54, 56-59, 61, 62, 64-67, 71, 72, 74, 75, 77, 81, 82, 87, 89, 133, 138 T total term, electronic term, 25, 131 T transmittance, 36 T1/2 half life, 24, 64, 116 T tera (SI prefix), 91 T tesla (SI unit), 27, 89, 141 Ti terabinary (prefix for binary), 91 Torr torr (unit of pressure), 81, 131, 138 u displacement vector, 42 u velocity, 13, 45, 90, 148 S u Bloch function, 43 u electric mobility, 73 . u estimated standard /ricertäinty, 151-154 u lattice direction index, 44 u speed, 13, 81, 90(93^5 u ungerade symmetry label, 32, 33 u unified atomic mass unit, 9, 22, 47, 92, 94, 111, 112, 116, 117, 121, 136 ua astronomical unit (unit of length), 9, 92, 94, 136( U cell potential, 71 U electric potential difference, 16, 17, 44, 74 U electrode potential, 71 U expanded uncertainty, 151-154 U internal energy, 55-57, 61, 65 U "enzyme unit", 89 v velocity, 13-16, 43, 45, 73, 90. 148*" v lattice direction index, 44 v rate of reaction, 63, 64, 68^69 > v specific volume, volume, 6, 13, 14, 90 v speed, 13, 81, 82, 93, 95 v vibrational auantumJmmbeV. 25, 26, 33, 51 vap vaporization (subscript), 60 vit vitreous substance/ 54 ' V electric potential, 16, 75, 89, 140 V potential energy, 14, 18, 23, 27, 29, 66, 96 V volume, 5-8, 13-17, 23, 34, 37, 41, 43, 45, 47, 48, 65, 66, 81, 90, 92, 111, 133, 13*^9, 146, 147 V volt (SLuWT 9, 89, 94, 134, 140 w velocity, 13, 90, 148 w lattice direction index, 44 w mass fraction, 47, 49, 97, 98 w radiant energy density, 34 w speed, 13, 81, 90, 93, 95 w Work, 14, 56, 70, 89 W m degeneracy, statistical weight, 26, 45 W\ number of open adiabatic channels, 66, 67 number of states, 45, 66, 67 W radiant energy, 34, 36 ^ty weight, 14 u n transmission coefficient, 65, 66 v A thermal conductivity tensor, 43 v A absolute activity, 46, 57, 58 ve A angular momentum quantum number (component), 30 £ A decay constant, 24 £ A mean free path, 44, 65, 80, 81 molar conductivity of an ion, 54, 73 thermal conductivity, 43, 44, 81, 90 van der Waals constant, 77 wavelength, 3, 34-38, 42-44, 131 lambda (unit of volume), 136 angular momentum quantum number (component), 30-32 molar conductivity, 6, 54, 73, 90, 132 electric dipole moment, 17, 23, 26, 35, 39, 41, 95, 141, 146 chemical potential, 5, 6, 8, 43, 44, 46, 57, 58, 60-62 electric mobility, 73 electrochemical potential, 71 dynamic friction factor, 15 Joule-Thomson coefficient, 57 magnetic dipole moment, 17, 23, 95, 115, 121, 133, 142 mean, 151 mobility, 43 permeability, 5, 17, 81, 90 reduced mass, 14, 64-66, 95 Thomson coefficient, 43 viscosity, 15, 81 electrochemical potential, 71 permeability of vacuum, magnetic constant, 16, 17, 28, m, 133, 143, 144, 147 Bohr magneton, 23, 27, 29, 95, 111, 112, 115, 133, 142 electron magnetic moment, 23, 111 nuclear magneton, 23, 29, 111, 115, 116, 121, 142 proton magnetic moment, 112, 116 micro (SI prefix), 91 micron (unit of length), 135 muon, 8, 50, 115, 116 stoichiometric number matrix, 53 charge number of cell reaction, 71 Debye frequency, 43 frequency, 13, 23, 25, 28, 29, 33-36, 41, 68, 81, 88, 89, 129 kinematic viscosity, 15, 90, 138 stoichiometric number, 53, 58, 61, 63, 71-74 Debye wavenumber, 43 wavenumber (in vacuum), 25, 26, 34-39, 131 vibrational state symbol, 33 electron neutrino, 50, 115 Coriolis coupling constant, 26 extent of reaction, advancement, 48, 49, 53, 58, 60, 63, 67 191 £ magnetizability, 23, 95, 142, 147 E E partition function (grand canonical ensemble), 46 U E standardized resonance frequency, 29 £ 7t internal vibrational angular momentum, 30 t tt surface pressure, 77 t it symmetry label, 32, 33 it ratio of circumference to diameter, 32, t 34-37, 42, 43, 45, 64, 65, 70, 95, 103, t 112, 134 t it pion, 115 t t it osmotic pressure, 59 t it Peltier coefficient, 43 II product sign, 105 f II symmetry label, 31-33 f f p density matrix, 46 p resistivity tensor, 43 p acoustic factor (reflection), 15 p charge density, 16, 18, 23, 43, 44, 140, 4> 146-148 4> p cylindrical coordinate, 13 p density of states, 43, 45, 66, 67 p density operator, 46 p radiant energy density, 34, 90 4>rst ( p mass density, mass concentration, 6, 14, . 15, 38, 48, 51, 54, 81, 82, 97 J> A p reflectance, 36 w p resistivity, residual resistivity, 17, 43 Pa surface density, 14 <2> a conductivity tensor, 43 <2> a shielding tensor, 29 <2> a absorption cross section, 37, 38 ' <2> a area per molecule, 77 <2> a cross section, 24, 38, 64-66 a conductivity, electrical conductivity, 17, x 43, 44, 73, 140 a density operator, 46 x a normal stress, 15 x a reflection symmetry o£erator,"31, 32 x a shielding constant, 29 a order parameter, 42 \ x a spin quantum number (component), 30 Xe a standard deviation, 103, 151 Xm a spin quantum number (component), 30 a- Stefan-Boltzmann constant, 35, 112 a surface charge density, 6, 16, 71 tp a surface tension, 14, 56, 77, 78, 81, 90 tp a symmetry number, 46 cr2 variance, 151 IF a wavenumber, 34 lF a symmetry label, 32, 33 spin quantum number (component, 30, 32 film tension, 77 summation sign, 105 acoustic factor (transmissii^), M> characteristic time interval, relaxation time, 13, 43, 64 correlation time, 29 i mean life, lifetime, 24, 43, 44, 64, 66, 115, 116 shear stress, 15, 81 thickness of layer^7 Thomson coefficient, 43 transmittance, 36 fluidity, 15 plane angle, 13, 38, 89, 92, 136 volume fraction, 97, 98 electruSrt^atial, 16, 75, 89, 140, 146, 147 fugacity coefficient, 58 inner electric potential, 70, 71 molecular orbital, 19-21 osmotic coefficient, 59 quantum yield, 66, 67 spherical polar coordinate, 13, 18, 21, 96 vibrational force constant, 27 volume fraction, 48, 97, 98 wavefunction, 18, 46 work function, 80 heat flux, thermal power, 57, 81 magnetic flux, 17, 89, 142 potential energy, 14, 23 quantum yield, 66 radiant power, 34 work function, 43, 80 quadrupole interaction energy tensor, 24, 29 atomic orbital, 19, 21 electronegativity, 22 magnetic susceptibility, 17, 133, 142, 145, 147 surface electric potential, 70 electric susceptibility, 16, 145, 146 molar magnetic susceptibility, 17, 133, 142 outer electric potential, 70 wavefunction, 18, 20, 30, 43, 96 electric flux, 16 wavefunction, 18, 46 192 ijj angular frequency, angular velocity, 8, 13, 23, 34, 89, 98 wo Debye angular frequency, 43 uj degeneracy, statistical weight, 26, 45 lo harmonic (vibrational) wavenumber, 25, 26 lu solid angle, 13, 34, 36, 65, 89 £2 angular momentum quantum number (component), 30, 32 £2 nutation angular frequency, 28 Q partition function, 46 Q solid angle, 13, 34-36, 65, 89 £2 volume in phase space, 45 ft ohm, 89, 140 Special symbols % percent, 97, 98 %o permille, 97, 98 degree (unit of arc), 38, 92, 136 standard (superscript), 60, 71 minute (unit of arc), 92, 136 second (unit of arc), 92, 136 complex conjugate, 107 excitation, 50 pure substance (superscript), 60 I, activated complex (superscript), transition state (superscript), 60 oo infinite dilution (superscript), 60 e even parity symmetry label, 32 0 odd parity symmetry label, 32 [B] amount concentration, concentration of B, 48, 54, 69 Ar derivative with respect to extent a of reaction, 58, 60, 74 dim(Q) dimension of quantity Q, 4 V nabla, 16-18, 20, 43, 81, 96, ^^46, 147 [a]xe specific optical rotatory power, 37, 38 [Q] unit of quantity Q, 4 O 193 194 SUBJECT INDEX 195 196 When more than one page reference is given, bold print is used to indicate the general (defining) reference. Underlining is used to indicate a numerical entry of the corresponding physical quantity. Greek letters are ordered according to their spelling and accents are ignored in alphabetical ordering. Plural form is listed as singular form, where applicable. ab initio, 20 abbreviations, 157-164 abcoulomb, 139 absolute activity, 46, 57 absolute electrode potential, 71 absorbance, 36, 37, 39, 40 decadic, 5, 36, 37 napierian, 36, 37 absorbed dose (of radiation), 89, 139 absorbed dose rate, 90 absorbing path length, 37 absorptance, 36, 37 internal, 36 absorption, 33, 35, 37, 38, 41 net, 38 rate of photon, 67 spectral, 38 absorption band, 37, 41 absorption band intensity net integrated, 39 absorption coefficient, 6, 35-37, 40, 41 decadic, 36 integrated, 37, 39-41 linear decadic, 36 linear napierian, 36 molar, 6 molar decadic, 36, 37 molar napierian, 36 napierian, 36, 38 net, 37 absorption cross section integrated, 38 integrated net, 37 net, 37 absorption factor, 36 absorption factor (acoustic), 15 absorption factor (radiation), 15>* absorption index, 37 absorption intensity, 38-40 conventions for, 40 spectroscopic, 40 absorption line, 37 lorentzian, 67 absorption of speciesJ7 absorption spectrum, 36 abundance isotopic, 121 acceleration, 13, 90, 137 acceleration of free fall, 13, 78, 81 standard, 112, 137 accepted non-SI unit, 92 acceptor, 44 acceptor ionization energy, 43 I acid dissociation equilibrium constant for an, 59 acoustic factor, 15 acoustics, 14, 15, 98, 99 acre, 136 acronyms, 157-164 action, 14, 95, 138 # au of, 138 activated complex (superscript), 60 activation standard enthalpy of, 65 standard entropy of, 65 standard internal energy of, 65 volume of, 65 activation enelgy, 64 Arrhenius, 64 activitvVr, 58, 70, 72 ansVlme, 46, 57 catalytic, 89 mean ionic, 70 radioactive, 24, 89, 139 Jrelative, 57, 71 , unit, 74 activity coefficient, 58, 59, 61, 70, 75 mean ionic, 70 activity coefficient (concentration basis), 59 activity coefficient (Henry's law), 59 activity coefficient (molality basis), 59 activity coefficient (mole fraction basis), 59 activity coefficient (Raoult's law), 59 activity of an electrolyte, 70 activity of hydrogen ions, 75 addition in quadrature, 153 adjoint, see hermitian conjugate admittance (complex), 17 adsorbed amount, 77 adsorbed molecules number of, 78 adsorbed species, 54 adsorbed substance amount of, 78 adsorption, 78 reduced, 78 relative, 78 adsorption (subscript), 59 advancement, 48 affinity, 58 197 electron, 22 affinity of reaction, 58, 61 Alfven number, 82 a-particle, 8, 50, 115 amagat, 139 reciprocal, 139 American system of names, 98 amorphous solid, 54, 55 amount, 4, 47, 53, 54, 59, 89 chemical, 4, 45, 47, 53 excess, 77 Gibbs surface, 78 surface, 77 surface excess, 77, 78 amount concentration, 4, 37, 38, 48, 54, 62, 63, 81, 90 amount density, 139 amount fraction, 48, 49, 97 amount of adsorbed substance, 78 amount of substance, 4, 45, 47, 48, 51, 53, 56, 63, 71, 85, 86, 88, 89 amount of substance of photons, 68 amount-of-substance concentration, 4, 6, 37, 38, 48, 90 amount-of-substance fraction, 48, 97, 121, 133 Ampere law, 147 ampere, 4, 85, 86, 87, 139, 143 amplitude, 8, 98 scattering, 42 amplitude function, 18 amplitude level signal, 98 analysis of variance, 152 angle, 35 Bragg, 42 contact, 77 loss, 17 plane, 13, 38, 89 plane (degree), 92 plane (minute), 92 plane (second), 92 reciprocal unit cell, 42 rotation, 79 scattering, 65 solid, 13, 34, 36, 65, 89 unit cell, 42 weak mixing, 24, 111 angle of optical rotation, 37, 38 angstrom, 131, 135 angular frequency, 8, 13, 34, 89, 98 Debye, 43 Larmor, 23 nutation, 28 angular frequency unit, 28 angular fundamental translation vector, 42 angular momentum, 14, 18, 20, 23, 28, 30, 67, 95, 121, 138 electron orbital, 30 electron orbital plus spin, 30 electron spin, 30 internal vibrational, 30 nuclear orbital, 30 nuclear spin, 30 orbital, 32 reduced, 95 spin, 19, 28, 116 total, 32 angular momentum component, 30 angular momentum eigenvalue equation, 30 angular momentum operator, 19, 28, 30 symbol of, 30 angular momentum quantum number, 23, 26, 30, 67 angular reciprocal lattice vector, 42 angular velocity, 13, 89, 90 angular wave vector, 43 anharmonicity constant vibrational, 25 anode, 75 anodic partial current, 72 anodic transfer coefficient, 72 ANOVA, 152 anti-bonding interaction, 19 anti-symmetrized product, 20 anticommutator, 19 antineutrino, 115 antineutron, 50 antiparticle, 115 antiproton, 50 apparent (superscript), 60 aqueous-organic solvent mixture, 76 aqueous solution, 54, 55, 57, 74, 97 aqueous solution at infinite dilution, 54, 55 aqueous system, 74 arc length of, 13 arc function, 106 are, 136 area, 6, 13, 24, 34, 36, 78, 81, 90, 136 electric quadrupole moment, 121 infinitesimal, 13 nuclear quadrupole, 26 quadrupole, 141 specific surface, 77 surface, 49, 71, 77, 78 vector element of, 16 area element, 35 area hyperbolic function, 106 area per molecule, 77 areic, 6 areic charge, 6 Arrhenius activation energy, 64 assembly, 45 198 astronomical day, 137 astronomical unit, 9, 92, 94, 136 astronomy, 94 asymmetric reduction, 25 asymmetric top, 30, 31, 33 asymmetry parameter, 25 Wang, 25 asymmetry-doubling, 31 atmosphere, 131 litre, 137 standard, 62, 111, 138 atmospheric pressure, 58 atomic absorption spectroscopy, 97 atomic mass, 22, 47, 121 average relative, 120 relative, 47, 117-120 atomic mass constant, 22, 47, 111, 117 atomic mass evaluation, 121 atomic mass unit unified, 9, 22, 47, 92, 94, 111, 116, 117, 121, 136 atomic number, 22, 23, 49, 50, 117, 121 atomic orbital, 19 atomic-orbital basis function, 19, 21 atomic physics, 22 atomic scattering factor, 42 atomic spectroscopy, 14 atomic standard, 137 atomic state, 32 atomic unit, 4, 18, 20, 22, 24, 26, 94, 95, 96, 129, 132, 135, 143-145 symbol for, 94-96 atomic unit of magnetic dipole moment, 95 atomic units system of, 145 atomic weight, 47, 117-120 atomization, 61 energy of, 61 atomization (subscript), 59 atto, 91 au, 139 au of action, 138 au of force, 137 au of time, 137 average, 41, 49 average collision cross section, 64 average distance zero-point, 27 average lifetime, 64 average mass, 117 average molar mass, 77 average molar mass (z-average), 77 average molar mass (mass-average), 77 average molar mass (number-average), 77 average population, 46 average probability, 46 average relative atomic mass, 120 average speed, 45, 81 averaged collision cross section, 64, 65 Avogadro constant, 4, 9, 45, 47, 53, 111, 112 avoirdupois, 136 axis of quantization, 30 azimuthal quantum number, 30 ^\ \ band gap energy, 43 band intensity, 26, 38-40 bar, 48, 131, 138, 233 barn, 121,136 barrel (US), 136 base hydrolysis, 59 base of natural logarithm, 8^ base quantity, 4, 16, 85, 86 symbol of, 4 base unit, 34, 85-89, 93, 135, 143, 144 SI, 4, 85-88, 90,^1^ base-centred lattice, 44 basis function atomic-orbital, 19, 21 Bates-Guggenheim convention, 75 becquerel, 89<«N?r^ Beer-Lambert law, 35, 36 bel, 92,^8, 99» bending coordinate, 27 best estimate, 149 P-particle, 50, 115 billion, 98 bimolecular reaction, 68, 69 fclna^ multiple, 91 binominal coefficient, 105 biochemical standard state, 62 biot, 135, 139, 144 li\Jl38 centi, 91 centigrade temperature, 56 centimetre, 134, 135, 144 centipoise, 138 central coulomb field, 23 centre of mass, 31 centrifugal distortion constant, 25 centrifugation, 78 CGPM, xi, 85, 87, 88, 92 CGS unit, 96, 135/^38^ , chain-reaction mechanism, 68 characteristic Debye length, 78 characteristic impedance, 111 characteristic number, 5 4* transport, 82 characteristic temperature, 46 characteristic terrestrial isotopic compos/ion, 117, 120 characteristic time interval, 13 characteristic (Weiss) temperature, 43 ^ charge, 16, 18, 20, 43, 71, 72, 95, lMJJIllffy areic, 6 effective, 21 electric, 4, 16, 50, 70, 89, 134, 135, 139, 144 electron, 18 electronic, 19 electrostatic unit of, 135 electroweak, 24 elementary, 8, 9, 22, 24, 26, 70, 94-96, 111, 145 ion, 73 point of zero, 7i fc proton, 22, 70 A reduced, 95 charge (au) proton, 139 charge densityj¥^L8, 23, 43, 44, 140 electric, ©0^ surface, 6, 71 charge ArmbJ, 20, 49-51, 70, 71, 73 iorHc^s charge number of cell reaction, 71 charge order, 19 charge partition, 74 charge transfer, 71, 72, 74 tfliei^'ical amount, 4, 45, 47 chemical composition, 47, 51, 71 chemical-compound name, 8 .^chemical element, 49 symbol for, 8, 9, 49, 50, 113, 117-120 chemical energy, 73 chemical equation, 49, 53 bimolecular elementary reaction, 68 elementary reaction, 68, 69 general, 53 stoichiometric, 52, 60, 63, 68, 69 unimolecular elementary reaction, 68 chemical equilibrium, 71, 74 local, 72 chemical formula, 49-51 chemical kinetics, 63, 68 chemical potential, 5, 6, 8, 44, 46, 57, 62 standard, 57, 61, 62 chemical reaction, 53, 60, 63, 73 equation for, 52 transition state theory, 65 chemical shift, 29 chemical substance, 51 chemical symbol for element, 50 chemical thermodynamics, 56-62 chemically modified electrode, 70 200 chemistry quantum, 18-21 chemistry (quantum), 94 choice of standard state, 58 choice of unit, 3 CIAAW, 117 CIPM, 9, 56, 94 circular cross-section, 87 circular frequency, 44 Larmor, 23 circumference, 112 classical electrostatics, 70 classical mechanics, 14-15 clausius, 138 Clausius-Mossotti formula, 40 closed-shell species, 21 closed-shell system, 20 CME, 70 CODATA, 95, 111, 115 coefficient absorption, 6, 35-37, 40, 41 activity, 58, 59, 61, 70, 75 decay, 8, 98 diffusion, 43 Einstein, 35, 37, 39 expansivity, 57 extinction, 37 fugacity, 58 Hall, 43 Joule-Thomson, 57 Lorenz, 43 mass transfer, 72 molar absorption, 6 osmotic, 59 Peltier, 43 pressure, 56 pressure virial, 57 sedimentation, 77, 78 Thomson, 43 transmission, 66 van der Waals, 57 virial, 57 coefficient of heat transfer, 81 coefficient of thermal expansion, 57 coherent derived units, 85 coherent SI units, 85 coherent system, 93 coherent system of units, 85, 93 coherent unit, 93 collision, 65 individual, 65 | collision cross section, &4j averaged, 64, 65 collision density, 65 collision diameter, 64 collision energy, 66 4* translational, 65 collision frequency, 64, 65 collision frequency factor, 65 collision model hard sphere, 64 collision number, 65 collision partner, 64 collision theory, 65 collisions total number of, 65 colloid chemistry, 77, 78 combined standard uncertainty, 152 combustion, 61 standard enthalpy of, 61 combustion reaction (subscript), 59 commutator, 19 complete set of SI units, 85 complex admittancejfT^ complex coefficient, 46 complex conjugate, 18, 107 complex conjugate matrix, 107 complex impedance, 17 complex ion/51"> complex mechanism, 68 complex molar polarizability, 41 complex number, 107 complex refractive index, 37, 40 complex relative permittivity, 40 component quantum number, 30 composite mechanism, 68 compressibility isentropic, 56 isothermal, 43, 56 compressibility factor, 57 compression factor, 57 compression modulus, 15 concentration, 38, 48, 59, 62, 64, 67, 70, 72, 90, 132 amount, 4, 37, 38, 48, 54, 62, 63, 81, 90 amount-of-substance, 4, 6, 37, 38, 48, 90 critical coagulation, 78 critical micellisation, 78 excited state, 67 mass, 38, 48, 97 mean ionic, 70 number, 43, 45, 48, 63, 64 reduced surface excess, 78 relative surface excess, 78 standard, 62, 72 substance, 4, 48 surface, 48, 63, 77 surface excess, 77 total surface excess, 77 concentration at the interface, 72 concentration change rate of, 63 concentration perturbation, 64 201 concentration vector, 67 condensed phase, 40, 41, 48, 54, 58 conductance, 17 electric, 89 specific, 73 thermal, 81 conductivity, 6, 17, 73, 90, 132, 140, 147 electric, 81 electrical, 44 equivalent, 73 ionic, 54, 73 molar, 6, 54, 73, 90, 132 molecular, 73 thermal, 81, 90 conductivity of an ion molar, 73 conductivity tensor, 43 thermal, 43 conductor, 87 electron, 73 ionic, 73 metallic, 74 conductor element, 143 confidence level of, 154 confidence interval, 154 conjugate transpose, 107 connectivity formula, 51, 52 conservation matrix, 53 constant atomic mass, 22, 47, 111, 117 Avogadro, 4, 9, 45, 47, 53, m, 112 Boltzmann, 36, 45, 46, 64, 81, 111 Coriolis coupling, 26 Coriolis C-, 26, 30 dielectric, 16 effective hamiltonian, 26 electric, 16, 111, 143 Faraday, 70, 111 fine-structure, 22, 95, m, 144, 145^ force, 27 fundamental physical, 9, 95, 9^1*1/112, 133 gravitational, 14 heliocentric gravitational, 94 Henry's law, 58, 59 hyperfine coupling, 27 / Madelung, 43 magnetic, 17, 111, 143 mathematical, 7, 103, 112 Michaelis, 66, 67 microscopic, 87^ molar mass, 47 Planck, 7, 9, 22, 34, 95, 111 radiation, 36, 112 rate, 63-65, 67-69 rotational, 25-27 Rydberg, 22, HI shielding, 29 Stefan-Boltzmann, 35, 112 time, 13 van der Waals, 77 van der Waals-Hamaker, 77 constant of motion, 67 contact angle, 77 contact potential difference, 74 conventions in chemical thermodynamics, conventions in electrochemistry, 73 conversion rate of, 63, 67, 89 conversion factors, 129, 134, 135, 145 energy, 234 pressure, 233 conversion of units, 129 conversion tables for units, 135-142 convolution of funofao!lyl07 coordinate, 42 bending, 27 cartesian (space), 13, 21 cylindrical, 13 dimensionless normal, 27 dimensionless vibrational, 27 electron, 20, 96 fractional, 42 generalized, 13, 14, 45 internal, 27 internal vibrational, 27 ^hass adjusted vibrational, 27 molecule-fixed, 31 normal, 27 space-fixed, 31 spherical polar, 13, 21 stretching, 27 symmetry, 27 symmetry vibrational, 27 vibrational normal, 27 coordinate perpendicular to surface, 80 coordinate representation, 18 core hamiltonian, 20 Coriolis coupling constant, 26 Coriolis (^-constant, 26, 30 correlation time, 29 corrosion nomenclature, 70 coulomb, 89, 134, 139 coulomb field central, 23 coulomb integral, 19, 20 Coulomb law, 146 coulomb operator, 20, 21 Coulomb's modulus, 15 coupling constant Coriolis, 26 dipolar, 28 202 Fermi, 111 hyperfine, 27 nuclear quadrupole, 29 nuclear spin-spin, 28 quadrupole, 29 reduced, 28 reduced nuclear spin-spin, 28 spin-orbit, 25 spin-rotation, 28 coupling path, 28 covariance, 152 coverage factor, 152, 153 Cowling number, 82 critical coagulation concentration, 78 critical micellisation concentration, 78 cross product, 107 cross section, 24, 38 averaged collision, 64, 65 collision, 64 differential, 65 integrated, 39 integrated absorption, 38 integrated net absorption, 37 net absorption, 37 reaction, 65 total, 65 cross-section circular, 87 crystal liquid, 54, 55 crystal axis, 40 crystal face, 44 crystal lattice, 42 crystal lattice symbol, 44 crystalline, 54, 61 crystalline solid, 55 cubic expansion coefficient, 43, 56, 81 cubic metre, 136 cumulative number of states, 45 curie, 139 Curie relation, 133 Curie temperature, 43 curl of a vector field, 107 current, 87, 145 anodic partial, 72 cathodic, 70 cathodic partial, 72 electric, 4, 16, 72, 85, 86, 95, 139, 143, 144 electromagnetic unit of, 135 exchange, 72 faradaic, 72 particle, 65 zero, 71 current density, 72, 147 electric, 16, 18, 72, 90 exchange, 72 probability, 18 thermal, 44 cycles per second, 89 cylindrical coordinate, 13 dalton, 9, 22, 47, 92, 94, 136 damped linear oscillator, 8 day, 92, 137 astronomical, 137 debye, 26, 39, 141 Debye angular frequency, 43 Debye angular wavenumber, 43 Debye cut-off wavelength, 43 Debye frequency, 43 Debye length, 77 characteristic, 78 Debye screening length, 78 Debye temperature, 43, 46, 80 Debye wavenumber, 43 Debye-Hiickel theory, 75 Debye-Waller factor, 42 deca, 91 decadic absorbance, 5, 36, 37 decadic absorption coefficient, 36 decadic logarithm, 63, 98, 99 decay exponential, 67 radioactive, 64 rate of, 36 decay coefficient, 8, 98 decay constant, 24 decay rate constant, 24 decay time, 64 deci, 91 decibel, 98, 99 decimal marker, 103 decimal multiple of units, 6, 85, 91 decimal prefixes, 85 decimal sign, 103 decimal submultiple of units, 6, 85, 91 degeneracy, 26, 39, 45 nuclear spin, 26 vibrational, 26 degenerate electronic state, 31 degenerate modes, 26 degenerate species, 31 degenerate vibrational state, 31 degree, 92, 136 degree Celsius, 89, 138 degree of certainty, 103 degree of dissociation, 49, 98 degree of freedom, 151, 152 degree of ionization, 49 degree of plane angle, 38 degree of reaction, 48 degree Rankine, 138 del operator, 107 203 density, 14, 81, 90 amount, 139 charge, 16, 18, 23, 43, 44, 140 collision, 65 current, 72, 147 electric current, 16, 18, 72, 90 energy, 35, 90 flux, 81 ion, 43 mass, 6, 14, 38, 48, 51, 54, 90, 97 number, 43-45, 48, 81 probability, 18, 151 probability current, 18 relative, 14 surface, 14 density matrix, 46 density matrix element, 21, 46 density of radiation energy, 148 density of states, 43, 45, 66, 67 density of states of transition state, 66 density operator, 46 deposition of species, 77 derived quantity, 4 determinant of a square matrix, 107 deuteron, 50, 115 diameter, 13, 112 collision, 64 dielectric constant, 16 dielectric effects, 40, 41 dielectric medium, 17 dielectric polarization, 16, 40, 146 differential cross section, 65 diffusion rate of, 74 diffusion coefficient, 43, 44, 72, 81, 90 thermal, 81 diffusion layer thickness, 72 diffusion length, 43 diffusivity thermal, 81 dilute solution, 58, 97 dilution infinite, 54, 55, 61 dilution (subscript), 59 dimension, 4 symbol of, 4 dimension one, 36, 82, 9j^|) dimensionless, 13 dimensionless normal coordinate, 27 dimensionless quantity, 5, 11, 18, 97-99 dimensionless vibrational coordinate, 27 dipolar coupling constant, 28 dipolar interaction tensor, 28 dipole, 17, 2014fN magnetic, 147 dipole layer, 70 dipole length, 26, 141 electric, 26 dipole moment, 17, 24, 26, 28, 134 electric, 17, 23, 26, 39, 41, 95, 141_^6 magnetic, 17, 23, 95, 121, 133, 142 molecular, 26 molecular magnetic, 133 dipole moment (molecule) electric, 26 transition, 26 dipole moment operator, 35 dipole moment per volume electric, 16 magnetic, 17 dipole moment vector, 16 dipole transition electric, 35, 39 J dipole vector, 17 -Dirac delta distriJiiiJawi, 106 Dirac delta fundfllln, 1#6 disintegration (rate) constant, 24 disintegration energy, 24 dispersion expected, 149 displacement electric, 16, 141, 146 non-rationalized, 141 vibrational, 27 displacement (subscript), 59 displacement vector, 42 dissipation factor (acoustic), 15 dissociation degree of, 49, 98 dissociation energy, 22, 96 bond, 22 dissociation energy (ground state), 22 dissociation energy (potential minimum), 22 dissolution, 59 distance, 13, 15, 17, 18, 20, 26, 44, 65 equilibrium, 27 ground-state, 27 interatomic, 27 interatomic equilibrium, 27 interatomic (internuclear), 27 mean Earth-Sun, 92 reduced, 95 substitution structure, 27 zero-point average, 27 distance parameter effective, 27 distribution function speed, 45 divergence of a vector field, 107 donor, 44 donor ionization energy, 43 204 dose equivalent, 89, 139 dose equivalent index, 89 dose of radiation absorbed, 89, 139 dose rate absorbed, 90 double harmonic approximation, 41 duration, 13 dyn,134 dynamic friction factor, 15 dynamic viscosity, 15, 90, 138 dyne, 137 effective charge, 21 effective distance parameter, 27 effective hamiltonian, 19 effective hamiltonian constant, 26 effective magnetic flux density, 29 effective mass, 43 effective parameter, 112 effective vibrational hamiltonian, 26 efficiency, 90, 97 effusion rate of, 81 eigenfunction, 21 eigenvalue equation angular momentum, 30 einstein, 68 Einstein coefficient, 35, 37, 39 Einstein temperature, 46 Einstein transition probability, 37, 39 elastic scattering, 65 elasticity modulus of, 15 electric capacitance, 89 electric charge, 4, 16, 50, 70, 89, 134, 135, lWM&y electric charge density, 90 electric conductance, 89 electric conductivity, 81 electric constant, 16, 111, 143 electric current, 4, 16, 72, 85, 86, 95, 139, 143, 144 electric current density, 16, 18, 72, 90 electric dipole length, 26 electric dipole moment, 17, 23, 26, 39, 41, 95, 141, 146 electric dipole moment (molecule), 26 electric dipole moment per volume, 16 electric dipole transition, 35~39J electric displacement, 16Tll^J46 electric field, 24, 95, 104, 146 electric field gradient, 29, 141 electric field gradient tensor, 24 electric field strength, 7, 16, 24, 26, 73, 90, 134, 140, 145-147 electric flux, 16 electric mobility, 73 electric permittivity, 145 electric polarizability, 95 electric polarizability of a molecule, 24 electric polarization, 16 electric potential, 16, 70, 75, 89, 140 inner, 70 outer, 70 surface, 70 electric potential difference, 16, 44, 74 electric quadrupole moment, 95, 141 electric quadrupole moment area, 121 electric resistance, 17, 89, 140 electric susceptibility, 16, 145, 146 electric tension, 16, 89, 140% electrical conductivity, 44 * electrical energy, 73 electricity, 16-17 quantity of, 16 • electrochemical biosensor, 70 electrochemical cel\oSy71, 73-75 electrochemical cell reaction, 52, 74 electrochemical engineering, 70 electrochemical potential, 71 electrochemical reaction, 71, 72, 74, 75 electrochemical transfer coefficient, 72 electrocMmisyy, 70-73 conventions in, 73 electrode, 73 md^fye, 75 positive, 75 electrode potential, 71, 74, 75 absolute, 71 equilibrium, 71, 72 standard, 71, 74 electrode reaction, 71, 72, 74 electrokinetic phenomena, 70 electrokinetic potential, 73 electrolytic cell, 73, 75 electromagnetic force, 143, 144 electromagnetic potential, 147, 148 electromagnetic quantity, 144 electromagnetic radiation, 34-41 intensity of, 35 electromagnetic theory, 143, 146 electromagnetic unit, 129, 144 electromagnetic unit of current, 135 electromagnetic unit system, 143 electromotive force, see cell potential electron, 8, 50, 115 electron affinity, 22 electron antineutrino, 50 electron charge, 18 electron conductor, 73 electron configuration, 8, 32 electron coordinate, 20, 96 electron energy, 44 electron magnetic moment, 23, 111 205 electron mass, 9, 22, 29, 94, 95, 111, 136 electron mean free path, 80 electron neutrino, 50, 115 electron number, 52, 71 electron number (of an electrochemical reaction), 7 electron orbital, 28 electron orbital angular momentum, 30 electron orbital operator, 25 electron orbital plus spin angular momentum, 30 electron paramagnetic resonance, 27 electron spin angular momentum, 30 electron spin multiplicity, 32 electron spin operator, 25, 27 electron spin resonance, 27 electron spin wavefunction, 19 electron velocity, 44 electron wavefunction, 18 electron work function, 43 electronegativity, 22 electronic arrangement, 51 electronic charge, 19 electronic energy, 44 total, 20, 21 electronic energy barrier, 64 electronic potential, 23, 64 electronic term, 25 electronic term symbol, 51 electronic transition, 37 electronvolt, 9, 92, 94, 132, 137 electrostatic force, 143, 144 electrostatic interaction energy, 43 electrostatic potential, 70, 146 electrostatic potential (of a phase), 70 electrostatic unit, 144 electrostatic unit of charge, 135 electrostatic unit system, 143 electroweak charge, 24 element standard atomic weight, 117-120 symbol for chemical, 49, 50, 113, 117-120 elementary charge, 8, 9, 22, 24, 26, 70, 94-96, 111 145 elementary entity, 4, 53, 88, 89 elementary particles symbols for, 8, 22, 50, 113, 115 elementary reaction, 52, 65, 68, 69 elementary step, 52 elements isotopic composition, 117, 122-128 relative atomic mass, 117-120 elongation relative, 15 emf, see electromotive force emission, 33, 38 induced, 35,37-39 spontaneous, 35, 67 stimulated, 35 emissivity, 35 emittance, 35 empirical formula, 51 emu, 135, 139, 141-145 non-rationalized, 133 emu system, 144 emu unit system, 143 energy, 14, 19, 20, 22, 32, 33, 36, 45, 56, 67, 88, 89, 92, 94, 95, 132, 137, 146, 147, 234 acceptor ionization, 43 activation, 64 Arrhenius activation, 64 band gap, 43 bond dissociation, 22 chemical, 73 collision, 66 disintegration, 24 dissociation, 22, 96 donor ionization, 43 electrical, 73 electron, 44 electronic, 44 electrostatic interaction, 43 excitation, 131, 132 Fermi, 43 Gibbs, 6, 56, 74 Hartree, 22, 95, 111 Helmholtz, 56, 78 internal, 55, 56 ionization, 22 kinetic, 5, 14, 93, 94 molar, 90, 129, 234 one-electron orbital, 20 partial molar Gibbs, 6, 57, 71 potential, 14, 23 radiant, 34 reduced, 95 specific, 90 standard Gibbs, 71 standard reaction Gibbs, 58 threshold, 64, 67 total, 65 total electronic, 20, 21 translational collision, 65 energy (dipole) potential, 26 energy barrier electronic, 64 energy conversion factors, 234 energy density, 35, 90 radiant, 34 spectral radiant, 34, 35 energy density of radiation, 148 energy eigenfunction, 45 energy flow 206 rate of radiation, 148 energy level, 38, 39 energy of activation standard Gibbs, 66 standard internal, 65 energy of atomization, 61 energy parameter, 19 reciprocal, 46 energy per area, 35 energy per time radiant, 34 energy tensor nuclear quadrupole interaction, 24 energy unit, 129 energy-related unit, 129 enplethic, 6 enplethic volume, 6 enplethy, 4, 6, 45 ensemble, 46, 67 canonical, 45, 46 grand canonical, 46 microcanonical, 46 enthalpy, 4, 6, 8, 56 molar, 6, 8 standard partial molar, 57 standard reaction, 5, 58 enthalpy of activation standard, 65 enthalpy of combustion standard, 61 enthalpy of formation standard, 55 standard molar, 58 enthalpy of reaction molar, 60 enthalpy of vaporization molar, 60 entitic, 6 entity, 44, 47, 50, 51, 53, 54, 72, 73, 88 elementary, 4, 53, 88, 89 formula, 54 mass of, 47 product, 53 reactant, 53 entropy, 4, 56, 90, 138 equilibrium (maximum), 46 molar, 55, 90, 139 specific, 90 standard partial molar, 57 standard reaction, 58 statistical, 46 entropy change rate of, 63 entropy of activation standard, 65 entropy of formation standard, 61 entropy of reaction molar, 60 entropy per area interfacial, 78 entropy unit, 139 enzyme, 67 enzyme catalysis, 67 enzyme unit, 89 enzyme-substrate complex, 67 EPR, 27 equation stoichiometric, 49, 52, 60, 63, 68, 69 equation of state van der Waals, 57 equations of electromagnetic theory, 143, 146 equilibrium, 52, 58, 71, 74, 75 thermal, 39 equilibrium (maximum) entropy, 46 equilibrium bond length, 96 equilibrium charge partition, 74 equilibrium constant, 58 standard, 58, 61 thermodynamic, 58 equilibrium constant (acid dissociation), 59 equilibrium constant (base hydrolysis), 59 equilibrium constant (concentration basis), 58 equilibrium constant (dissolution), 59 equilibrium constant (fugacity basis), 58 equilibrium constant (molality basis), 58 equilibrium constant (pressure basis), 58 equilibrium constant (water dissociation), 59 equilibrium distance, 27 interatomic, 27 equilibrium electrode potential, 71, 72 equilibrium macroscopic magnetization, 28 equilibrium position vector, 42 equilibrium potential, 71, 72, 74 equilibrium signal intensity, 29 equivalent conductivity, 73 erg, 134, 135, 137, 138, 142, 144 ESR, 27 estimated probability, 154 estimated standard deviation, 152 estimated variance, 152 esu, 134, 135, 139-141, 144, 145 non-rationalized, 133 esu system, 143, 144 esu unit system, 143 etendue, 35, 36 Euler number, 82 evaporation (subscript), 60 exa, 91 exabinary, 91 exbi, 91 excess amount, 77 207 excess quantity (superscript), 60 exchange current, 72 exchange current density, 72 exchange integral, 20 exchange operator, 20, 21 excitation, 50 excitation energy, 131, 132 molar, 132 excitation symbol, 50 excited electronic state, 8, 50 excited state, 32 excited state concentration, 67 excited-state label, 33 expanded uncertainty, 151-153 expansion coefficient cubic, 43, 56, 81 linear, 56 expansivity coefficient, 57 expectation value, 23, 24 expectation value (core hamiltonian), 20 expectation value (operator), 18 expected best estimate, 149 expected dispersion, 149 expected value, 151 exponential decay, 67 exponential function, 106 exposure, 90 extensive, 6 extensive quantity, 6, 57, 81 extensive quantity (divided by area), 81 extensive quantity (divided by volume), 81 extensive thermodynamic quantity, 60 extent of reaction, 48, 53, 60 extinction, 37 extinction coefficient, 37 /-value, 37 Fabry-Perot cavity, 36 face-centred lattice, 44 factor absorption, 36 acoustic, 15 (acoustic) absorption, 15 (acoustic) dissipation, 15 (acoustic) reflection, 15 (acoustic) transmission,!^ atomic scattering, 42 | collision frequency, 65 compressibility, 57 compression, 57 Debye-Waller, 4jL dynamic friction, 15 frequency, 64/* normalization, 18, 20, 21 pre-exponential, 63, 64 quality, 36 (radiation) absorption, 15 reflection, 36 structure, 42 symmetry, 72 transmission, 36 factorial, 105 Fahrenheit temperature, 138 farad, 89 faradaic current, 72 Faraday constant, 70, 111 Faraday induction law, 147 femto, 91 femtometre, 121 fermi, 135 Fermi coupling constant, li^ Fermi energy, 43 Fermi sphere, 44 field, 98 electric, 24, 95,^^146 magnetic, 17,ij^ytl field gradient electric, 29, 141 field gradient tensor, 29 field level, 98, 99 field strength^ ele.fcic^jjl 16, 24, 26, 73, 90, 134, 140, 145-147 magnetic, 17, 90, 142, 147 film, 7^7 film tension, 77 film thickness, 77 »fine^ructure constant, 22, 95, 111, 144, 145 finesse, 36 first-order kinetics generalized, 67 first-order rate constant, 66, 67 first-order reaction, 72 Fischer projection, 51, 52 flow, 81 flow rate, 81 fluence, 35 fluence rate, 35 fluid phase, 54 fluidity, 15 fluorescence, 67 fluorescence rate constant, 66 flux, 72, 81 electric, 16 heat, 81 luminous, 89 magnetic, 17, 89, 142 probability, 18 radiant, 89 reflected, 15 sound energy, 15 transmitted, 15 flux density, 81 effective magnetic, 29 208 heat, 81, 90 magnetic, 17, 28, 81, 89, 95, 141, 145-147 number, 81 particle, 44 radiofrequency (magnetic), 28 static magnetic, 28 flux density of mass, 81 flux of mass, 81 Fock operator, 20, 21 matrix element of, 21 fonts for symbol, 7 foot, 135 force, 14, 87, 89, 95, 104, 134, 137, 138, 144, 146-148 au of, 137 electromagnetic, 143, 144 electromotive, see cell potential electrostatic, 143, 144 infinitesimal electromagnetic, 143 lineic, 144 Lorentz, 148 moment of, 14, 90 thermoelectric, 43, 44 force constant, 27 vibrational, 27 formal potential, 72 formation, 52, 55, 58, 61 formation reaction (subscript), 60 formula, 50, 51 chemical, 49-51 connectivity, 51, 52 empirical, 51 molecular, 51 resonance structure, 51, 52 stereochemical, 51, 52 structural, 51 formula entity, 54 formula matrix, 53 formula symbol, 54 formula unit, 47, 51, 70 Fourier number (mass transfer), 82 fraction amount, 48, 49, 97 amount-of-substance, 48, 97, 121, 133 mass, 47, 49, 97, 98 mole, 48, 59, 62, 81, 121 volume, 48, 97, 98 fractional coordinate, 42 fractional population, 38, 39 fractions, 97 franklin, 134, 135, 139, 144 free path electron mean, 80 mean, 44, 65, 80, 81 free spectral range, 36 frequency, 13, 29, 34-36, 41, 68, 81, 88, 89, 129 angular, 8, 13, 34, 89, 98 circular, 44 collision, 64, 65 Debye, 43 Larmor, 23 resonance, 29 standardized resonance, 29 transition, 25 frequency factor, 64 frequency shift, 87 frequency standard, 87 frequency unit, 25, 26 angular, 28 friction factor dynamic, 15 Froude number, 82 fugacity, 58, 74 fugacity coefficient, 58 function amplitude, 18 atomic-orbital basis, 19, 21 Gibbs, 56 . Hamilton, 14 Heaviside, 45, 67, 107 Helmholtz, 56 Lagrange, 14 Legendre, 18 Massieu, 56 Planck, 56 spherical harmonic, 18 time-dependent, 18 work, 43, 80 fundamental physical constant, 9, 95, 96, 111-112, 133 fundamental translation vector, 42 angular, 42 fundamental vibration, 26 fundamental wavenumber vibrational, 26 fusion (subscript), 60 FWHM, 36, 67 ^-factor, 23, 27, 28, 111 Lande, 23, 27, 28, 111 nuclear, 23 gal, 136, 137 gallon (UK), 136 gallon (US), 136 Galvani potential difference, 71, 72 galvanic cell, 73, 75 gamma, 136 gamma function, 107 gap energy band, 43 gas, 38, 39, 54, 74, 81, 139 ideal, 39, 61, 66 real, 48, 139 van der Waals, 57 209 gas constant, 45 molar, 45, 111 gas mixture, 48, 61, 88 gas phase, 40, 41, 55, 61 gauss, 141, 142 Gauss law, 146 Gaussian, 139, 141 Gaussian probability distribution, 151, 154 Gaussian system, 41, 133, 135, 143, 144 Gaussian-type orbital, 21 Gaussian unit system, 143 general chemical equation, 53 general chemistry, 47-55 general rules for symbol, 5 generalized coordinate, 13, 14, 45 generalized first-order kinetics, 67 generalized momentum, 14, 45 Gibbs dividing surface, 77, 78 Gibbs energy, 6, 56, 74 partial molar, 57, 71 standard, 71 standard reaction, 58 Gibbs energy of activation standard, 66 Gibbs function, 56 Gibbs surface, 77, 78 Gibbs surface amount, 78 gibi, 91 giga, 91 gigabinary, 91 glide operation, 44 gon, 136 gradient, 44 electric field, 29, 141 gradient of a scalar field, 107 gradient tensor, 29 grain, 136 gram, 91, 136 grand canonical ensemble, 46 Grashof number (mass transfer), 82 gravitation Newton constant of, 112 gravitational constant, 14 heliocentric, 92, 94 gray, 89, 139 Greek alphabet, 179 Greek letter, 8, 9, 19, 32, 55** Gregorian year, 137 ground electronic state, 32 ground state, 8, 22, 32, 33, 61 ground-state distance, 27 ground-state label, 33 Griineisen parameter, 43 GTO, 21 gyromagnetic ratio, 23, 27, 29 proton, 112 shielded proton, 112 half cell, 73, 74 half life, 24, 64, 116 Hall oefficient, 43 Hamilton function, 14 hamiltonian, 18, 31 core, 20 effective, 19 effective vibrational, 26 hyperfine coupling, 27 rotational, 25 hamiltonian operator, see hamiltonian hard sphere collision model, 64 hard sphere radius, 64 ^ harmonic (vibrational) wavenumber, 25, 26 harmonic approximation double, 41 Harned cell, 75 , 76'< fi Hartmann number, 82 hartree, 9, 132, 137, 145 Hartree energy, 22, 95, 111 Hartree-Fock equation, 21 Hartree-Fock SCF, 20 Hartree/Fock tiheory, 19 Hartree-Fock-Roothaan SCF, 21 heat, 5/KJ3, 89 heat capacities ratio of, 57 heat capacity, 5, 56, 90, 138 molar, 90, 139 specific, 90 heat capacity (constant pressure), 5, 6, 8, 56 specific, 6, 81 heat capacity (constant volume), 8, 43, 55, 56 ' heat flux, 81 heat flux density, 81, 90 heat flux vector, 44 heat transfer coefficient of, 81 Heaviside function, 45, 67, 107 hectare, 136 hecto, 91 hectopascal, 48 height, 13 heliocentric gravitational constant, 92, 94 helion, 50, 115 helion magnetic moment shielded, 116 Helmholtz energy, 56, 78 Helmholtz function, 56 henry, 89 Henry's law, 59 Henry's law constant, 58, 59 Hermann-Mauguin symbol, 31, 44 hermitian conjugate, 19, 107 hertz, 6, 88, 89 210 high energy, 34 high pressure limit, 66 HMO, see Hiickel (molecular orbital) theory horse power imperial, 138 metric, 138 hot-band transition, 41 hour, 92, 137 Hiickel (molecular orbital) theory, 19 Hiickel secular determinant, 19 hydrogen-like atom, 23 hydrogen-like wavefunction, 18 hyper-polarizability, 17, 24, 95 hyper-susceptibility, 16 hyperbolic function, 106 hyperfme coupling constant, 27 hyperfme coupling hamiltonian, 27 hyperfme level, 87 ideal (superscript), 60 ideal gas, 39, 61, 66 molar volume of, 111 identity operator, 31 IEC, 6, 91, 92, 136 illuminance, 89 imaginary refractive index, 37, 40 immersion (subscript), 60 impact parameter, 65 impedance, 17, 70, 98 characteristic, 111 complex, 17 imperial horse power, 138 inch, 135, 138 incident intensity, 38 incoming particle, 50 indirect mechanism, 68 indirect spin coupling tensor, 28 individual collision, 65 induced emission, 35, 37-39 induced radiative transition, 38 inductance, 17, 89, 142 mutual, 17 self-, 17, 142, 147 induction magnetic, 17, 27 inertial defect, 25 inexact differential, 56 infinite dilution, 54, 55, 61 infinite dilution (superscnp|\^(D infinitesimal electromagnetic force, 143 infinitesimal variation, 8 infrared spectra, 40 \ inner electric pota^ial, W inner potential, 71 inner potential difference, 71 integral coulomb ,\^J0 exchange, 20 inter-electron repulsion, 20 one-electron, 20, 21 overlap, 19 resonance, 19 two-electron, 21 two-electron repulsion, 19, 20 integrated absorption coefficient, 37, 39-41 integrated absorption cross section, 38 integrated cross section, 39 integrated net absorption cross section, 37 integration element, 18 intensity, 34-36, 38 absorption, 38-40 band, 26, 38-40 incident, 38 line, 26, 38-40 • luminous, 4, 34, 85, 86, 88 photon, 34 radiant, 34, 35, 88 signal, 29 J spectral, 35, 36 transmitted, 38 intensity of electromagnetic radiation, 35 intensive, 6, 81 interacHgrnmergy electrostatic, 43 interaction tensor dipolar, 28 interatomic equilibrium distance, 27 interatomic (internuclear) distance, 27 inter-electron repulsion integral, 20 interface, 71 Jnterfacial entropy per area, 78 interfacial layer, 77, 78 interfacial tension, 77, 78 internal absorptance, 36 internal coordinate, 27 internal energy, 55, 56 internal energy of activation standard, 65 internal energy of atomization standard, 61 internal vibrational angular momentum, 30 internal vibrational coordinate, 27 international calorie, 137 international ohm mean, 140 US, 140 international protype of the kilogram, 87 International System of Units (SI), 85 International temperature, 56 international volt mean, 140 US, 140 intrinsic number density, 44 211 inverse of a square matrix, 107 inversion operation, 31, 44 inversion operator, 31 ion, 50 ion charge, 73 ion density, 43 ionic activity mean, 70 ionic activity coefficient mean, 70 ionic charge number, 49 ionic concentration mean, 70 ionic conductivity, 54, 73 ionic conductor, 73 ionic molality mean, 70 ionic strength, 59, 70, 76 ionic strength (concentration basis), 59, 70 ionic strength (molality basis), 59, 70 ionization degree of, 49 ionization energy, 22 acceptor, 43 donor, 43 ionization potential, 22 irradiance, 34, 35, 90 spectral, 35 irrational (ir), see non-rationalized isentropic compressibility, 56 ISO, 6, 11, 13, 14, 16, 17, 22, 25, 34, 42, 45, 47, 56, 58, 81, 92, 97, 136 ISO 1000, 7 ISO 31, 7, 48 ISO/IEC 80000, 7 ISO/TC 12, xi isobaric nuclide, 49 isobars, 49 isoelectric point, 78 isothermal compressibility, 43, 56 isotope, 49, 117, 120 isotope mixture, 88 isotope-isomer, 52 isotopic abundance, 121 isotopic composition, 51, 120 ' isotopic composition of the elements, 117, 122-128 isotopic nuclide, 49 isotopologues, 51 isotopomers, 51 isotropic medium, 16 ISQ, 4, 16, 143 italic fonts, 7 ITS, 56 IUPAC, ix, xi, 4, 6, 11, 18, 25, 29, 34, 48, 58, 62, 63, 77, 117 IUPAP, xi, 11, 13, 14, 16, 18, 22, 25, 34, 42, 45, 47, 56, 81 joule, 6, 89, 93, 137 Joule-Thomson coefficient, 57 Julian year, 136, 137 ^-doubling, 31 kat, 89 katal, 89 kelvin, 56, 85, 86, 87, 89, 138 i kerma, 89 kibi, 91 kilo, 91 kilobinary, 91 kilodalton, 94 kilogram, 4, 85, 86, 87, 88, 91, 136, 143 kilogram-force, 137 # kilometre, 91 kilopond, 137 kilotonne, 92 kilovolt, 6 kinematic viscosity, 15, 90, 138 kinetic energy, 5, 14, 93, 94 kinetic energy operator, 18 kinetic radiative transition rate, 40 kinetics, 33, 50, 63 chemical, 63, 68 Knudsen number, 82 Kronecker delta, 106 ^-doubling, 31 label for symmetry species, 31 Lagrange function, 14 Lagrangian, 45 rlambda, 136 A-doubling, 31 Lande g-factor, 23, 27, 28, 111 langmuir, 81 Laplacian operator, 107 Larmor angular frequency, 23 Larmor circular frequency, 23 laser cavity, 36 laser physics, 34 Latin letter, 32 lattice, 79 lattice constant, 42 lattice direction, 44 lattice plane spacing, 42 lattice symbol crystal, 44 lattice vector, 42 angular reciprocal, 42 Bravais, 42 reciprocal, 42, 80 lattice wave, 43 LCAO-MO theory, 21 Legendre function, 18 212 length, 3, 4, 6, 13, 20, 42, 45, 53, 81, 85-87, 92, 94, 95, 131, 135, 136, 143-145, 147 absorbing path, 37 characteristic Debye, 78 Debye, 77 Debye screening, 78 diffusion, 43 dipole, 26, 141 electric dipole, 26 equilibrium bond, 96 path, 13, 36, 38 reciprocal unit cell, 42 unit cell, 42 length of arc, 13 level, 45 energy, 38, 39 field, 98, 99 hyperfine, 87 A-doubled, 31 macroscopic, 49, 51, 52 microscopic, 49, 50, 52 orbital energy, 19 power, 98, 99 quantum, 46 rotational, 30 level of confidence, 151, 154 level of field quantity, 92 level of power quantity, 92 level width, 24 Levi-Civita symbol, 106 Lewis number, 82 lifetime, 24, 44, 64, 121 average, 64 natural, 66 light, 35, 87 speed of, 9, 13, 16, 25, 34, 43, 95, wavelength of, 38 light gathering power, 35, 36 light second, 136 light source, 38 light transmitted, 37 light year, 136 line intensity, 26, 38-40 line strength, 38, 39 line width, 36 line width at half maximun/fsfw linear combination of functions, 21 linear combinations of basis functions, 21 linear decadic absorption coefficient, 36 linear expansion coefficient,Ho linear molecule, 30-32 linear momentum, 95 linear napierian absorption coefficient, 36 linear strain, 15 linear top, 301 lineic, 6 y v lineic force, 144 lineshape, 36 linewidth, 36, 67 natural, 66 predissociation, 66 liquid, 16, 27, 38, 40, 41, 54, 73 liquid crystal, 54, 55 liquid junction, 73 liquid junction potential, 72, 74,'T*L liquid phase, 41, 55 liquid state, 61 litre, 6, 92, 136 litre atmosphere, 137 logarithm decadic, 63, 98, 99* napierian, 98 logarithmic function* lOtP"^ logarithmic quantities, 99 logarithmic quantity, 98 logarithmic ratio, 92 logical operator, 107 longitudinal cavity mode, 36 longitudinal relaxation time, 23 longitudinal spin-lattice relaxation time, 29 Lorentzfebrce, 1148 Lorentz gauge, 148 Lorentz local field, 40 Lorentz-Lorenz formula, 40 Lorenz coefficient, 43 Loschmidt constant, see Avogadro constant loss angle, 17 lumen, 89 ^yminescence, 36, 37 luminous flux, 89 'luminous intensity, 4, 34, 85, 86, 88 luminous quantity, 34 lux, 89 Mach number, 82 macroscopic level, 49, 51, 52 macroscopic magnetization equilibrium, 28 Madelung constant, 43 magnetic constant, 17, 111, 143 magnetic dipole, 147 magnetic dipole moment, 17, 23, 95, 121, 133, 142 atomic unit of, 95 molecular, 133 magnetic dipole moment per volume, 17 magnetic field, 17, 29, 141 see magnetic flux density, 17 magnetic field strength, 17, 90, 142, 147 magnetic flux, 17, 89, 142 magnetic flux density, 17, 28, 81, 89, 95,141,145-147 effective, 29 radiofrequency, 28 static, 28 213 magnetic induction, 17, 27 magnetic moment, 23, 115, 116 electron, 23, 111 nuclear, 29, 122-128 proton, 112 shielded helion, 116 shielded proton, 112, 116 magnetic permeability, 145 magnetic quantum number, 30 magnetic Reynolds number, 82 magnetic susceptibility, 17, 133, 142, 145, 147 molar, 17, 133, 142 magnetic vector potential, 17, 146, 147 magnetism, 16-17 magnetizability, 23, 95, 142 magnetization, 17, 28, 147 magnetization (volume), 142 magnetogyric ratio, see gyromagnetic ratio magneton Bohr, 23, 29, 95, 111, 115, 133, 142 nuclear, 23, 29, 111, 121, 142 mass, 4, 6, 7, 14, 18, 20, 28, 45, 48, 77, 81, 85-89, 92-96, 111, 115,116,133,135,136,143,144 atomic, 22, 47, 121 average molar, 77 centre of, 31 effective, 43 electron, 9, 22, 29, 94, 95, m, 136 mean molar, 47 molar, 47, 54, 117, 133 molecular, 51 neutron, 111 proton, 29, 111 reduced, 8, 14, 64, 65, 95 relative atomic, 47, 117-120 relative molar, 47 relative molecular, 47 SI base unit of, 91 mass adjusted vibrational coordinate, 27 mass concentration, 38, 48, 97 mass constant atomic, 22, 47, 111, 117 molar, 47, 117 mass density, 6, 14, 38, 48, 51, 54, 90, 97 mass distribution, 14 mass excess, 22 mass fraction, 47, 49, 97, 98 mass number, 22, 49, 50, 120, 121 mass of atom, 22, 47 mass of entity, 47 mass transfer Fourier number for, 82 Grashof number for, 82 Nusselt number for, 82 Peclet number for, 82 Stanton number for, 82 steady-state, 72 mass transfer coefficient, 72, 81 mass unit unified atomic, 9, 22, 47, 92, 94, 111, 116, 117, 121, 136 massic, 6 massic volume, 6 Massieu function, 56 massless, 70 mathematical constant, 7, 103, 112 mathematical function, 103, 105 mathematical operator, 7, 105 symbol of, 7, 8 mathematical symbol, 7, 8, 103-108 printing of, 103, 104 matrix, 5, 7, 79, 104, 107 complex conjugate, 107 conservation, 53 density, 46 determinant of a square, 107 formula, 53 inverse of square, 107 rate (coefficient), 66 scattering, 65 stoichiometric number, 53 trace of a square, 107 transpose of, 107 unit, 107 unitary, 65 matrix element, 19, 21, 67, 79, 107 density, 21, 46 Fock operator, 21 overlap, 21 matrix element of operator, 18 matrix equation, 67 matrix for superlattice notation, 80 matrix notation, 33, 79 matrix notation (chemical equation), 53 matrix quantity, 65 maxwell, 142 Maxwell equations, 144, 145, 147 Maxwell relation, 143, 144 Mayan year, 137 mean, 151, 152 standard deviation of the, 151 mean Earth-Sun distance, 92 mean free path, 44, 65, 80, 81 mean international ohm, 140 mean international volt, 140 mean ionic activity, 70 mean ionic activity coefficient, 70 mean ionic concentration, 70 mean ionic molality, 70 mean life, 24, 64, 115, 116 mean molar mass, 47 mean molar volume, 47 214 mean relative speed, 64 mixing angle mean value, 105 weak, 24,111 measurand, 153 mixing of fluids (subscript), 60 measure of uncertainty, 153 mixture, 8, 46, 59 measured quantity, 151 gas, 48, 61, 88 measurement equation, 153 isotope, 88 mebi, 91 mobility, 43 mechanics electric, 73 classical, 14-15 mobility ratio, 43 quantum, 18-21 modulus statistical, 66 bulk, 15 mechanism compression, 15 catalytic reaction, 68 Coulomb's, 15 chain-reaction, 68 shear, 15 complex, 68 Young's, 15 composite, 68 modulus of elasticity, 15 indirect, 68 modulus of the Poynting vector, 35 stepwise, 68 molal solution, 48 mega, 91 molality, 48, 49, 59, 62, 70, 75, 133 megabinary, 91 mean ionic, 70 megadalton, 94 standard, 61, 62, 75 megaelectronvolt, 92, 116 molality basis, 75 megahertz, 6 molar, 6, 47, 48 melting (subscript), 60 partial, 57 metallic conductor, 74 molar absorption coefficient, 6 metre, 3, 6, 53, 85, 86, 87, 135, 143 molar conductivity, 6, 54, 73, 90, 132 cubic, 136 molar conductivity of an ion, 73 square, 136 molar decadic absorption coefficient, 36, 37 metric horse power, 138 molar energy, 90, 129, 234 metric pound, 136 molar enthalpy, 6, 8 metric tonne, 136 standard partial, 57 Michaelis constant, 66, 67 molar enthalpy of formation micro, 91 standard, 58 microcanonical ensemble, 46 molar enthalpy of reaction, 60 microfacet, 79 molar enthalpy of vaporization, 60 microgram, 8 molar entropy, 55, 90, 139 micromolar, 48 standard partial, 57 micron, 135 molar entropy of reaction, 60 microscopic constant, 87 molar excitation energy, 132 microscopic level, 49, 50, 52 molar gas constant, 45, 111 microscopic quantity, 17 molar Gibbs energy migration velocity, 73 partial, 6, 57, 71 mile, 135 molar heat capacity, 90, 139 nautical, 92, 135 molar magnetic susceptibility, 17, 133, 142 Miller indices, 44, 79, 80 molar mass, 47, 54, 117, 133 Miller-index vector, 79 average, 77 milli, 91 mean, 47 millibar, 48 relative, 47 millilitre, 92 molar mass constant, 47, 117 millimetre of mercury, 138 molar napierian absorption coefficient, 36 millimicron, 135 molar optical rotatory power, 37, 38 millimolar, 48 molar polarizability, 17, 41 million complex, 41 part per, 98 molar quantity, 6, 56 minute, 92, 137 partial, 6, 57 minute (plane angle), 92, 136 molar refraction, 37 215 molar refractivity, 37 molar susceptibility non-rationalized, 142 molar volume, 5, 6, 17, 41, 47, 55, 57, 90, 139 mean, 47 partial, 57 molar volume of ideal gas, 111 molarity, 48 mole, 4, 6, 43, 53, 59, 85, 86, 88, 98 mole fraction, 48, 59, 62, 81, 121 mole fraction (gas), 48 molecular conductivity (of equivalent entities), 73 molecular dipole moment, 26 molecular formula, 51 molecular magnetic dipole moment, 133 molecular mass, 51 relative, 47 molecular momentum vector, 45 molecular orbital, 19-21 occupied, 21 molecular partition function (per volume), 66 molecular partition function (transition state), 66 molecular point group, 32 molecular position vector, 45 molecular quadrupole moment, 24 molecular quantity, 37 molecular spectroscopy, 14 molecular spin orbital, 20 molecular state, 32 molecular velocity vector, 45 molecular weight, 47 molecularity, 68 molecule-fixed coordinate, 31 moment magnetic, 23, 115, 116 transition, 39 moment of force, 14, 90 moment of inertia, 14 principal, 25 momentum, 14, 45 angular, 14, 18, 20, 23, 28, 30, 67,^^21, 138 generalized, 14, 45 linear, 95 spin angular, 19, 28, 116 momentum operator, 18 momentum transfer, 42, 112 momentum vector molecular, 45 monomeric form, 54 monomolecular reaction, 68 MS scheme, 111 muon, 8, 50, 115 muonium, 50, 116 mutual inductance, 17 myon neutrino, 115 nabla operator, 18, 107 4* name chemical-compound, 8 name of a physical quantity, 3, 5, 53 nano, 91 nanometre, 3, 6 napierian absorbance, 36, 37 napierian absorption coefficient, 36, 38 napierian logarithm, 98 natural lifetime, 66 natural linewidth, 66 nautical mile, 92, 135 Neel temperature, 43 negative electrode, 75 nematic phase, 54 neper, 8, 92, 98, 99 Nernst equation, 72, 75 Nernst potential, 72» , Nernst slope, 76 net absorption, 38 net absorption coefficient, 37 net absorption cross section, 37 net forward reaction, 52 net integrated absorption band intensity, 39 net reaction, 68 neutrino, 115 electron, 50, 115 electron anti, 50 iVpOTnj.15 tau, 115 neutron, 8, 23, 49, 50, 115 scattering of, 42 ^^ftron mass, 111 neutron number, 22, 24 newton, 87, 89, 137 Newton constant of gravitation, 112 NMR, 28, 29 nomenclature, 9 biochemical, 9 inorganic, 9 macromolecular, 9 organic, 9 non-aqueous system, 74 non-rationalized, 145 non-rationalized displacement, 141 non-rationalized emu, 133 non-rationalized esu, 133 non-rationalized molar susceptibility, 142 non-rationalized quantity, 145 non-rationalized system, 133 non-rationalized unit, 129, 145 non-SI unit, 129, 135 accepted, 92 normal coordinate, 27 dimensionless, 27 normal distribution, 154 normal mode, 33 216 normal probability distribution, 151, 153 normal stress, 15 normalization factor, 18, 20, 21 normalized wavefunction, 18 nuclear chemistry, 34, 89, 116 nuclear g-factor, 23 nuclear interaction weak, 24 nuclear magnetic moment, 29, 122-128 nuclear magnetic resonance, 28 nuclear magneton, 23, 29, 111, 121, 142 nuclear orbital angular momentum (rotational), 30 nuclear Overhauser effect, 29 nuclear Overhauser enhancement, 29 nuclear physics, 22, 116 nuclear quadrupole area, 26 nuclear quadrupole coupling constant, 29 nuclear quadrupole interaction energy tensor, 24 nuclear quadrupole moment, 24, 28 nuclear reaction, 50 nuclear spin, 29 nuclear spin angular momentum, 30 nuclear spin degeneracy, 26 nuclear spin operator, 28 nuclear spin quantum number, 121 nuclear spin-spin coupling, 28 nuclear spin-spin coupling constant reduced, 28 nucleon number, 22, 49 nuclide, 49, 50, 113, 121 isobaric, 49 isotopic, 49 nuclides properties of, 121-128 null space, 53 number, 8, 13, 24, 63 Alfen, 82 atomic, 22, 23, 49, 50, 117, 121 characteristic, 5 charge, 20, 49-51, 70, 71, 73 collision, 65 complex, 107 Cowling, 82 electron, 52, 71 Euler, 82 Fourier, 82 Froude, 82 Grashof, 82 Hartmann, 82 ionic charge, 49 Knudsen, 82 Lewis, 82 Mach, 82 magnetic Reynolds, 82 mass, 22, 49, 50, 120, 121 neutron, 22, 24 nucleon, 22, 49 Nusselt, 82 oxidation, 50 Peclet, 82 Prandtl, 82 proton, 20, 22, 24, 49, 121 quantum, 21, 27, 28, 30, 32, 33/*, tty 103 Rayleigh, 82 Reynolds, 5, 82 Schmidt, 82 Sherwood, 82 Stanton, 82 stoichiometric, 48, 49, 52, 53, 60, 61, 71, 72, 74 Strouhal, 82 symmetry, 46 transport, 73 transport (characteristic), 82 Weber, 82 number concentration, 43, 45, 48, 63, 64 number density, jj&S, 48, 81 intrinsic, 44 specific, 44 number flux density, 81 number matrix stoichiometric, 53 number of adsorbed molecules, 78 number of anions, 70 number of atoms, 42, 50, 53 number of cations, 70 number of collisions total, 65 number of defined events, 67 number of electrons, 19, 20, 71, 72 number of entities, 6, 45, 47, 89 ''number of measurements, 151 number of moles, 4, 53 number of open adiabatic reaction channels, 66 number of particles, 68 number of photons absorbed, 67 number of protons, 121 number of quantum states, 45 number of states, 44, 66, 67 cumulative, 45 number of states (transition state), 66, 67 number of vibrational modes, 44 numbers printing of, 103 numerical value of a physical quantity, 3, 133 Nusselt number (mass transfer), 82 nutation angular frequency, 28 oblate top, 25 occupied molecular orbital, 21 oersted, 142 ohm, 89, 140 mean international, 140 US international, 140 217 one-electron integral, 20, 21 one-electron orbital, 32 one-electron orbital energy, 20 one-electron wavefunction, 44 on-shell scheme, 111, 112 operator, 18, 96, 103 angular momentum, 19, 28, 30 coulomb, 20, 21 del, 107 density, 46 dipole moment, 35 electron orbital, 25 electron spin, 25, 27 exchange, 20,21 expectation value of, 18 Fock, 20, 21 hamiltonian, 18 hermitian conjugate of, 19 identity, 31 inversion, 31 kinetic energy, 18 Laplacian, 107 logical, 107 mathematical, 7, 8, 105 matrix element of, 18 momentum, 18 nabla, 18, 107 nuclear spin, 28 parity, 31 permutation, 31 permutation-inversion, 31 reflection, 31 rotation, 31 rotation-reflection, 31 space-fixed inversion, 31 symmetry, 31 vector, 30 operator inverse, 106 operator symbol, 30 operator symbol (reaction in general), 58 optical rotation, 38 angle of, 37, 38 optical rotatory power molar, 37, 38 specific, 37, 38 optical spectroscopy, 32, 36 orbital Gaussian-type, 21 molecular, 19-21 molecular spin, 20 Slater-type, 21 orbital angular momentum, 32 orbital energy one-electron, 20 orbital energy level, 19 order bond, 19,21 charge, 19 order (Legendre function), 18 order of reaction, 66, 72 overall, 63 partial, 63, 64 order of reflection, 42 order parameter, 42 oscillator damped linear, 8 oscillator strength, 37 osmole, 59 osmotic coefficient, 59 osmotic coefficient (molality basis), 59 osmotic coefficient (mole fraction basis), 59 osmotic pressure, 59 ounce, 136 ounce (avoirdupois), 136 ounce (troy), 136 outer electric potential, 70 outer potential difference, 70 outgoing particle, 50 overall order of reaction, 63 Overhauser effect nuclear, 29 Overhauser enhancement nuclear, 29 overlap integral, 19 overlap matrix element, 21 overpotential, 72 oxidation number, 50 oxidation rate constant, 72 parity, 31, 32, 121 parity operator, 31 parity violation, 31 parsec, 136 part per billion, 98 part per hundred, 98 part per hundred million, 98 part per million, 98 part per quadrillion, 98 part per thousand, 98 part per trillion, 98 partial molar, 57 partial molar Gibbs energy, 6, 57, 71 partial molar quantity, 6, 57 partial molar volume, 57 partial order of reaction, 63, 64 partial pressure, 5, 38, 48, 63 partial reaction order, 64 particle, 42, 45, 54, 65, 88, 115 a-, 8, 50, 115 P-, 50, 115 incoming, 50 outgoing, 50 particle current, 65 218 particle current per area, 65 particle flux density, 44 particle position vector, 42 particle spin quantum number, 115 particle symbol, 8, 22, 50, 113, 115 particle trajectory, 65 particles properties of, 115-116 partition, 31 partition function, 39, 45, 46, 67 partition function (per volume), 67 partition function (transition state) molecular, 66 pascal, 89, 138 path length, 13, 36, 38 PDG, 111, 113, 115 pebi, 91 Peclet number (mass transfer), 82 Peltier coefficient, 43 per mille, 98 percent, 97, 121 period, 13 periodic, 13 permeability, 8, 17, 81, 90 magnetic, 145 relative, 5, 17, 145, 147 permeability of vacuum, see magnetic constant permille, 97, 98 permittivity, 16, 41, 90 electric, 145 relative, 16, 40, 145, 146 relative complex, 40 permittivity of vacuum, see electric constant permutation, 31 permutation group, 31 permutation inversion, 31 permutation operator, 31 permutation-inversion operator, 31 peta, 91 petabinary, 91 pH, 5, 70, 75, 76 definition of, 70, 75 measurement of, 75, 76 quantity, 75 symbol, 70, 75 pH standards, 75 phase condensed, 40, 41, 48, 54, 58 fluid, 54 nematic, 54 phase of current, 17 phase of potential difference, 17 phase space volume in, 45 phonon, 44 photo-electrochemical energy conversion, 70 photochemical yield, 66 photochemistry, 34, 35, 63-67 photodissociation, 31 photon, 50, 67, 88, 115 photon absorption rate of, 67 photon intensity, 34 photon quantity, 34 physical constant fundamental, 9, 95, 96, 111-112, 133 physical quantity, 3, 4, 8, 85, 92-94, 96, 131, 135, 140, 143 dimensionless, 97 extensive, 81 intensive, 81 name of, 3, 5, 53 numerical value of, 3, 133 product of, 7 quotient of, 7 SI unit for, 85 symbol of, 5, 7-9, 11, 70, 97, 131 table of, 11 unit of, 3, 18, 19 value of, 3, 129 physical quantity as unit, 94 physical quantity in SI unit, 143 pico, 91 picomole, 98 pion, 115 planar conjugated hydrocarbons, 19 Planck constant, 7, 9, 22, 34, 95, 111 Planck constant divided by 2n, 22, 34, 111, 145 Planck function, 56 plane angle, 13, 38, 89, 92, 136 plane angle (degree), 92 plane angle (minute), 92, 136 plane angle (second), 92, 136 plane of polarization, 38 plane of rotation, 13 plasma physics, 22 point charge, 17 point group, 103 point of zero charge, 78 poise, 138 polar coordinate spherical, 13, 21 polarizability, 24, 134, 135, 140, 141 complex molar, 41 electric, 95 molar, 17, 41 polarizability of a molecule electric, 24 polarizability volume, 141 polarization, 141 dielectric, 16, 40, 146 electric, 16 219 volume, 141 Polo-Wilson equation, 41 polymeric form, 54 polytropy, 45 population average, 46 position vector, 13 molecular, 45 particle, 42 positive electrode, 75 positron, 50, 115 positronium, 116 potential cell, 16, 71, 89 chemical, 5, 6, 8, 44, 46, 57, 62 electric, 16, 70, 75, 89, 140 electrochemical, 71 electrode, 71, 74, 75 electrokinetic, 73 electromagnetic, 147, 148 electronic, 23, 64 electrostatic, 70, 146 equilibrium, 71, 72, 74 formal, 72 inner, 71 ionization, 22 liquid junction, 72, 74, 76 Nernst, 72 reversible, 72 standard, 74, 75 standard cell, 74 standard chemical, 57, 61, 62 surface, 70 C-, 73 potential (of cell reaction) standard, 71, 74 potential barrier, 94 potential difference, 94 electric, 16, 44, 74 Galvani, 71, 72 inner, 71 outer, 70 Volta, 70 potential difference (electrochemical cell), 74 potential energy, 14, 23 potential energy (dipole), 26 potential energy difference, 14 potential of adiabatic channel, 67 potential of electrochemical cell reaction, 74 pound,136 pound (avoirdupois), 136 pound (metric), 136 pounds per square inch, 138 power, 14, 22, 98, 138 light gathering, 35, 36 radiant, 34 resolving, 36 thermal, 57, 81 power amplifier, 99 power level, 98, 99 signal, 98 sound,99 power per area radiation, 35 power transmitted, 36 Poynting vector, 17, 35, 148 ppb, 98 pph, 98 pphm, 98 ppm, 97, 98 ppq, 98 ppt, 98 Prandtl number, 82 pre-exponential factor, 63, 64 predissociation linewidth, 66 predissociation rate constant, 67 prefix (symbol), 91 pressure, 6, 14, 39, 48, 58, 81, 89, 131, 138 atmospheric, 58 osmotic, 59 partial, 5, 38, 48, 63 rate of change of, 8 reference, 99 standard, 48, 61, 62, 74 standard state, 62 surface, 77 total, 48 vapor, 60, 131 pressure change rate of, 63 pressure coefficient, 56 relative, 56 pressure conversion factors, 233 pressure dependence (of the activity), 58 pressure dependence (unimolecular reaction), 64 pressure level sound, 99 pressure unit, 129 pressure virial coefficient, 57 primary standard (pH), 75, 76 primary symmetry axis, 32 primitive lattice, 44 principal component, 29 principal moment of inertia, 25 principal quantum number, 23 printing of mathematical symbols, 103, 104 printing of numbers, 103 probability, 45, 46, 65, 153, 154 average, 46 Einstein transition, 37, 39 estimated, 154 transition, 65 220 probability current density, 18 probability density, 18, 151 probability distribution, 151, 154 Gaussian, 151, 154 normal, 151, 153 rectangular, 151, 153 Student-t, 151, 154 triangular, 151 probability flux, 18 process, 60 transport, 63 process symbol, 59 product entity, 53 product of unit, 7 prolate top, 25 propagation of uncertainty, 153 propagation vector, 43 properties of chemical elements, 117-120 properties of nuclides, 121-128 properties of particles, 115-116 proton, 8, 23, 29, 49, 50, 115 proton charge, 22, 70 proton charge (au), 139 proton gyromagnetic ratio, 112 shielded, 112 proton magnetic moment, 112 shielded, 112, 116 proton mass, 29, 111 proton number, 20, 22, 24, 49, 121 pulsatance, 34 pure phase, 38, 59, 61 pure solvent, 132 pure state, 46 pure substance, 56, 61 pure substance (superscript), 60 quadrupole area, 141 quadrupole coupling constant, 29 quadrupole interaction energy tensor, 24 . nuclear, 24 quadrupole moment, 121-128 electric, 95, 141 molecular, 24 nuclear, 24, 28 quadrupole moment (of a molecule), 23 quadrupole moment (of a nucleus), 24 quality factor, 36 quanta, 50 quantity base, 4, 16, 85, 86 derived, 4 dimensionless, \, 1)^18, 97-99 electromagnetic, 144 extensive, 6, 57, 81 extensive thermodynamic, 60 luminous! 34^ molar, 6,%^^ partial molar, 6, 57 photon, 34 physical, see physical quantity reduced, 96 specific, 6, 56 standard, 58 tensor, 14, 15, 28, 104 quantity calculus, 3, 129, 131-133/* quantity of dimension one, 97 quantity of electricity, 16 quantum chemistry, 18-21, 94 quantum level, 46 quantum mechanics, 18-21 quantum number, 21, 27, 28, 30, 32, 33, 51, 67, 103 angular momentum, 23, 26, 30, 67 azimuthal, 30 component, 30 • magnetic, 30 nuclear spin, 121 particle spin^^^S principal, 23 vibrational, 26, 33 quantum yield, 66, 67 quasi-equilibrium theory, 67 quotient of unit, 7 absort rad (absorbed dose of radiation), 139 radian, 8, 13, 89, 98,136 radiance, 34-36 radiant energy, 34 radiant energy density, 34 spectral, 34, 35 radiant energy per time, 34 radiant excitance, 34, 35 rtadiant exposure, 35 radiant flux, 89 radiant intensity, 34, 35, 88 radiant power, 34 radiation constant, 36, 112 radiation energy flow rate of, 148 radiation power per area, 35 radiative transition rate kinetic, 40 radical, 50 radioactive activity, 139 radioactive decay, 64 radiofrequency (magnetic) flux density, 28 radius, 13, 70 Bohr, 9, 22, 95, m hard sphere, 64 rank of tensor, 16 Rankine temperature, 138 Raoult's law, 59 rate, 81 rate (coefficient) matrix, 66 rate coefficient, see rate constant 221 rate constant, 63-65, 67-69 first-order, 66, 67 fluorescence, 66, 67 oxidation, 72 predissociation, 67 reduction, 72 specific, 66 rate constant of fluorescence, 67 rate constant of unimolecular reaction (high pressure), 64 rate constant of unimolecular reaction (low pressure), 64 rate equation, 67 rate law, 63, 68, 69 rate of change, 35 rate of change of pressure, 8 rate of change of quantity, 63 rate of concentration change, 63 rate of conversion, 63, 67, 89 rate of decay, 36 rate of diffusion, 74 rate of effusion, 81 rate of entropy change, 63 rate of photon absorption, 67 rate of pressure change, 63 rate of radiation energy flow, 148 rate of reaction (amount concentration), 63 rate of reaction (number concentration), 63 ratio circumference to diameter, 112 ratio of heat capacities, 57 rational, 145 rationalized, 145 rationalized unit, 145 Rayleigh number, 82 reactance, 17 reactant entity, 53 reactants zero-point level, 67 reaction, 52 bimolecular, 68, 69 electrochemical, 71, 72, 74, 75^ monomolecular, 68 nuclear, 50 symbol for, 58 termolecular, 68 trimolecular, 68, 69 unimolecular, 68, 69 reaction cross section, 65 reaction enthalpy, 69 standard, 5, 58 reaction entropy standard, 58 * reaction equation, 52, 61, 63, 69 reaction in general (subscript), 58, 60 reaction order, 63, 66, 72 partial, 64 4* reaction quantity standard, 60 reaction quotient, 58, 61 reaction rate, 68, 69 real gas, 48, 139 reciprocal energy parameter, 46 reciprocal lattice vector, 42, 80 reciprocal substrate basis vector, 80 reciprocal superlattice basis vector>§0 i reciprocal unit cell angle, 42 reciprocal unit cell length, 42 reciprocal wavelength, 25, 90 rectangular probability distribution, 151, 153 reduced adsorption, 78 reduced angular momentum, 95 reduced charge, 95 reduced coupling constant, 28 reduced distance, 95 reduced energy, 95 reduced limiting sedimentation coefficient, 78 reduced mass, 8, 14, 64, 65, 95 reduced nuclear spin-spin coupling constant, 28 reduced quantity, 96 reduced surface excess concentration, 78 reductiqarate/constant, 72 reference pressure, 99 reference state, 61 reference state (of an element), 62 reflectance, 36 reflected flux, 15 reflection, 36, 37 reflection factor, 36 reflection factor (acoustic), 15 reflection loss, 36 reflection operation, 44 reflection operator, 31 refraction, 41 molar, 37 refraction index, 34 refractive index, 34, 40, 41 complex, 37, 40 imaginary, 37, 40 refractivity molar, 37 relative activity, 57, 71 relative adsorption, 78 relative atomic mass, 47, 117 relative atomic mass of the elements, 117-120 relative combined standard uncertainty, 152 relative density, 14 relative elongation, 15 relative molar mass, 47 relative molecular mass, 47 relative permeability, 5, 17, 145, 147 relative permittivity, 16, 40, 145, 146 complex, 40 222 relative pressure coefficient, 56 relative speed mean, 64 relative standard uncertainty, 152 relative surface excess concentration, 78 relative uncertainty, 97 relative value, 97 relaxation, 28 relaxation time, 13, 23, 29, 43, 64 longitudinal, 23 longitudinal spin-lattice, 29 spin-lattice, 29 spin-spin, 29 transverse, 23 (transverse) spin-spin, 29 rem, 139 repetency, 25, 90, 129, 131 repulsion integral inter-electron, 20 two-electron, 19, 20 residual resistivity, 43 resistance electric, 17, 89, 140 thermal, 81 resistivity, 17 residual, 43 resistivity tensor, 43 resolution, 36 resolving power, 36 resonance, 29 electron paramagnetic, 27 electron spin, 27 nuclear magnetic, 28 resonance frequency, 29 standardized, 29 resonance integral, 19 resonance structure, 51, 52 response factor, 76 retarded van der Waals constant, 77 reversible potential, 72 Reynolds number, 5, 82 magnetic, 82 rhombohedral lattice, 44 Roman fonts, 7 röntgen, 139 röntgen equivalent man, see rem rotation specific, 38 rotation of a vector field, 107 rotation operation, 44 rotation operator, 31 rotation-reflection operator, 31 rotational band, 33 rotational constant, 25-27 rotational constant (in frequency), 25 rotational constant (in wavenumber), 25 rotational hamiltonian, 25 rotational level, 30 rotational state, 30, 51 rotational temperature, 46 rotational term, 25 rotational transition, 33 rotatory power molar optical, 37, 38 specific, 38 specific optical, 37, 38 RRKM, 67 rydberg, 137 Rydberg constant, 22, 111 SACM, 67 salt bridge, 74 scalar product, 107 scattering, 36, 37 elastic, 65 scattering amplitude, 42 scattering angle, 65 scattering factor atomic, 42 scattering matrix, 65 scattering of neutron, 42 SCF, 20, 21 Schmidt number, 82 Schonflies symbol, 31 screw operation, 44 second, 6, 85, 86, 87, 92, 98, 136, 137, 143, 144 light, 136 second (of arc), 92, 136 second virial coefficient, 57 second-rank tensor, 5, 16, 17 secondary standard (pH), 76 sedimentation velocity of, 78 sedimentation coefficient, 77, 78 reduced limiting, 78 self-consistent field theory, 20, 21 self-inductance, 17, 142, 147 semiconductor electrochemistry, 70 semiconductor space-charge theory, 78 SHE, 71, 74 shear modulus, 15 shear strain, 15 shear stress, 15 shear stress tensor, 81 shear viscosity, 81, 90 Sherwood number, 82 shielded helion magnetic moment, 116 shielded proton gyromagnetic ratio, 112 shielded proton magnetic moment, 112, 116 shielding constant, 29 shielding parameter, 21 shielding tensor, 29 SI, 4, 85 223 SI base unit, 4, 85-88, 90, 91 symbol for, 86 SI derived unit, 85, 89-91 SI prefix, 85, 91, 92, 94 SI unit, 96 SI unit system, 143 Siemens, 89 sievert, 89, 139 signal amplitude level, 98 signal intensity, 29 signal power level, 98 signal transmission, 98 single-crystal surface, 79 Slater determinant, 20 Slater-type orbital, 21 solid, 27, 28, 38, 40, 54, 79 amorphous, 54, 55 crystalline, 55 solid absorbent, 77 solid angle, 13, 34, 36, 65, 89 solid phase, 55, 59, 61 solid state, 42-44, 61 solubility, 48, 49 solubility product, 59 solute, 40, 48, 59, 61, 62, 133 solution, 38, 40, 48, 52, 54, 57, 61, 72, 74, 75, 132, 133 aqueous, 54, 55, 57, 74, 97 dilute, 58, 97 molal, 48 solution (subscript), 60 solution at infinite dilution aquaeous, 54, 55 solvent, 29, 37, 48, 59, 61, 133 pure, 132 solvent mixture, 76 sound speed of, 13, 81 sound energy flux, 15 sound power level, 99 sound pressure level, 99 space and time, 13 space charge, 78 space coordinate cartesian, 13, 21 space-fixed cartesian axis, 35, 39 space-fixed component, 30, 35 space-fixed coordinate, 31 space-fixed inversion operator, 31 space-fixed origin, 31 specific, 6 specific conductance, 73 specific energy, 90 specific entropy, 90 specific heat capacity, 90 specific heat capacity (constant pressure), 6, 81 specific number density, 44 specific optical rotatory power, 37, 38 specific quantity, 6, 56 specific rate constant, 66 specific rotation, 38 specific rotatory power, 38 specific surface area, 77 specific volume, 6, 14, 90 spectral absorption, 38 spectral density of vibrational modes, 43 spectral intensity, 35, 36 spectral irradiance, 35 spectral line, 37 spectral radiant energy density, 34, 35 spectroscopic absorption intensity, 40 spectroscopic transition, 33 spectroscopy, 25-33, 36 atomic, 14 atomic absorption, 97 molecular, 14 optical, 32, 36 spin-resonance, 23 spectrum absorption, 36 speed, 13, 81, 90, 93, 95 average, 45, 81 mean relative, 64 speed distribution function, 45 speed of light, 9, 13, 16, 25, 34, 43, 95, m, 143-145 speed of sound, 13, 81 spherical harmonic function, 18 spherical polar coordinate, 13, 21 spherical symmetry, 145 spherical top, 30 spin nuclear, 29 spin angular momentum, 19, 28, 116 spin-lattice relaxation time, 29 longitudinal, 29 spin multiplicity, 32 electron, 32 spin operator electron, 25, 27 nuclear, 28 spin-orbit coupling constant, 25 spin quantum number nuclear, 121 spin-resonance spectroscopy, 23 spin-rotation coupling constant, 28 spin-rotation interaction tensor, 28 spin-spin coupling constant nuclear, 28 reduced nuclear, 28 spin-spin relaxation time, 29 spin statistical weight, 45 spin wavefunction, 19, 29 224 electron, 19 spontaneous emission, 35, 67 square metre, 136 staggered arrangement, 51 standard atomic, 137 standard (pH) primary, 75, 76 secondary, 76 standard acceleration of free fall, 137 standard atmosphere, 62, 111, 138 standard atomic weight, 47, 117-120 standard buffer, 76 standard cell potential, 74 standard chemical potential, 57, 61, 62 standard concentration, 62, 72 standard condition, 61, 71 standard deviation, 103, 151 standard deviation of the mean, 151 standard electrode potential, 71, 74 standard enthalpy of activation, 65 standard enthalpy of combustion, 61 standard enthalpy of formation, 55 standard entropy of activation, 65 standard entropy of formation, 61 standard equilibrium constant, 58, 61 standard Gibbs energy, 71 standard Gibbs energy of activation, 66 standard hydrogen electrode (SHE), 71, 74 standard internal energy of activation, 65 standard (internal) energy of atomization, 61 standard molality, 61, 62, 75 standard molar enthalpy of formation, 58 standard partial molar enthalpy, 57 standard partial molar entropy, 57 standard potential, 74, 75 standard potential (of cell reaction), 71, 74 standard pressure, 48, 61, 62, 74 standard quantity, 58 standard reaction enthalpy, 5, 58 standard reaction entropy, 58 standard reaction Gibbs energy, 58\ standard reaction quantity, 60 standard state, 58, 61, 62, 66, 71 biochemical, 62 choice of, 58 standard state (gas phase), 61 standard state (liquid or solid state), 61 standard state pressure, 62 standard state (solute), 61 standard (symbol), 57, 60, 66, 71 standard uncertainty, 103, 151, 152 combined, 152 relative, 152 relative combined, 152 standardized resonance frequency, 29 Stanton number (mass transfer), 82 statcoulomb, 144 state function, see wavefunction state of aggregation, 52, 54 static magnetic flux density, 28 stationary state, 18 statistical entropy, 46 statistical mechanics, 66 statistical thermodynamics, 45-46^ statistical weight, 26, 45 spin, 45 steady-state mass transfer, 72 Stefan-Boltzmann constant, 35, 112 stepped surface, 79 stepwise mechanism, 68 steradian, 13, 88, 89 stereochemical formula, 51, 52 stimulated emission, 35 STO, 21 Stockholm convention, 73 stoichiometric chemical equation, 60, 68, 69 stoichiometric equation, 49, 52, 60, 63, 68, 69 stoichiometric number, 48, 49, 52, 53, 60, 61, 71, 72, 74 stoichiometric number matrix, 53 stoichiometric reaction equation, 60 stoichiometry, 69 stokes, 138 strain bulk, 15 linear, 15 shear, 15 volume, 15 stress, 89 normal, 15 shear, 15 stress tensor shear, 81 stretch-bend interaction, 27 stretching coordinate, 27 Strouhal number, 82 structural arrangement, 51 structural formula, 51 structure factor, 42 Student-t probability distribution, 151, 154 sublimation (subscript), 60 substance concentration, 4, 48 substitution structure distance, 27 substrate basis vector, 79, 80 sum of states, 66 sum over states, see partition function superlattice basis vector, 80 surface, 79 single-crystal, 79 stepped, 79 surface amount, 77 225 surface area, 49, 71, 77, 78 specific, 77 surface basis vector, 79 surface charge density, 6, 71 surface chemistry, 77 surface concentration, 48, 63, 77 surface coverage, 77, 80 surface density, 14 surface density of charge, 16 surface electric potential, 70 surface excess, 77, 78 surface excess amount, 77, 78 surface excess concentration, 77 surface layer, 77 surface potential, 70 surface pressure, 77 surface structure, 79-80 surface tension, 14, 56, 77, 78, 81, 90 susceptance, 17 susceptibility electric, 16, 145, 146 magnetic, 17, 133, 142, 145, 147 molar magnetic, 17, 133, 142 svedberg, 137 symbol fonts for, 7 general rules for, 5 Hermann-Mauguin, 31, 44 mathematical, 7, 8, 103-108 Schonflies, 31 symbol of angular momentum operator, 30 atomic unit, 94-96 base quantity, 4 chemical element, 8, 9, 49, 50, 113, 117-120 chemical thermodynamics, 56-62 crystal lattice, 44 decimal (sub)multiple, 91 dimension, 4 directions in crystal, 44 excitation, 50 mathematical operator, 7, 8 particle, 8, 22, 50, 113, 115 physical or chemical process, 59-61 physical quantity, 5, 7-9, 11, 70, 97, 131 planes in crystals, 44 prefix, 91 reaction, 58, 60 SI base unit, 86 SI derived unit, 89-91 standard state, 57, 60, 66, 71 surface structure, 80 symmetry operator, 31 symmetry species, 103 term, 32 unit, 3, 4-6, 83, 92 symbolic notation (chemical equation), 53 symmetric group, 31 symmetric reduction, 25 symmetric top, 30, 33 oblate, 30 prolate, 30 symmetry, 8, 16, 24, 31 permutation-inversion, 31 spherical, 145 symmetry axis, 30 symmetry coordinate, 27 symmetry factor, 72 symmetry number, 46 symmetry operation, 31, 44 symmetry operator, 31 symmetry species, 32 label for, 31 symbol for, 103 system Gaussian, 41, 133, 135, 143, 144 system of atomic units, 145 Tafel slope, 72 tau neutrino, 115 tebi, 91 telecommunications, 98 temperature, 6, 35, 38, 39, 41, 56, 61-64, 87, 129, 133 Celsius, 38, 46, 56, 89, 111, 138 centigrade, 56 characteristic, 46 characteristic (Weiss), 43 Curie, 43 Debye, 43, 46, 80 Einstein, 46 Fahrenheit, 138 International, 56 Neel, 43 Rankine, 138 rotational, 46 thermodynamic, 4, 56, 81, 85-87, 89, 138 vibrational, 46 Weiss, 43 temperature dependence (unimolecular reaction), 64 tension electric, 16, 89, 140 film, 77 interfacial, 77, 78 surface, 14, 56, 77, 78, 81, 90 tensor, 5, 7, 8, 17, 23, 24, 28, 44 conductivity, 43 electric field gradient, 24 field gradient, 29 gradient, 29 indirect spin coupling, 28 quadrupole interaction energy, 24 rank of, 16 226 resistivity, 43 second-rank, 5, 16, 17, 104 shear stress, 81 shielding, 29 spin-rotation interaction, 28 thermal conductivity, 43 tensor quantity, 14, 15, 28, 104 tera, 91 terabinary, 91 term, 34 electronic, 25 rotational, 25 total, 25 vibrational, 25 term formula vibrational, 26 term symbol, 32, 117-120 electronic, 51 term symbol (atomic state), 32 term symbol (molecular state), 32 term value, 25 termolecular reaction, 68 tesla, 89, 141 thermal averaging, 42 thermal conductance, 81 thermal conductivity, 81, 90 thermal conductivity tensor, 43 thermal current density, 44 thermal diffusion coefficient, 81 thermal diffusivity, 81 thermal equilibrium, 39 thermal expansion coefficient, 57 thermal power, 57, 81 thermal resistance, 81 thermochemical calorie, 58, 137 thermodynamic equilibrium constant, 58 thermodynamic temperature, 4, 56, 81, 85-87, 89, 138 thermodynamics chemical, 56-62 statistical, 45-46 thermoelectric force, 43, 44 thickness, 13, 77 diffusion layer, 72 film, 77 thickness of layer, 77 third virial coefficient, 57 Thomson coefficient, 43 three-body collision, 65™ threshold energy, 64, 67 throughput, 35 time, 4, 8, 13, 81*A5-8V92, 95, 98, 135, 137, 143, 144 au of, 137 correlation, 29 decay, 64 1 V longitudinal relaxation, 23 relaxation, 13, 23, 29, 43, 64 transverse relaxation, 23 time constant, 13 time-dependent function, 18 time dimension (NMR), 29 time integral, 14 time integral of fluence rate, 35 time integral of irradiance, 35 time interval, 35, 64 characteristic, 13 tonne, 92 tonne (metric), 136 top asymmetric, 30, 31, 33 linear, 30 oblate, 25 prolate, 25 spherical, 30 / symmetric, 30, 33 torque, 14 Torr, 81, 131, f!^J33 torr, 131, 13^~ total angular momentum, 32 total cross section, 65 total electronic energy, 20, 21 total energy, 65 total excess, 78 total pressure, 48 »tüta^purface excess concentration, 77 total term, 25 total wavefunction, 20 trace of a square matrix, 107 transfer coefficient anodic, 72 cathodic, 72 electrochemical, 72 mass, 72, 81 transition, 33, 38, 39, 41 electronic, 37 hot-band, 41 rotational, 33 spectroscopic, 33 vibrational, 33 vibronic, 33 transition between phases (subscript), 60 transition dipole moment (molecule), 26 transition frequency, 25 transition moment, 39 transition probability, 65 Einstein, 37, 39 transition rate kinetic radiative, 40 transition state, 64 density of states, 66 (molecular) partition function, 66 227 number of states, 66, 67 partition function, 67 zero-point level, 67 transition state (superscript), 60 transition state theory, 65-67 transition wavelength, 37 transition wavenumber, 25 translation vector (angular) fundamental, 42 translational collision energy, 65 transmission, 40 transmission coefficient, 66 transmission factor, 36 transmission factor (acoustic), 15 transmittance, 36 transmitted flux, 15 transmitted intensity, 38 transport (characteristic) number, 82 transport number, 73 transport process, 63 transport properties, 81-82 transpose of matrix, 107 transverse relaxation time, 23 (transverse) spin-spin relaxation time, 29 triangular probability distribution, 151 trigonometric function, 105 trillion, 98 trimolecular reaction, 68, 69 triple point (subscript), 60 triton, 50, 115, 116 tropical year, 137 troy, 136 two-electron integral, 21 two-electron repulsion integral, 19, 20 type A evaluation, 152 type B evaluation, 152, 153 uncertainty, 75, 76, 103, 149-154 expanded, 151-153 relative, 97 standard, 103, 151, 152 thermodynamic data, 62 uncertainty budget, 149, 152 uncorrelated measurements, 153>* unified atomic mass unit, 9, 22, 47, 92, 94, 111, 116, 117, 121, 136 unimolecular reaction, 68, 69 Unit, see enzyme unit unit angular frequency, 28 astronomical, 9,JB2, 94, a36 atomic, 4,18, 20^4, 26, 94, 95, 96, 129,132, 135, 143^M5 base, 34, 85-89, 93, 135, 143, 144 CGS, 96, l&5£hy choice of, 3 coherent, 93 decimal multiple of, 6, 85, 91 decimal submultiple of, 6, 85, 91 electromagnetic, 129, 144 electrostatic, 144 energy, 129 energy-related, 129 entropy, 139 enzyme, 89 frequency, 25, 26 non-rationalized, 129, 145 pressure, 129 product of, 7 quotient of, 7 rationalized, 145 SI base, 4, 85-88, 9^^^! SI derived, 85, 89-91 symbol for, 3, b-&S$& unit cell angle, 42 unit cell length, 42,' unit matrix, 107 l unit of physical quantity, 3, 18, 19 unit step function, 107 unit system / electromagnetic, 143 electrostatic, 143 emu, 143 esuy^3 Gaussian, 143 SI, 143 unit vector, 13 JUi^^y matrix, 65 US international ohm, 140 international volt, 140 Valence electrons, 52 value of physical quantity, 3, 129 van der Waals coefficient, 57 van der Waals constant, 77 retarded, 77 van der Waals equation of state, 57 van der Waals gas, 57 van der Waals-Hamaker constant, 77 vapor, 54 vapor pressure, 60, 131 vaporization (subscript), 60 variance, 151 vector, 5, 7, 13, 16, 104, 107 dipole, 17 dipole moment, 16 fundamental translation, 42 lattice, 42 Miller-index, 79 position, 13 Poynting, 17, 35, 148 substrate basis, 79, 80 superlattice basis, 80 surface basis, 79 228 unit, 13 vector operator, 30 vector potential magnetic, 17, 146, 147 vector product, 107 velocity, 13, 90, 148 angular, 13, 89, 90 electron, 44 migration, 73 velocity distribution function, 45 velocity of sedimentation, 78 velocity vector molecular, 45 vibration fundamental, 26 vibrational angular momentum internal, 30 vibrational anharmonicity constant, 25 vibrational coordinate dimensionless, 27 internal, 27 mass adjusted, 27 symmetry, 27 vibrational degeneracy, 26 vibrational displacement, 27 vibrational force constant, 27 vibrational fundamental wavenumber, 26 vibrational ground state, 26 vibrational hamiltonian effective, 26 vibrational modes, 25 number of, 44 spectral density of, 43 vibrational normal coordinate, 27 vibrational quantum number, 26, 33 vibrational state, 33, 51 vibrational symmetry coordinate, 27 vibrational temperature, 46 vibrational term, 25 vibrational term formula, 26 vibrational transition, 33 vibronic transition, 33 virial coefficient, 57 viscosity, 15, 81 bulk, 81 dynamic, 15, 90, 138 , kinematic, 15, 90, 138 shear, 81, 90 vitreous substance, 54 volt, 89, 94, 134, 140 mean international, 140 US international, 140 Volta potential difference, 70 volume, 6, 13, 63, 81, 90, 92, 136 enplethict 6 ^ massic, o\ mean molar, 47 molar, 5, 6, 17, 41, 47, 55, 57, 90, 139 ^ molar (ideal gas), 111 partial molar, 57 polarizability, 141 specific, 6, 14, 90 volume fraction, 48, 97, 98 volume in phase space, 45 volume magnetization, 142 volume of activation, 65 volume polarization, 141 volume strain, 15 volumic, 6 W-boson, 111, 115 Wang asymmetry parameter, 25 watt, 88, 89, 138 wave equations, 148 J wave vector angular, 43 , wavefunction, 18 electron, 18 hydrogen-like, 18 normalized, 18 one-electron, 44 spin, 19, 29 total, 20 wave]^^L 3, 34-36, 42, 44, 131 Debye cut-off, 43 reciprocal, 25, 90 transition, 37 wavelength of light, 38 wavenumber, 25, 27, 34-36, 39, 41, 90, 129, 131 Debye, 43 harmonic (vibrational), 25, 26 transition, 25 vibrational fundamental, 26 wavenumber (in a medium), 34 wavenumber (in vacuum), 34 weak mixing angle, 24, 111 weak nuclear interaction, 24 weber, 89, 142 Weber number, 82 weight, 14 atomic, 47, 117-120 molecular, 47 spin statistical, 45 statistical, 26, 45 Weinberg parameter, 111 Weiss temperature, 43 Wood label (notation), 79 work, 14, 56, 70, 89 work function, 43, 80 electron, 43 work function change, 80 x unit, 135 229 yard, 135 year, 24, 137 Gregorian, 137 Julian, 136, 137 light, 136 Mayan, 137 tropical, 137 yield, 97 photochemical, 66 quantum, 66, 67 yobi, 91 yocto, 91 yotta, 91 yottabinary, 91 Young's modulus, 15 Z-boson, 111, 115 zebi, 91 zepto, 91 zero current, 71 zero matrix, 53 zero-point average distance, 27 zero-point level (reactants), 67 zero-point level (transition state), 67 ^-potential, 73 zetta, 91 zettabinary, 91 230 231 232 PRESSURE CONVERSION FACTORS o psi Pa kPa bar atm Torr psi 1 Pa = 1 = 10"3 = 10~5 ps 9.869 23 x 10" -6 ps 7.500 62xl0~3 ^ 1.450 38 x io-4 1 kPa = 103 = 1 = io-2 ps 9.869 23 x 10- -3 ps 7.500 62 ' 0.145 038 1 bar = 105 = 102 = 1 ps 0.986 923 ps 750.062 ps 14.5038 1 atm = 101 325 = 101.325 = 1.013 25 = 1 = 760 ps 14.6959 1 Torr ps 133.322 ps 0.133 322 ps 1.333 22> <10~3 ps 1.315 79x io- -3 = 1 ps 1.933 68 x io-2 1 psi ps 6894.76 ps 6.894 76 ps 6.894 76 >