REVIEW
Confusing Quantitative Descriptions of BrønstedÀLowry AcidÀBase
Equilibria in Chemistry Textbooks – A Critical Review and Clarifications for
Chemical Educators
by Erich C. Meistera
), Martin Willekeb
), Werner Angstc
), Antonio Tognia
), and Peter Walde*b
)
a
) Department of Chemistry and Applied Biosciences (D-CHAB), ETH Zürich, Wolfgang-Pauli-Strasse
10, CH-8093 Zürich
b
) Department of Materials (D-MATL), ETH Zürich, Wolfgang-Pauli-Strasse 10, CH-8093 Zürich
(e-mail: peter.walde@mat.ethz.ch)
c
) Department of Environmental Systems Science (D-USYS), ETH Zürich, Universitätstrasse 16,
CH-8092 Zürich
In chemistry textbooks, the pKa,H2O value of water in the solvent water at 258C is sometimes given as
14.0, sometimes as 15.7. This is confusing. The particular chemical reaction considered is the one in which
water as BrønstedÀLowry acid reacts with water as BrønstedÀLowry base in water as solvent to yield
equal concentrations of hydrated oxonium and hydroxide ions, H3Oþ
(aq) and HOÀ
(aq), respectively.
This reaction is also known as the self-ionization of water for which the equilibrium constant is
abbreviated as Kw with its known value of 10À14.0
at 258C, i.e., pKw(258C) ¼ 14.0. Identical values for
pKa,H2O and pKw at a fixed temperature appear reasonable, since Ka,H2O and Kw refer to one and the same
reaction. Therefore, reasons for the apparent disagreement between the thermodynamically correct pKa
value for water (14.0 at 258C) and the value reported in most organic chemistry textbooks (15.7) should
be discussed when teaching acidÀbase chemistry. There are good arguments for introducing, from the
very beginning, the concepts of activity and thermodynamic standard states when teaching quantitative
aspects of chemical equilibria. This also explains in a straightforward way why all thermodynamic
equilibrium constants, including Kw, are dimensionless, and why pKH3Oþ(258C) ¼ 0.
1. Introduction. – The treatment of chemical reaction equilibria – in particular the
equilibrium that exists if a BrønstedÀLowry acid reacts with water as a base, or if a
BrønstedÀLowry base reacts with water as an acid – is part of all general chemistry
textbooks that are used, e.g., at universities for first-year students of chemistry, biology,
and other related disciplines (Fig. 1). The reasons for this are obvious: i) the
description and understanding of chemical reaction equilibria is an essential part of the
fundament on which chemistry or biology as scientific disciplines are built; and ii) the
treatment of BrønstedÀLowry acidÀbase reactions in aqueous solution is a particularly
important application of the general concepts of chemical equilibria which are usually
first outlined in a chemistry course for reactions taking place in the gas phase, i.e., in the
absence of any solvent.
Helvetica Chimica Acta – Vol. 97 (2014) 1
 2014 Verlag Helvetica Chimica Acta AG, Zürich
Since many years, there exists an apparent inconsistency between the pKa value of
H2O as BrønstedÀLowry acid in water as solvent, if tabulated values reported in some
of the general-chemistry textbooks are compared with the values given in organicchemistry
textbooks. Arbitrarily chosen examples of general-chemistry textbooks
report that pKa,H2O(258C) ¼ 14.0 (see [4 – 11]). Apart from notable exceptions [12],
organic-chemistry textbooks generally report a pKa value for H2O at 258C of 15.7 or
15.74, (see, e.g., [13 – 22]). This is confusing, and even more so, if in one and the same
textbook both values are given, a value of 14.0 in the chapter on Acid and Base
Strength and a value of 15.7 in the chapter Structure and Reactivity: Organic Acids and
Bases [23].
The aim of this review is to clarify this dissatisfying apparent discrepancy which was
pointed out and discussed previously in a number of articles and personal statements
published in chemical education journals [3][24 – 39]. In only a few textbooks, e.g.,
[40][41], the authors make the critical readers aware of the existence of different pKa
values for water: a thermodynamically meaningful value (pKa,H2O(258C) ¼ 14.0) which
is fully compatible with the standard Gibbs free energy for the reaction (see below in
Sects. 3 and 4 (Fig. 2), and a value (15.7 or 15.74) which originally was calculated by
taking into account the value of Kw(258C) and the molar concentration of water; see
Appendix). This latter approach has its roots in those years when the BrønstedÀLowry
acidÀbase concepts were applied to carbon acids and to other weakly acidic organic
molecules with low water solubility, on the estimation of their acidity, and the
concomitant development of relative acidity scales [43 – 46].
Helvetica Chimica Acta – Vol. 97 (2014)2
Fig. 1. According to Johannes Nicolaus Brønsted (1879 – 1947; left) and Thomas Martin Lowry (1874 –
1936; right), an acid is a molecule that is able to donate (release) a proton, and a base is a molecule
that is able to accept (bind) a proton [1][2]. This is the definition of a BrønstedÀLowry acid and a
BrønstedÀLowry base (see [3]). Photographs from Encyclopedia Britannica Online, accessed February
22, 2013, http://www.britannica.com (Brønsted); and from Trans. Faraday Soc. 1936, 32, 1657 (Lowry).
One of the reasons for the different reported pKa values are the different
conventions used when defining Ka, and the different standard states used for the
solvent water in its liquid state, H2O(l). They are either i) pure water (thermodynamically
correct, i.e., compatible with tabulated thermodynamic data), which is consistent
with pKa,H2O(258C) ¼ 14.0, or ii) one mole water per liter total volume (1m). Similarly, a
straightforward thermodynamically correct treatment of H3Oþ
(aq) as BrønstedÀ
Lowry acid in water yields pKa,H3Oþ ¼ 0, while in many organic chemistry textbooks a
value of À 1.7 or À 1.74 is given, which again is not compatible with tabulated
thermodynamic data.
An additional inconsistency and confusion currently exists about the dimension of
the equilibrium constant Kw as reported in chemistry textbooks, irrespective whether
the textbook is on general or organic chemistry. The thermodynamically correct
constant Kw(258C) ¼ 10À14.0
¼ Ka,H2O(258C) is dimensionless, as all thermodynamic
equilibrium constants, including Ka for any type of BrønstedÀLowry acid. This is
immediately obvious if one applies the straightforward thermodynamic conventions. It
is difficult to understand why these well-elaborated thermodynamic conventions are
Helvetica Chimica Acta – Vol. 97 (2014) 3
Fig. 2. The work of Josiah Willard Gibbs (1839 – 1903) was fundamental for the development of chemical
thermodynamics and for a large part of physical chemistry. One of the most important publications of
Gibbs was [42]. Photograph from C. S. Hastings, Biographical memoir of Josiah Willard Gibbs 1839 –
1903, Natl. Acad. Sci. U.S.A., Biogr. Mem. 1909, 6, 373 – 393.
not used in contemporary chemistry textbooks, or why reference to the thermodynamic
conventions is only made in side remarks, often in such a way that they appear dubious
and more irritating than useful (see Appendix).
In this contribution, we describe different conventions used in chemistry textbooks
for the quantitative thermodynamic description of BrønstedÀLowry acidÀ
base reaction equilibria. Particularly, we outline the clear and convincing advantages
of introducing and applying thermodynamically correct conventions from the very
beginning when teaching this important topic of general chemistry.
For some of the readers, we may be occasionally a bit too trivial, for which we
apologize. However, we always try to be clear, dealing with arguments that are
straightforward, always based on scientifically reasonable grounds, and hopefully easy
to understand. With this, we try to contribute to an improved and consistent teaching of
BrønstedÀLowry acidÀbase equilibria. We are convinced that clarifications in textbooks
would help avoiding all the on-going confusion and unnecessary discussions that
exist in this area of chemistry education since several decades. It is time to reconsider
and to rewrite and improve certain chapters in chemistry textbooks, as it would be for
the benefit of those students who are interested in chemistry and for the benefit of
chemistry as scientific discipline as a whole.
2. The Classical Example: Acetic Acid Dissolved in Water as Solvent. – Before
discussing the particular case in which water reacts as BrønstedÀLowry acid with water
as BrønstedÀLowry base, we will discuss in detail the behaviour of acetic acid in water,
the classical example which is frequently used in chemistry textbooks to outline the
BrønstedÀLowry acidÀbase concepts. This example refers to the reaction that occurs if
a small amount of acetic acid is added to water, typically 3.0 g (50 mmol) acetic acid
dissolved in a total volume of 1 l at an assumed – and usually not explicitly mentioned –
pressure of 1 bar or 1 atm. If the acetic acid molecules (CH3COOH) come into contact
with the water molecules (H2O), a reaction between the acetic acid molecules (a
BrønstedÀLowry acid) and the water molecules takes place in such a way that the water
molecules act as BrønstedÀLowry base. This results in a net transfer of a proton (Hþ
)
from a small part of the CH3COOH molecules to some of the H2O molecules, so that
acetate ions (CH3COOÀ
) and an equal amount of oxonium ions form (H3Oþ
; also
called hydronium ions). The net proton transfer process is very fast and often described
with Lewis formula (Figs. 3 and 4)1).
Helvetica Chimica Acta – Vol. 97 (2014)4
1) Curved arrows are often used to indicate the formal electron flow, the movement of an electron
pair bound to a H2O molecule to a H-atom bound to a CH3COOH molecule, as shown in Fig. 4,a,
on the left hand side. With this curved arrow convention [13][18][19][52 – 56], also called arrow
pushing [55][57] or pushing electrons convention [46], the formal attack of the acid by the base is
illustrated. The convention is that the curved arrow begins where the electrons are originally
localized, at the nucleophilic, electron-rich part of the base (H2O for the forward reaction), and
points towards the electrophilic center of the acid (the H-atom of the carboxy group for the forward
reaction). While a new OÀH bond is formed to yield H3Oþ
, the existing OÀH bond in CH3COOH is
cleaved, again indicated with a curved arrow. The curved arrows on the right hand side of the
reaction in Fig. 4,a, show the formal electron flux for the backward reaction (from right to left).
Using arrows in opposite direction, i.e., from the proton of the BrønstedÀLowry acid to the
BrønstedÀLowry base (see for example [9][58 – 60]), is not recommended as it does not agree with
In the reaction considered in Fig. 4,a, the H2O molecules are assumed to be present
in large excess with respect to the CH3COOH molecules and with respect to the formed
CH3COOÀ
and H3Oþ
ions, i.e., water is at the same time BrønstedÀLowry base as well
as solvent. In the role of solvent, the H2O molecules hydrate all the dissolved molecules,
CH3COOH, CH3COOÀ
, and H3Oþ
, forming an aqueous shell around them. This is
usually emphasized by writing CH3COOH(aq), CH3COOÀ
(aq), and H3Oþ
(aq),
indicating that the species actually are dissolved in water and do not form a separate
phase. If we consider a closed system without any material exchange with the
environment, at any time at a temperature between 0 and 1008C, the vast majority of
the H2O molecules in this dilute solution is present as liquid water. This means that, at
258C, water is a liquid, indicated as H2O(l), i.e., the standard state of water is liquid.
The molar concentration of H2O(l) is cH2O ¼ 55.33m at 258C (with density rH2O ¼
0.9970 g/cm3
and molar mass MH2O ¼ 18.02 g/mol). Since the concentrations of the
Helvetica Chimica Acta – Vol. 97 (2014) 5
Fig. 3. In addition to his contributions to chemical thermodynamics
and the general concept of activities [47][48],
Gilbert Newton Lewis (1875 – 1946) introduced a very useful
convention for representing chemical structures, in particular
organic molecules, whereby pairs of dots indicate electron pairs.
Usually pairs of electrons forming a covalent bond are drawn
as line, and lone pairs (non-bonding) are either drawn as pairs
of dots or as lines [49 – 51]. See also [50][51]. Photograph
from http://www.msu.edu, accessed February 22, 2013.
the mentioned curved arrow formalism which is usually used for describing organic reaction
mechanisms. Another generally accepted convention is that the reaction equilibrium is represented
with two half arrows (Fig. 4,a). In contrast, a single arrow with two heads is used to indicate
existence of mesomeric structures (also called resonance structures; Fig. 4,b). The two reasonable
mesomeric structures of the CH3COOÀ
ion shown in Fig. 4,b, are of equal energy, and, therefore,
contribute equally to the real situation. The curved arrow convention is sometimes also used to
indicate how one obtains from one mesomeric structure another one [41][55][61], as indicated in
Fig. 4,b. This may help students to understand how the different mesomeric structures can be
obtained formally.
dissolved species are negligibly small, the concentration of H2O(l) in the reaction
mixture is always almost the same as the concentration of water in pure water2).
The chemical reaction occurring in water between CH3COOH(aq) and H2O(l) and
the resulting equilibrium situation is usually represented as shown in Eqn. 1:
CH3COOH(aq) þ H2O(l) > CH3COOÀ
(aq) þ H3Oþ
(aq) (1)
Another way of describing exactly the same reaction is given in Eqn. 1’:
CH3COOH(aq) > CH3COOÀ
(aq) þ Hþ
(aq) (1’)
In Eqn. 1’, the solvent, H2O(l), is not explicitly mentioned. The specification (aq)
indicates that the solvent is water. Note that Eqns. 1 and 1’ indicate that the
concentrations of H3Oþ
(aq) and Hþ
(aq) are the same, although the two species
H3Oþ
(aq) and Hþ
(aq) are, from a molecular point of view, not identical and not even
the only ones present (e.g., H5Oþ
2 (aq) or H7Oþ
3 (aq) and so on). Therefore, they all have
different standard Gibbs energies of formation (see below).
Note that, in principle, the equilibrium of any chemical reaction can be
quantitatively specified by using any type of rules to yield an equilibrium constant
which has a defined numerical value for the particular stoichiometric reaction
Helvetica Chimica Acta – Vol. 97 (2014)6
Fig. 4. a) Chemical formula of the relevant chemical species present at equilibrium upon dissolving
CH3COOH in H2O. In the forward reaction, CH3COOH molecules react with H2O to yield CH3COOÀ
and H3Oþ
ions; the formal electron flow for the forward reaction is shown with curved arrows on the left
hand side, and for the backward reaction on the right hand side. The two straight arrows with half heads
indicate that the system is a chemical equilibrium, i.e., the forward as well as the backward reactions take
place. b) Illustration of the two mesomeric structures of the CH3COOÀ
ion. The curved arrows indicate
here how one can arrive from one mesomeric structure to the other, by formally delocalizing electron
pairs.
2) For the sake of simplicity, in introductory chemistry textbooks, discussions of the reactions between
BrønstedÀLowry acids and BrønstedÀLowry bases, and their quantitative treatment are often
limited to room temperature, i.e., to 258C. Furthermore, it is assumed that the pressure p is
constant, p ¼ 1 bar, the standard pressure, although this is not always specifically emphasized.
considered at the particular temperature and pressure. The numerical values of such
equilibrium constants then not only depend on the stoichiometry of the reaction, but
also on how the equilibrium constant is actually defined, i.e., on the conventions used.
Therefore, the numerical value of the equilibrium constant depends on the used
measure of the composition (i.e., molar concentrations, mass fractions, etc.), and
whether relation to a particular reference state of the different species is made. If the
conventions applied are not clearly communicated, a correct interpretation and
comparison of the equilibrium constants is not possible. With different conventions
different equilibrium constants are obtained, with different numerical values and
different meanings. Depending on what one intends to express and to compare with a
certain equilibrium constant, some of the conventions used are more useful than others.
Confusion arises if equilibrium constants obtained using different conventions are
compared. This is actually the main reason for the apparent water pKa value problem
which is mentioned in Sect. 1, and which led to the writing of this review.
3. The Thermodynamic Acidity Constant of Acetic Acid in Water, Ka,CH3COOH, and
Its Negative Logarithm, pKa,CH3COOH. – On the basis of the definitions and conventions
of classical equilibrium thermodynamics developed for ideal gases and ideal solutions
(Raoults law and Henrys law; see [62 – 64]), for the particular reaction in Eqns. 1 and
1’, the relevant quantities which define the thermodynamic equilibrium constant are
the activities, ai,c of CH3COOH(aq), CH3COOÀ
(aq), H3Oþ
(aq) or Hþ
(aq), and ai,x of
H2O(l), The activity is a quantity which is dependent on the concentrations of all
species A1...n in the reaction mixture. For a given reaction between species listed in the
stoichiometric equation, the thermodynamic equilibrium constant K is defined by
taking the activities of these species into account. The numerical value for the
particular equilibrium constant at fixed pressure only depends on temperature.
Before proceeding with the definition of K for the reaction in Eqns. 1 and 1’, an
alternative and useful formalism is first mentioned, i.e., Eqns. 2 and 2’:
À CH3COOH(aq) À H2O(l) þ CH3COOÀ
(aq) þ H3Oþ
(aq) ¼ 0 (2)
À CH3COOH(aq) þ CH3COOÀ
(aq) þ Hþ
(aq) ¼ 0 (2’)
In such type of equations, the sign of the stoichiometric coefficients ni is
emphasized. It is negative for all species on the left hand side in Eqns. 1 and 1’ and
positive for all species on the right hand side of Eqns. 1 and 1’, i.e., ni is À 1 for
CH3COOH(aq), À 1 for H2O(l), þ 1 for CH3COOÀ
(aq), and þ 1 for H3Oþ
(aq).
A generalized formalism for any type of chemical reaction, including the discussed
reaction in Eqns. 2 and 2’, is given in Eqn. 3 (see e.g., [63]):
Xn
i¼1
niAi ¼ 0 ð3Þ
Here, Ai (A1, A2,...An) represents the n various chemical species appearing in the
stoichiometric reaction equation. The stoichiometric coefficient ni of the species Ai is
Helvetica Chimica Acta – Vol. 97 (2014) 7
defined as above, with ni < 0 for species on the left hand side of the reaction equation
(reactants) and ni > 0 for species on the right hand side (products), respectively.
Eqns. 2, 2’, and 3 are simple and useful for a straightforward mathematical treatment of
equilibrium equations.
The thermodynamic equilibrium constant K for the general reaction described in
Eqn. 3 follows from the general equilibrium condition for a closed system and is
defined as given in [63]3):
K ¼
Yn
i¼1
ani
i ð4Þ
whereby ai is the equilibrium activity of species Ai, as mentioned above.
Applying the definition for K in Eqn. 4 to the reaction formulated with Eqns. 1 and
2, one obtains for K the following expression:
KEqn:1 ¼ KEqn:2 ¼ aÀ1
CH3COOHðaqÞ;c Á aÀ1
H2OðlÞ;x Á aþ1
CH3COOÀ
ðaqÞ;c Á aþ1
H3Oþ
ðaqÞ;c
¼
aCH3COOÀ
ðaqÞ;c Á aH3Oþ
ðaqÞ;c
aCH3COOHðaqÞ;c Á aH2OðlÞ;x
ð5Þ
This is the correct form of the thermodynamic acidity constant for CH3COOH in
water, abbreviated as H2O
Ka,CH3COOH or simply Ka,CH3COOH, without explicit indication
that the solvent is water4). If not stated otherwise, the solvent is always water, and the
pressure is assumed to be p ¼ 1 bar. Note, that the activity of the solvent is referred to
its standard state which is the pure liquid (see below).
Applying the definition for K in Eqn. 4 to the reaction in Eqns. 1’ and 2’ leads to
Eqn. 5’:
KEqn:1’ ¼ KEqn:2’ ¼ aÀ1
CH3COOHðaqÞ;c Á aþ1
CH3COOÀ
ðaqÞ;c Á aþ1
Hþ
ðaqÞ;c ¼
aCH3COOÀ
ðaqÞ;c Á aHþ
ðaqÞ;c
aCH3COOHðaqÞ;c
ð50
Þ
Helvetica Chimica Acta – Vol. 97 (2014)8
3) In a closed system, the equilibrium condition requires that the stoichiometric sum of the chemical
potentials of the species vanishes, i.e.,
Xn
i¼1
nimi ¼ 0
The chemical potentials are given by mi ¼ m
i þ RT ln ai, with the standard chemical potentials m
i and
the activities ai. Note that these latter quantities depend on the choice of the standard state. Upon
substitution, we arrive at Eqn. 4.
4) If water as solvent is replaced with another solvent, i.e., dimethyl sulfoxide (DMSO, (CH3)2SO), the
chemical reaction considered is different as well as the equilibrium constant, DMSO
Ka,CH3COOH. In this
case, CH3COOH as BrønstedÀLowry acid reacts with DMSO as BrønstedÀLowry base in DMSO as
solvent according to the chemical reaction: À CH3COOH(dmso) À (CH3)2SO(l) þ
CH3COOÀ
(dmso) þ (CH3)2SOHþ
(dmso) ¼ 0, whereby (dmso) indicates that the dissolved species
are solvated by DMSO, in analogy to (aq) in the case of aqueous solutions.
Now, let us turn to the definition of the activities. Within the thermodynamic
convention that is compatible with tabulated thermodynamic data for chemical
compounds (see below), the activities of the dissolved species, i.e., the activities of
CH3COOH(aq), CH3COOÀ
(aq), and H3Oþ
(aq) or Hþ
(aq), are related to the molar
concentrations of the dissolved species ci and the standard concentration c8 (1 mol/l ¼
1m) according to Eqn. 6 [62 – 64]:
ai;c ¼ gi;c Á
ci
c
ð6Þ
whereby gi,c is the dimensionless activity coefficient of Ai on the basis of molar
concentration. The activity coefficient can be interpreted as an adjustment factor that
relates the actual, real behavior of a species to the ideal behavior. In most generalchemistry
textbooks, the molar concentration ci of species Ai is denoted with square
brackets, i.e., cCH3COOH(aq) ¼ [CH3COOH(aq)]. Note, however, that chemistry-textbook
authors, when dealing with BrønstedÀLowry acidÀbase equilibria, often use the
brackets for both molar concentrations as well as for molar concentrations divided by
1 mol/l. Therefore, in one and the same textbook, even within the same chapter,
[CH3COOH(aq)] may stand for a quantity with the unit mol/l, or it may stand for a
dimensionless quantity – which is another point of potential confusion.
In this treatise, ci stands for the molar concentration of species Ai and it has the unit
mol/l ¼ m, while ci/c8 is dimensionless. For strongly diluted, ideal solutions, gi,c ¼ 1 is
valid and, therefore, ai,c ¼ ci/c8. Here, and in most of the general-chemistry textbooks,
this ideal, diluted state is assumed to prevail, since it simplifies all further discussions
and calculations (gi,c ¼ 1). There is, however, no conceptual problem at all to consider
for all equations and calculations values of gi,c=1, although the situation becomes more
complicated. For example, in electrolyte solutions one has to deal with average, and not
individual, activity coefficients due to the interdependence of oppositely charged
dissolved species. This is usually outlined in detail in physical-chemistry textbooks and
is highly relevant in chemistry courses dedicated to students of environmental sciences,
since the concentrations of acids and bases, and other dissolved species in natural
waters, e.g., sea water, may be so high that considering them as diluted solutions
would be inappropriate [12]. Again, for the sake of simplicity, as it is done in most
general-chemistry textbooks, we assume here that gi,c ¼ 1.
It is very important to note that the thermodynamic standard states of dissolved
species and of the solvent are different, and this has to be taken into account whenever
activities and equilibrium constants are used. Furthermore, since these quantities are
related to other thermodynamic properties such as, e.g., the Gibbs energies of
formation of the solvated species and of the solvent, which are tabulated according to
defined standard states, the thermodynamic conventions have to be applied accordingly.
Arbitrariness and sloppiness in this subject are the key points for all the confusing
discussions in the literature.
To be compatible with tabulated thermodynamic data (Table), the activity of the
liquid solvent, i.e., the activity of H2O(l) for the reaction considered here, corresponds
to the convention given in Eqn. 7.
asolvent;x ¼ gsolvent;x Á xsolvent ð7Þ
Helvetica Chimica Acta – Vol. 97 (2014) 9
whereby gsolvent,x is the dimensionless activity coefficient of the solvent on the basis of
mole fraction (for ideal solutions, gsolvent,x ¼ 1 for all concentrations); and xsolvent is the
mole fraction of the solvent (xsolvent ¼ nsolvent/ntotal,solution; nsolvent being the amount of
solvent and ntotal,solution the total amount of species in the solution). For highly diluted
solutions, xsolvent ¼ 1 is an appropriate assumption. Therefore, with this convention and
assuming diluted solutions, the activity of the solvent is aH2O(l),x ¼ 1. This is what should
be kept in mind.
Taking into account the two definitions in Eqns. 6 and 7, and assuming that gi,c ¼ 1
for all dissolved species and gH2O(l),x ¼ 1 for the solvent, one obtains with Eqns. 5 or 5’
the expression given in Eqn. 8 for the thermodynamic equilibrium constant K ¼
Ka,CH3COOH for the BrønstedÀLowry acidÀbase reaction formulated with Eqns. 1 or 1’:
Ka;CH3COOH ¼
cCH3 COOÀ ðaqÞ
c
Á
cH3 Oþ ðaqÞ
c
cCH3 COOHðaqÞ
c
¼
cCH3 COOÀ ðaqÞ
c
Á
cHþ ðaqÞ
c
cCH3 COOHðaqÞ
c
ð8Þ
Obviously, Ka,CH3COOH is dimensionless. Furthermore, both reactions in Eqns. 1 and
1’ yield the same equilibrium constant, since cH3Oþ(aq) ¼ cHþ(aq) . This is reasonable since
both chemical reactions describe the same equilibrium reaction and are formulated
with the same stoichiometry.
Note that the numerical value of the molar concentration of the solvent water,
cH2O(l) ¼ 55.33m at 258C, must not be considered in the equilibrium expression (Eqn. 8),
since the thermodynamic standard state of the solvent is the pure solvent, i.e., xH2O(l) ¼ 1
rather than cH2O(l) ¼ 1m.
With the definition
pK ¼ À log10 K ¼ À log K ð9Þ
and, accordingly, pKa,CH3COOH ¼ Àlog Ka,CH3COOH, Eqn. 8 can be transformed to Eqn. 10:
Table. Values for the Standard Gibbs Energy of Formation, DfG8i (258C), for Selected Species Ai in Their
Standard States, as Used in This Treatise, from [5][11][62]
Species Ai Standard state DfG8i (258C) [kJ/mol]
CH3COOH(aq) solution, c ¼ 1m À 396.46
CH3COOÀ
(aq) solution, c ¼ 1m À 369.31
H2O(l) pure solvent, x ¼ 1 À 237.13
Hþ
(aq) solution, c ¼ 1m 0
H3Oþ
(aq) solution, c ¼ 1m À 237.13
HOÀ
(aq) solution, c ¼ 1m À 157.24
Helvetica Chimica Acta – Vol. 97 (2014)10
log Ka;CH3COOH ¼ ÀpKa;CH3COOH ¼ log
cCH3 COOÀ ðaqÞ
c
Á
cH3 Oþ ðaqÞ
c
cCH3 COOHðaqÞ
c
0
B
B
@
1
C
C
A
¼ log
cCH3 COOÀ ðaqÞ
c
cCH3 COOHðaqÞ
c
0
B
B
@
1
C
C
A þ log
cH3Oþ
ðaqÞ
c
 
¼ log
cCH3COOÀ
ðaqÞ
cCH3COOHðaqÞ
 
þ log
cH3Oþ
ðaqÞ
c
 
ð10Þ
Again, it is assumed that all activity coefficients are gi ¼ 1, as mentioned above.
The pH value is defined as the negative logarithm to base 10 of the activity of
H3Oþ
(aq) or Hþ
(aq) [62 – 64] (see Eqn. 11 and [65]).
pH ¼ À log aH3Oþ
ðaqÞ;c
 
¼ À log gH3Oþ
ðaqÞ;c Á
cH3Oþ
ðaqÞ
c
 
¼ À log gHþ
ðaqÞ;c Á
cHþ
ðaqÞ
c
 
ð11Þ
For gH3Oþ(aq),c ¼ 1 and gHþ(aq),c ¼ 1 (see above), one obtains Eqn. 12:
pH ¼ À log
cH3Oþ
ðaqÞ
c
 
¼ À log
cHþ
ðaqÞ
c
 
ð12Þ
With the definition of pH in Eqn. 12, Eqn. 10 yields the well-known equation of
Henderson and Hasselbalch (Fig. 5), which relates the acidity constant and the
equilibrium concentrations to the pH value:
pH ¼ pKa;CH3COOH þ log
cCH3COOÀ
ðaqÞ
cCH3COOHðaqÞ
 
ð13Þ
with cCH3COOÀ(aq) þ cCH3COOH(aq) ¼ cCH3COOH,total, the constant pKa,CH3COOH can be determined
experimentally through simple titration experiments, in which an aqueous acetic
acid solution is titrated with a solution of a strong base, typically a solution of hydroxide
ions, HOÀ
(aq), obtained by previously dissolving NaOH(s) in water (see Fig. 6). The
pH value is measured as a function of added amount of NaOH. The pKa,CH3COOH value is
the pH value of the solution at which the equilibrium concentration of CH3COOH(aq)
and CH3COOÀ
(aq) are equal, i.e., cCH3COOÀ(aq)/cCH3COOH(aq) ¼ 1, which means that
log(cCH3COOÀ(aq)/cCH3COOH(aq)) ¼ 0. Since the entire experimental titration curve in Fig. 6
can be fitted with Eqn. 13, experiments and theory are in full agreement with each
other, confirming that the assumptions made are reasonable, i.e., gi ¼ 1. An
Helvetica Chimica Acta – Vol. 97 (2014) 11
Helvetica Chimica Acta – Vol. 97 (2014)12
Fig. 6. Titration of 0.1 l of a cCH3COOH,total ¼ 0.1m aqueous CH3COOH solution with a 1.0m solution of
aqueous NaOH. The pH value of the solution was measured (open circles) at 208C with a Metrohm 691
pH meter equipped with a Metrohm pH glass electrode calibrated at pH 4.0 and pH 7.0. The solid line
results from a simulation on the basis of the HendersonÀHasselbalch equation (Eqn. 13), and the selfionization
of water (Eqn. 21), with pKa,CH3COOH ¼ 4.63 and Kw ¼ 6.8 · 10À15
(at 208C). There is excellent
agreement with the ideal model.
Fig. 5. Lawrence Joseph Henderson (1878 – 1942; left) and Karl Albert Hasselbalch (1874 – 1962; right).
See also [66 – 68]. Photographs from J. W. Severinghaus, P. Astrup, J. F. Murray, Blood gas analysis and
critical care medicine, Am. J. Respir. Crit. Care Med. 1998, 157, S114 – S122 (Henderson); and F.
Sgambato, S. Prozzo, E. Sgambato, R. Sgambato, L. Milano, Il centenario del pH (1909 – 2009) – parte
seconda. Ma era proprio necessario sostituire lequazione di Henderson con quella di HendersonHasselbalch?,
Ital. J. Med. 2011, 5, 215 – 226 (Hasselbalch).
extrapolation of measured pKa values obtained with different acetic acid concentrations
to an infinitely diluted solution yields pKa,CH3COOH(258C) ¼ 4.76, which means
that Ka,CH3COOH(258C) ¼ 10À4.76
¼ 1.74 · 10À5
, a value which was obtained in the 1920s and
1930s with a series of very careful experiments [69 – 71].
In all textbooks which tabulate the pKa value for acetic acid in water, there is a
rather good agreement about its numerical value; it varies between pKa,CH3COOH ¼ 4.76
(e.g., [8][72]) and pKa,CH3COOH ¼ 4.74 (e.g., [5][73]) at 258C.
As already mentioned above, the conventions used so far are thermodynamically
meaningful and fully compatible not only with titration experiments, but also with
tabulated thermodynamic data. To demonstrate this latter compatibility, Ka,CH3COOH is
calculated from the standard reaction Gibbs energy, DrG8(T), for the reactions in
Eqn. 1 and 1’, by using Eqn. 14 [62 – 64].
KðTÞ ¼ e
ÀDr G ðTÞ
RT ð14Þ
which is equivalent to Eqn. 14’:
DrG
ðTÞ ¼ ÀRT ln KðTÞ ð140
Þ
R is the gas constant (8.3145 J KÀ1
molÀ1
) and T the absolute temperature (in Kelvin,
K).
The magnitude of DrG8(258C) can be calculated from tabulated values of the
standard Gibbs energy of formation for each species Ai of the reaction, DfG
i (258C),
(see the Table), as shown in Eqn. 15:
DrG
ðTÞ ¼
Xn
i¼1
ni Á DfG
i ðTÞ ð15Þ
For the reaction of Eqn. 1
DrG8(258C)Eqn.1 ¼ ÀDfG8CH3COOH(aq)(258C) À DfG8H2O(l)(258C) þ
DfG8CH3COOÀ(aq)(258C) þ DfG8H3Oþ(aq)(258C)
¼ À(À 396.46 kJ/mol) À (À237.13 kJ/mol) þ
(À 369.31 kJ/mol) þ (À237.13 kJ/mol) ¼ 27.15 kJ/mol (16)
Note, that the tabulated values for the solvated species and for the solvent again
refer to their respective standard states, i.e., c ¼ 1m (solvated species) and x ¼ 1
(solvent).
For the reaction of Eqn. 1’
DrG8(258C)Eqn.1’ ¼ ÀDfG8CH3COOH(aq)(258C) þ DfG8CH3COOÀ(aq)(258C)
þ DfG8Hþ(aq)(258C) ¼ À(À 396.46 kJ/mol) þ (À 369.31 kJ/mol)
þ 0 kJ/mol ¼ 27.15 kJ/mol (16’)
Helvetica Chimica Acta – Vol. 97 (2014) 13
Both calculations yield exactly the same value for DrG8(258C) ¼ 27.15 kJ/mol. This
is reasonable since Eqns. 1 and 1’ are stoichiometrically equivalent representations of
one and the same reaction, as discussed above.
With this calculated value for DrG8(258C), the thermodynamic equilibrium
constant K(258C) for the reactions of Eqns. 1 and 1’ can be calculated with Eqn. 14:
Kð25 
CÞEqn:1 ¼ Kð25 
CÞEqn:1’ ¼ e
ÀDr G ð25  CÞ
RT
¼ e
À27150 J molÀ1
8:3145 J KÀ1 molÀ1 Á298 K ¼ eÀ10:966
¼ 1:74 Á 10À5
¼ 10À4:76
ð17Þ
Since K(258C)Eqn.1 ¼ K(258C)Eqn.1’ ¼ Ka,CH3COOH(258C), the acidity constant for
acetic acid in water at 258C, as calculated from the standard reaction Gibbs energy
DrG8(258C), is 1.74 · 10À5
, and accordingly pKa,CH3COOH(258C) ¼ 4.76. This value is in
very good agreement with the value used for simulating experimental data (Fig. 6),
confirming the statement made above that the conventions used in this Sect. and the
definition of K are fully compatible with tabulated thermodynamic data. Any type of
BrønstedÀLowry acidÀbase reaction can be treated in exactly the same way, including
the particular case of the reaction of water as BrønstedÀLowry acid with water as
BrønstedÀLowry base in water as solvent, as outlined in the following Sect.
4. The Thermodynamic Acidity Constant of Water. – Following exactly the same
conventions as described in Sect. 3 for the reaction of acetic acid with water, the
thermodynamic equilibrium constant for the reaction of water as BrønstedÀLowry acid
with water as BrønstedÀLowry base in water as solvent, as formulated in Eqn. 18 can
easily be obtained.
H2O(l) þ H2O(l) > HOÀ
(aq) þ H3Oþ
(aq) (18)
Eqn. 18 can also be expressed as in Eqn. 19
À 2 H2O(l) þ HOÀ
(aq) þ H3Oþ
(aq) ¼ 0 (19)
For reasons discussed above, Eqns. 18 and 19 can also be formulated as in Eqns. 18’
and 19’:
H2O(l) > HOÀ
(aq) þ Hþ
(aq) (18’)
À H2O(l) þ HOÀ
(aq) þ Hþ
(aq) ¼ 0 (19’)
Eqns. 18, 18’, 19, and 19’ are equivalent in the sense that they describe in a
stoichiometrically equivalent way the same reaction, known as self-ionization of
water, autodissociation of water, or autoprotolysis of water. The thermodynamic
equilibrium constant for the reaction as formulated in Eqns. 18 and 19 is given in
Eqn. 20:
Helvetica Chimica Acta – Vol. 97 (2014)14
KEqn:18 ¼ aÀ2
H2OðlÞ;x Á aþ1
HOÀ
ðaqÞ;c Á aþ1
H3Oþ
ðaqÞ;c ¼
aHOÀ
ðaqÞ;c Á aH3Oþ
ðaqÞ;c
a2
H2OðlÞ;x
ð20Þ
With the conventions outlined above, in particular asolvent,x ¼ aH2O(l),x ¼ 1, and
assuming that all activity coefficients of the dissolved species are 1 (gi,c ¼ 1), one
obtains Eqn. 21:
KEqn:18 ¼
cHOÀ
ðaqÞ
c
Á
cH3Oþ
ðaqÞ
c
ð21Þ
Accordingly, the thermodynamic equilibrium constant for the reaction in Eqns. 18’
and 19’ yields
KEqn:18’ ¼ aÀ1
H2OðlÞ;x Á aþ1
HOÀ
ðaqÞ;c Á aþ1
Hþ
ðaqÞ;c ¼
aHOÀ
ðaqÞ;c Á aHþ
ðaqÞ;c
aH2OðlÞ;x
ð20’Þ
and by again taking into account that aH2O(l),x ¼ 1 and assuming gi,c ¼ 1, one obtains
KEqn:18’ ¼
cHOÀ
ðaqÞ
c
Á
cHþ
ðaqÞ
c
ð21’Þ
Since cH3Oþ(aq) ¼ cHþ(aq) , K must have the same value for the reactions formulated in
Eqns. 18 and 19, or Eqns. 18’ and 19’. This particular thermodynamic equilibrium
constant is usually abbreviated as Kw.
Experimentally, it was found by electrochemical measurements that at 258C
cH3Oþ(aq) ¼ 10À7.0
m [74] (Fig. 7), i.e., cH3Oþ(aq)/c8 ¼ 10À7.0
. This means that cHþ(aq)/c8 ¼ 10À7.0
,
as well as cHOÀ(aq)/c8 ¼ 10À7.0
. This latter relation is due to the fact that for each H3Oþ
(aq)
formed from H2O though the reaction shown in Eqns. 18 or 18’, one HOÀ
(aq) is
obtained. This is the actual meaning of Eqns. 18 and 18’. With these experimental data,
one can easily calculate that Kw(258C) ¼ 10À7.0
· 10À7.0
¼ 10À14.0
[75], as mentioned in
basically all general-chemistry textbooks, although there is no general consensus on
whether the water self-ionization constant has dimensions or not. The constant may
have dimensions only if the conventions used for defining the constant are different
from the thermodynamic conventions outlined here (see below and Appendix).
Kw(258C) can be calculated in the same way as outlined above for Ka,CH3COOH(258C)
by taking into account tabulated values for DfG8i (258C) for the relevant chemical
species of the reaction, as given in the Table (see Eqns. 22 and 22’ for the reactions of
Eqns. 18 and 18’).
DrG8(258C)Eqn.18 ¼ À2 DfG8H2O(l)(258C) þ DfG8HOÀ(aq)(258C) þ DfG8H3Oþ(aq)(258C)
¼ À2 (À237.13 kJ/mol) þ (À157.24 kJ/mol) þ (À237.13 kJ/mol)
¼ 79.89 kJ/mol (22)
Helvetica Chimica Acta – Vol. 97 (2014) 15
DrG8(258C)Eqn.18’ ¼ ÀDfG8H2O(l)(258C) þ DfG8HOÀ(aq)(258C) þ DfG8Hþ(aq)(258C)
¼ À(À237.13 kJ/mol) þ (À157.24 kJ/mol) þ 0 kJ/mol
¼ 79.89 kJ/mol (22’)
Independent of whether the autoprotolysis of water is formulated with Eqns. 18, 18’,
or Eqns. 19, 19’, one obtains DrG8(258C) ¼ 79.89 kJ/mol. With this and applying
Eqn. 14, one obtains Kw(258C) ¼ 10À14.0
.
Kwð25 
CÞ ¼ e
ÀDr G ð25  CÞ
RT
¼ e
À79890 J molÀ1
8:3145 J KÀ1 molÀ1 Á298 K
¼ eÀ32:243
¼ 9:93 Á 10À15
% 10À14:0
ð23Þ
Therefore, pKw(258C) ¼ 14.0.
Since Eqns. 18, 19, and Eqns. 18’, 19’ represent a chemical reaction in which water
reacts as BrønstedÀLowry acid with water as BrønstedÀLowry base, Kw is also the
acidity constant for water. Therefore, Kw(258C) ¼ 10À14.0
means that Ka,H2O(258C) ¼
Fig. 7. Augusta Marie Unmack (1896 – 1990; left), assistant of Niels Janniksen Bjerrum (1879 – 1958;
right) – one of the pioneers of physical chemistry –, analyzed aqueous solutions with a hydrogen electrode
and determined from these measurements the ionization constant of water [74]. See [76 – 78]. Photographs
from the Royal Library, Copenhagen, Denmark (Unmack,  Royal Library Copenhagen, reprinted with
permission); and N. Bohr, J. A. Christiansen, K. J. Pedersen, et al. (Eds.), Niels Bjerrum. Selected
Papers, edited by friends and co-workers on the occasion of his 70th birthday the 11th of March, 1949,
Einar Munksgaard, Copenhagen, 1949 (Bjerrum).
Helvetica Chimica Acta – Vol. 97 (2014)16
10À14.0
and pKa,H2O(258C) ¼ Àlog Ka,H2O(258C) ¼ 14.0 ¼ pKw(258C) ¼ Àlog Kw(258C).
Therefore, the thermodynamically correct pKa value for the dissociation of water in the
solvent water at a temperature 258C and an assumed pressure of p ¼ 1 bar is 14.0. This
value can be found in a number of general-chemistry textbooks [4 – 11]. Often,
however, pKa tables in introductory-chemistry textbooks do not have an entry for H2O
(or H3Oþ
), probably to avoid discussing the kind of question we address in this treatise.
As will be also demonstrated in Sect. 6, the thermodynamically correct value for the
pKa,H3Oþ(258C) is 0.
Full compatibility of pKw(258C) ¼ pKa,H2O(258C) ¼ 14.0 with tabulated thermodynamic
data is further illustrated by considering the autodissociation of water as the sum
of two redox half-cell reactions under standard conditions, the reduction of water to
hydrogen gas and to two hydrated hydroxide ions, and the oxidation of hydrogen gas to
two hydrated protons, as shown in Eqns. 24 and 25.
2 H2O(l) þ 2 eÀ
> H2(g) þ 2 HOÀ
(aq) (24)
H2(g) > 2 Hþ
(aq) þ 2 eÀ
(25)
The sum of Eqns. 24 and 25 yields
2 H2O(l) > 2 HOÀ
(aq) þ 2 Hþ
(aq) (26)
Divisions on both sides of Eqn. 26 by 2 gives
H2O(l) > HOÀ
(aq) þ Hþ
(aq) (27)
At 258C and standard pressure p ¼ 1 bar, the tabulated standard reduction
potential E8red for the half-cell reaction of Eqn. 24 is À 0.8277 V [75]. The standard
oxidation potential E8ox for the half-cell reaction of Eqn. 25 is 0.0000 V by definition
[75], since all tabulated standard reduction potentials are related to this reaction, the
standard hydrogen electrode. Therefore, the standard potential DE8 for the reaction of
Eqn. 26 or 27 is DE8Eqn.26 ¼ DE8Eqn.27 ¼ E8red,Eqn.24 þ E8ox,Eqn.25 ¼ À0.8277 Vþ 0.0000 V¼
À0.8277 V.
The relationship between DE8 and DrG8 is given as [62 – 64]:
DrG8 ¼ À ne · F · DE8 (28)
F being the Faraday constant (96485 C molÀ1
¼ 96485 J VÀ1
molÀ1
), and ne the number
of electrons involved in the redox half-cell reactions leading to the net reaction. For the
reaction in Eqn. 27, ne ¼ 1.
For the reaction of Eqn. 27, one obtains Eqn. 29
DrG8(258C)Eqn.27 ¼ À1 · 96485 J VÀ1
molÀ1
· (À0.8277 V)
¼ 79860 J/mol ¼ 79.86 kJ/mol (29)
Helvetica Chimica Acta – Vol. 97 (2014) 17
DrG8(258C)Eqn.27 is identical with the value DrG8(258C)Eqn.18’ calculated on the basis of
the tabulated DfGi8(258C) values for the individual chemical species. This yields
Kw(258C) ¼ Ka,H2O(258C) ¼ 10À14.0
(see Eqn. 23), and pKw(258C) ¼ pKa,H2O(258C) ¼
14.0.)
As already mentioned repeatedly above, like all other thermodynamic equilibrium
constants, Kw is a dimensionless constant. However, in several textbooks, this constant
is given with units (mol2
lÀ2
¼ m2
) [9][16][46][60][79 – 82] (see Appendix). Interestingly,
somewhat strange appears the translation of an American general-chemistry
textbook [83a] into German [83b] which resulted in a change of the units of Kw from
dimensionless in the original edition to unit mol2
lÀ2
in the German version (similar
changes were made for other equilibrium constants, e.g., for the solubility product, Ksp).
This change was apparently done on purpose and not by mistake, in order to adapt to
local customs, as quoted in the introduction to the German edition. Chemistry as
scientific discipline should be language- and country-independent. Uncertainty about
the proper dimensions is also evident if, in one particular case, various editions of one
and the same textbook are compared: Kw was dimensionless in an early edition [84],
while in a later edition [20], Kw is given in units mol2
lÀ2
.
5. Calculation of Equilibrium Constants for BrønstedÀLowry AcidÀBase Reactions
in Which the Base Is not Water. – Knowing the pKa values for BrønstedÀLowry acids in
water, i.e., knowing the equilibrium constants for reactions between a BrønstedÀLowry
acid and water as BrønstedÀLowry base, allows calculation of equilibrium constants for
aqueous BrønstedÀLowry acidÀbase equilibria in which the base is different from
water. The equilibrium constant for such reactions can be calculated from the
individual acidity constants Ka of the two acids involved in the equilibrium. To illustrate
this, let us consider the titration experiment discussed above and shown in Fig. 6 (see
Eqn. 30). Hydrated acetic acid, CH3COOH(aq), and hydrated hydroxide ions,
HOÀ
(aq), react to yield hydrated acetate ions, CH3COOÀ
(aq), and H2O(l). As
indicated with (aq), the reaction takes place in H2O(l) as solvent.
CH3COOH(aq) þ HOÀ
(aq) > CH3COOÀ
(aq) þ H2O(l) (30)
The acid on the left hand side of Eqn. 30 is CH3COOH(aq), its corresponding base on
the right hand side is CH3COOÀ
(aq); the acid on the right hand side is H2O(l), its
corresponding base on the left hand side HOÀ
(aq). The chemical reaction represented
with Eqn. 30 can be considered as the sum of two separate BrønstedÀLowry acidÀbase
reactions, both being reactions of BrønstedÀLowry acids with H2O(l) as BrønstedÀLowry
base; the equilibrium constant, K(258C)Eqn.30 for the reaction of Eqn. 30
can then be obtained as the product of the reaction constants of the two separate
reactions. The two separate reactions are given in Eqns. 1 and 31:
CH3COOH(aq) þ H2O(l) > CH3COOÀ
(aq) þ H3Oþ
(aq) (1)
H3Oþ
(aq) þ HOÀ
(aq) > H2O(l) þ H2O(l) (31)
Helvetica Chimica Acta – Vol. 97 (2014)18
The sum of the two reactions yields Eqn. 30. Note that the reaction of Eqn. 31 is the
inverse of the reaction of Eqn. 18.
In Eqn. 1, CH3COOH(aq) is the BrønstedÀLowry acid and H2O(l) the BrønstedÀ
Lowry base, in Eqn. 31 H3Oþ
(aq) is the BrønstedÀLowry acid and HOÀ
(aq) the
BrønstedÀLowry base. Since, at 258C, KEqn.1 ¼ Ka,CH3COOH ¼ 10À4.76
and KEqn.31 ¼ 1/
Ka,H2O ¼ 1/Kw ¼ 1/10À14.0
¼ 1014.0
, one obtains KEqn.30 ¼ KEqn.1 · KEqn.31 ¼ 10À4.76
· 1014.0
¼
109.24
.
The same value results from the tabulated thermodynamic data for the standard
Gibbs energy of formation, DfG8i (258C), for the different species of Eqn. 30 (see the
Table):
DrG8(258C)Eqn.30 ¼ ÀDfG8CH3COOH(aq)(258C) À DfG8HOÀ(aq)(258C) þ
DfG8CH3COOÀ(aq)(258C) þ DfG8H2O(l)(258C)
¼ À(À 396.46 kJ/mol) À (À157.24 kJ/mol) þ (À 369.31 kJ/mol) þ (À237.13 kJ/mol)
¼ À52.74 kJ/mol (32)
Kð25 
CÞEqn:30 ¼ e
ÀDr G ð25  CÞEqn:30
RT ¼ e
52740 J molÀ1
8:3145 J KÀ1 molÀ1 Á298 K
¼ e21:286
¼ 1:76 Á 109
¼ 109:24
ð33Þ
Note that the numerical value of the equilibrium constant of 1.76 · 109
is a large
number, since a weak acid, CH3COOH(aq), reacts with a strong base, HOÀ
(aq). This is
the basis for any type of weak acidÀstrong base titration experiment.
6. The Thermodynamic Acidity Constant of the Oxonium Ion (H3Oþ
) in Water. –
To conclude the straightforward application of the thermodynamic conventions for the
quantitative treatment of BrønstedÀLowry acidÀbase equilibria, let us ask the
question: What is the thermodynamic pKa value of H3Oþ
in water as solvent? The
answer must be clear: pKa,H3Oþ ¼ 0.
The reaction considered is
H3Oþ
(aq) þ H2O(l) > H2O(l) þ H3Oþ
(aq) (34)
with
Ka;H3Oþ ¼
aH2OðlÞ;x Á aH3Oþ
ðaqÞ;c
aH3Oþ
ðaqÞ;c
Á aH2OðlÞ;x
¼ 1 ð35Þ
Ka,H3Oþ ¼ 1 means DrG8 ¼ 0 (see Eqns. 14 and 14’). Therefore, pKa,H3Oþ ¼ Àlog Ka,H3Oþ ¼
Àlog 1 ¼ 0, independent of temperature. This means that, whenever a H2O molecule
attacks an oxonium ion, an oxonium ion and a molecule of H2O result, irrespective of
whether this process takes place at 258C, 508C, or any other temperature.
Helvetica Chimica Acta – Vol. 97 (2014) 19
In summary, a description of BrønstedÀLowry acidÀbase reactions within the
framework of the thermodynamic conventions yields pKa,H2O(258C) ¼ 14.0 ¼
pKw(258C) and pKa,H3Oþ ¼ 0.
7. Final Remarks and a Suggestion. – As outlined repeatedly above, apparent
inconsistency in chemistry textbooks not only exists about the value of the pKa of H2O
or H3Oþ
in water at 258C, but there is also an apparent inconsistency about the
dimension of equilibrium constants, since comparison is made between constants that
are defined differently. Therefore, a clear communication of the conventions applied is
of outmost importance. A further observation that actually adds to the confusion in
chemistry textbooks is the change in conventions on going from one chapter to the
other, even if the topics of the two chapters may be directly related. This is particularly
evident when moving from the treatment of reactions occurring in the gas phase
between gaseous molecules to the treatment of reactions occurring in solution; either i)
between dissolved species (the solvent being an inert medium not directly taking part
in the chemical reactions); or ii) between dissolved species and the solvent molecules
(e.g., the BrønstedÀLowry acidÀbase reactions discussed extensively in this treatise).
Such a change in conventions is unnecessary, although it persists in chemistry textbooks
since decades. It appears difficult to alter past customs to a uniform treatment of
chemical reaction equilibria.
We are convinced that the concepts of activities and standard states – with all the
consequences – could be introduced from the very beginning when teaching general
chemistry. Certainly, changing the way the students get introduced into the quantitative
treatment of chemical reaction equilibria – in particular BrønstedÀLowry acidÀbase
reactions – would mean a considerable effort to modify certain chapters in generalchemistry
textbooks. However, this can certainly be accomplished and should be
accomplished. We feel that specific improvements in textbooks and in the teaching are
indeed necessary. The fundamental concepts of chemistry should be independent of the
textbook used, independent of the country in which chemistry is taught, and
independent of the written and spoken language.
Improvements as we suggest here would be mainly for the benefit of chemistry
students and for a better general reputation of chemistry as central, scientific discipline.
The proposed changes are based on well-established physicochemical considerations,
as outlined extensively in Sect. 3 – 6. There is no doubt that such changes can be made,
but they clearly have to go beyond cosmetic modifications.
The thermodynamically correct treatment of chemical reaction equilibria as
outlined in Sects. 3 to 6 is straightforward, useful, and transparent, and should be
easy to understand. Even without a previous detailed education in chemical
thermodynamics, a student should be able to follow the arguments presented. There
are at the end only a few concepts and conventions to consider and apply. The main
rules are the following.
1) First, a chemically and stoichiometrically correct equation describing a
particular chemical reaction in which one is interested in should be formulated.
In this equation, the states of the chemical species involved have to be clearly
indicated. The states are either gaseous (g), liquid (l), solid (s), or dissolved. If
Helvetica Chimica Acta – Vol. 97 (2014)20
the solvent is water, the dissolved species are indicated by adding (aq). The
solvent used has to be clear from the equation.
2) Any chemical equilibrium can be represented with the generalized Eqn. 3 in
which the stoichiometric coefficients and the species involved are ni and Ai
Xn
i¼1
niAi ¼ 0 ð3Þ
Examples are Eqns. 2 and 2’ in Sect. 3, and Eqns. 19 and 19’ in Sect. 4.
3) The equilibrium constant K is then defined for the as formulated reaction
according to Eqn. 4.
K ¼
Yn
i¼1
ani
i ð4Þ
whereby ai is the activity of species Ai and ni is the stoichiometric coefficient of
Ai. Examples are Eqns. 5 and 5’ in Sect. 3, and Eqn. 20 and 20’ in Sect. 4.
4) The activities of the different types of species involved in the equilibrium are
defined depending on their states, as follows.
For dissolved species Ai as shown in Eqn. 6
ai;c ¼ gi;c Á
ci
c
ð6Þ
gi,c being the activity coefficient of the dissolved species Ai on the basis of molar
concentration, ci being the molar concentration of the dissolved species Ai, and
c8 being the standard concentration which is 1 mol lÀ1
¼ 1m.
For liquid solvents as in Eqn. 7
asolvent;x ¼ gsolvent;x Á xsolvent ð7Þ
gsolvent,x being the activity coefficient of the solvent on the basis of mole fraction,
xsolvent being the mole fraction of the solvent (xsolvent ¼ nsolvent/ntotal,solution, nsolvent
being the amount of solvent and ntotal,solution being the total amount of substances
in the solution). In dilute solutions, xsolvent ¼ 1.
For gaseous species Ai, as in Eqn. 36
ai;p ¼ gi;p Á
pi
p
ð36Þ
Helvetica Chimica Acta – Vol. 97 (2014) 21
gi,p being the activity coefficient of gaseous species Ai on the basis of pressure in
units of bar, pi being the partial pressure of the gas Ai (pi ¼ xi · ptotal, with xi being
the mole fraction of gas Ai), and p8 being the standard pressure which is 1 bar.
For solids as in Eqn. 37
asolid;x ¼ gsolid;x Á xsolid ð37Þ
gsolid,x being the activity coefficient of the solid on the basis of mole fraction
xsolid ¼ nsolid/ntotal,solid. For a pure solid phase, xsolid ¼ 1. Note that the standard state
of the solid is its most stable polymorphic form at the temperature considered.
5) Since in introductory-chemistry textbooks, the discussion of chemical equilibria
of any type usually is limited to ideal systems in which, for example, the
concentrations of all dissolved species are low, all activity coefficients take a
value of 1, i.e., gi,c ¼ 1, gsolvent,x ¼ 1, gi,p ¼ 1, gsolid,x ¼ 1.
As a general summary of our rather detailed argumentations in this treatise, we
propose to introduce straightforward thermodynamic conventions and the concept of
activities at the very beginning when teaching reaction equilibria in a general-chemistry
course. This is independent of whether reactions occurring in the gas phase are
discussed, i.e., reactions occurring without any solvent, or whether the reactions take
place in a solvent with its participation in the reaction. When using tabulated pKa
values for organic molecules taken from organic-chemistry textbooks, students should
be made aware of the fact that the listed values often are based on conventions which
are different from the thermodynamic ones. Differently defined constants should not
be mixed-up.
The many fruitful discussions over the last years on the topic of this review with Prof. Walter Caseri,
D-MATL, ETH Zürich, Dr. Wolfram Uhlig, D-CHAB, ETH-Zürich, and Dr. Urs Hollenstein, D-CHAB,
ETH Zürich are highly appreciated. The authors are greateful to Prof. Sven E. Harnung, University of
Copenhagen, Denmark, and to Dr. Oliver Renn, Information Center Chemistry, Biology, Pharmacy, ETH
Zürich, for their help in finding a photograph of Augusta Marie Unmack.
REFERENCES
[1] J. N. Brønsted, Recl. Trav. Chim. Pays-Bas 1923, 42, 718.
[2] T. M. Lowry, Chem. Ind. 1923, 42, 43.
[3] G. B. Kauffman, J. Chem. Educ. 1988, 65, 28.
[4] M. Munowitz, Principles of Chemistry, 1st edn., W. W. Norton & Company, New York, 2000.
[5] D. W. Oxtoby, W. A. Freeman, T. F. Block, Chemistry: Science of Change, 4th edn., Thomson
Learning, 2003.
[6] J. C. Kotz, P. M. Treichel Jr., P. A. Harman, Chemistry & Chemical Reactivity, 5th edn., Thomson
Learning, 2003.
[7] T. R. Gilbert, R. V. Kirss, G. Davies, Chemistry: the Science in Context, W. W. Norton, New York,
2004.
[8] J. A. Bell, Chemistry: a Project of the American Chemical Society, W. H. Freeman, New York,
2005.
[9] G. Baars, Basiswissen Chemie, 2. Aufl., hep verlag, Bern, 2011.
[10] J. W. Moore, C. L. Stanitski, P. C. Jurs, Chemistry. The Molecular Science, 4th edn., Brooks Cole,
Belmont, 2011.
Helvetica Chimica Acta – Vol. 97 (2014)22
[11] D. W. Oxtoby, H. P. Gillis, N. H. Campion, Principles of Modern Chemistry, 7th edn., Brooks Cole,
Cengage Learning, 2012.
[12] R. P. Schwarzenbach, P. M. Gschwend, D. Imboden, Environmental Organic Chemistry, 2nd edn.,
John Wiley & Sons, Hoboken, 2003.
[13] J. B. Hendrickson, D. J. Cram, G. S. Hammond, Organic Chemistry, 3rd edn., McGraw-Hill
Kogakusha, Tokyo, 1970.
[14] A. Streitwieser, C. H. Heathcock, E. M. Kosower, Introduction to Organic Chemistry, 4th edn.,
Macmillan Publishing, New York, 1992.
[15] J. McMurry, Organic Chemistry, 5th edn., Brooks Cole, Pacific Grove, 2000.
[16] J. Clayden, N. Greeves, S. Warren, P. Wothers, Organic Chemistry, Oxford University Press,
Oxford, 2001.
[17] L. G. Wade Jr., Organic Chemistry, 5th edn., Pearson Education, Upper Saddle River, 2003.
[18] M. A. Fox, J. K. Whitesell, Organic Chemistry, 3rd edn., Jones and Bartlett Publishers, Sudbury,
2004.
[19] T. W. G. Solomons, C. B. Fryhle, Organic Chemistry, 8th edn., John Wiley & Sons, Hoboken, 2004.
[20] K. P. C. Vollhardt, N. E. Schore, Organic Chemistry, 5th edn., W. H. Freeman, New York, 2007.
[21] M. B. Smith, Organic Chemistry: an AcidÀbase Approach, CRC Press, Taylor & Francis Group,
Boca Raton, 2011.
[22] P. Y. Bruice, Organic Chemistry, 6th edn., Pearson Education, 2011.
[23] D. W. Oxtoby, H. P. Gillis, N. H. Nachtrieb, Principles of Modern Chemistry, 5th edn., Thomson
Learning, 2002.
[24] R. Starkey, J. Norman, N. Hinze, J. Chem. Educ. 1986, 63, 473.
[25] W. G. Baldwin, C. E. Burchill, J. Chem. Educ. 1987, 64, 1067.
[26] R. J. Thompson, J. Chem. Educ. 1990, 67, 220.
[27] W. G. Baldwin, C. E. Burchill, J. Chem. Educ. 1992, 69, 514.
[28] R. J. Thompson, J. Chem. Educ. 1992, 69, 515.
[29] A. Tapparo, J. Chem. Educ. 1992, 69, 515.
[30] H. L. Abrash, J. Chem. Educ. 2001, 78, 1496.
[31] W. Angst, D. Diem, R. Kummert, W. Leupin, J. Wirz, c þ b: Fachz. VSN 2004, 4/04, 27.
[32] R. Kummert, c þ b: Fachz. VSN 2003, 4/03, 14.
[33] M. Cosandey, c þ b: Fachz. VSN 2004, 3/04, 20.
[34] R. Kummert, c þ b: Fachz. VSN 2004, 2/04, 14.
[35] R. Kummert, c þ b: Fachz. VSN 2004, 3/04, 17.
[36] A. von Zelewsky, c þ b: Fachz. VSN 2004, 4/04, 38.
[37] D. Keeports, J. Chem. Educ. 2005, 82, 999.
[38] E. J. Behrman, J. Chem. Educ. 2006, 83, 1290.
[39] D. Keeports, J. Chem. Educ. 2006, 83, 1290.
[40] R. E. Dickerson, I. Geis, Chemistry, Matter, and the Universe: an Integrated Approach to General
Chemistry, W. A. Benjamin, Menlo Park, 1976.
[41] F. A. Carey, Organic Chemistry, 7th edn., McGraw Hill Higher Education, New York, 2008.
[42] J. W. Gibbs, Trans. Connecticut Acad. Arts Sci. 1875 – 1878, 3, 108 and 343.
[43] E. M. Arnett, Prog. Phys. Org. Chem. 1963, 1, 223.
[44] D. J. Cram, in Fundamentals of Carbanion Chemistry, Ed. D. J. Cram, Academic Press, London,
1965, Chapt. I, p. 1.
[45] R. A. Cox, Adv. Phys. Org. Chem. 2000, 35, 1.
[46] E. A. Anslyn, D. A. Dougherty, Modern Physical Organic Chemistry, University Science Books,
2006.
[47] G. N. Lewis, Proc. Am. Acad. Art Sci. 1907 – 1908, 43, 259.
[48] G. N. Lewis, J. Am. Chem. Soc. 1908, 30, 668.
[49] G. N. Lewis, J. Am. Chem. Soc. 1916, 38, 762.
[50] M. Calvin, J. Chem. Educ. 1984, 61, 14.
[51] G. E. K. Branch, J. Chem. Educ. 1984, 61, 18.
Helvetica Chimica Acta – Vol. 97 (2014) 23
[52] L. Jones, P. Atkins, Chemistry: Molecules, Matter, and Change, 4th edn., W. H. Freeman, New
York, 2000.
[53] A. Blackman, S. E. Bottle, S. Schmid, M. Mocerino, U. Wille, Chemistry, John Wiley & Sons
Australia, Milton, 2004.
[54] O. Lin Smith, V. Dragojlovic, Chem. Educator 2012, 17, 206.
[55] D. Klein, Organic Chemistry as a Second Language, 3rd edn., John Wiley & Sons, Hoboken, 2012.
[56] M. S. Silberberg, Chemistry: the Molecular Nature of Matter and Change, Global edn., McGrawHill,
New York, 2013.
[57] D. E. Levy, Arrow Pushing in Organic Chemistry: an Easy Approach to Understanding Reaction
Mechanisms, John Wiley & Sons, Hoboken, 2008.
[58] K. W. Whitten, R. E. Davis, M. L. Peck, G. G. Stanley, General Chemistry, 7th edn., Thomson
Learning, 2004.
[59] G. Baars, H. R. Christen, Allgemeine Chemie: Theorie und Praxis, Sauerländer Verlage, Aarau,
2005.
[60] M. Stieger et al., Elemente: Grundlagen der Chemie für Schweizer Maturitätsschulen, 3. korr.
Nachdruck, Klett und Balmer, Zug, 2010.
[61] W. O. Kermack, R. Robinson, J. Chem. Soc., Trans. 1922, 121, 427.
[62] P. W. Atkins, Physical Chemistry, 5th edn., Oxford University Press, Oxford, 1994.
[63] T. Engel, P. Reid, Physikalische Chemie, Pearson Education Germany, 2006; T. Engel, P. Reid,
Physical Chemistry, Pearson Education, San Francisco, 2006.
[64] H. Kuhn, H.-D. Fçrsterling, D. H. Waldeck, Principles of Physical Chemistry, 2nd edn., John Wiley
& Sons, Hoboken, 2009.
[65] C. G. McCarty, E. Vitz, J. Chem. Educ. 2006, 83, 752.
[66] H. N. Po, N. M. Senozan, J. Chem. Educ. 2001, 78, 1499.
[67] R. deLevie, Chem. Educator 2002, 7, 132.
[68] R. deLevie, J. Chem. Educ. 2003, 80, 146.
[69] H. S. Harned, B. B. Owen, J. Am. Chem. Soc. 1930, 52, 5079.
[70] D. A. MacInnes, T. Shedlovsky, J. Am. Chem. Soc. 1932, 54, 1429.
[71] H. S. Harned, R. W. Ehlers, J. Am. Chem. Soc. 1932, 54, 1350.
[72] R. Chang, K. A. Goldsby, Chemistry. 11th edn., McGraw-Hill, New York, 2013.
[73] S. S. Zumdahl, S. A. Zumdahl, Chemistry, 6th edn., Houghton Mifflin, Boston, 2003.
[74] N. Bjerrum, A. Unmack, Kgl. Danske Vidensk. Selsk. Math.-fys. Medd. 1929, 9 (1), 1 – 208.
[75] CRC Handbook of Chemistry and Physics, 93rd edn., 2012 – 2013; http://www.hbcpnetbase.com
[76] E. A. Guggenheim, Proc. Chem. Soc. 1960, 104.
[77] G. B. Kauffman, J. Chem. Educ. 1980, 57, 779.
[78] G. B. Kauffman, J. Chem. Educ. 1980, 57, 863.
[79] L. Pauling, Grundlagen der Chemie, Verlag Chemie, Weinheim, 1973.
[80] C. E. Mortimer, U. Müller, Chemie: das Basiswissen der Chemie, 8. Aufl., Georg Thieme Verlag,
Stuttgart, 2003.
[81] E. Riedel, Allgemeine und Anorganische Chemie, 8. Aufl., Walter de Gruyter, Berlin, 2004.
[82] A. Burrows, J. Holman, A. Parsons, G. Pilling, G. Price, Chemistry3
: Introducing Inorganic, Organic
and Physical Chemistry, Oxford University Press, Oxford, 2009.
[83] a) T. L. Brown, H. E. LeMay Jr., B. E. Bursten, Chemistry: The Central Science, 10th edn.,
Pearson Education, Upper Saddle River, 2006; b) T. L. Brown, H. E. LeMay, B. E. Bursten,
Chemie: Die zentrale Wissenschaft, 10. aktual. Aufl., Pearson, 2007.
[84] K. P. C. Vollhardt, N. E. Schore, Organic Chemistry, 2nd edn., W. H. Freeman, New York, 1994.
Received August 20, 2013
Helvetica Chimica Acta – Vol. 97 (2014)24
Appendix
Aim and Content. This Appendix contains two Notes which describe in detail how quantitative
aspects of BrønstedÀLowry acidÀbase reaction equilibria often are treated in chemistry textbooks, and
why these treatments frequently are confusing, and why they may lead to equilibrium constants which are
not compatible with tabulated thermodynamic data. The main reason for the confusion is the apparent
refusal of defining equilibrium constants on the basis of activities and the appropriate thermodynamic
standard states. We show that there are clear advantages of applying the concept of activities and
thermodynamic standard states when discussing quantitative aspects of chemical reaction equilibria (see
the main text).
The first Note is on molar-concentration-based reaction constants (Note A1) and the second one on
the molar-concentration-based acidity constant of water in water at 258C (Note A2). In this second Note,
we analyze the arguments which were brought up in literature for obtaining for the acidity constant of
water in water at 258C a value of 10À15.7
(or 10À15.74
) vs. the thermodynamically meaningful value of 10À14.0
(Sect. 4 of the main text).
General Remarks. All equations in this Appendix which do not appear in the main text are labelled
accordingly, e.g., Eqn. A1.
As mentioned in Sect. 3, in the strictly thermodynamic treatment of BrønstedÀLowry acidÀbase
reaction equilibria, there are clear conventions about the standard states which vary depending on the
nature of the species involved in the equilibrium. They are c8 ¼ 1m for dissolved species, p8 ¼ 1 bar for
gaseous reactants, and for the solvent the standard state is the pure liquid solvent. With the assumption of
strongly diluted solutions, i.e., gi,c ¼ 1 and gsolvent,x ¼ 1, the activity of dissolved species Ai is ai,c ¼ ci/c8, and
the activity of the solvent is always asolvent,x ¼ 1.
In principle, there are various ways to define an equilibrium constant (Notes 1 and 2), although such
arbitrarily defined constants may not necessarily be compatible with tabulated thermodynamic data, or
they may not be useful for a quantitative, comparative treatment of the equilibrium state. In any case, it is
important and absolutely neccessary that the rules used are clearly specified. If such specifications are not
made, it is likely that unnecessary confusion arises. This may lead to the assumption that there is an
inconsistency with other values of equilibrium constants for one and the same reaction, formulated with
the same stoichiometric equation, although, in fact, there is only an apparent inconsistency due to the
different rules and conventions used. All what follows originates from a disregard of the strictly
thermodynamic conventions, although this is often not explicitly made clear.
We hope that our explanations here are useful for chemical educators and help understanding how
BrønstedÀLowry acidÀbase reaction equilibria are actually treated in most chemistry textbooks.
Note A1. On the Molar-Concentration-Based Reaction Constants, Kc
and K’c
. In this Note, we explain
definitions of equilibrium constants of BrønstedÀLowry acidÀbase reactions for the following two cases:
Case i, in which the equilibrium constants are expressed in terms of molar concentrations of all
species involved (including the solvent water) without relation to a standard state; and
Case ii, in which the equilibrium constants are expressed in terms of dimensionless molar
concentrations of all species involved (including the solvent, e.g., water).
These cases are actually applied in many general-chemistry textbooks, and in (virtually) all organicchemistry
textbooks we consulted. Furthermore, if the textbook authors consider activity coefficients at
all, they are assumed to be 1, as we did in the strictly thermodynamic treatment outlined in Sects. 3 to 6.
With this, one can say that for Case ii, the equilibrium constant is defined by taking into account the
numerical value of molarity of all species [A1]. Note, that this procedure does not rely on a true
consideration of thermodynamic standard states but rather on the need to have dimensionless
concentrations.
Having clarified this point, we will first discuss Case ii. The equilibrium constant, Kc
CH3COOH for the
reaction in Eqn. 1 following the rules of Case ii can then be defined as in Eqn. A1:
Helvetica Chimica Acta – Vol. 97 (2014) 25
Kc
CH3 COOH ¼
cCH3 COOÀ ðaqÞ
1M
Á
cH3 Oþ ðaqÞ
1M
cCH3 COOHðaqÞ
1M
Á
cH2 OðlÞ
1M
ðA1Þ
In chemistry textbooks, Eqn. A1 often appears with the different chemical species put within square
brackets, as shown in Eqn. A1’:
Kc
CH3COOH ¼
CH3COOÀ
ðaqÞ½ Š Á H3Oþ
ðaqÞ½ Š
CH3COOHðaqÞ½ Š Á H2OðlÞ½ Š
ðA10
Þ
whereby the bracket nomenclature in this particular type of equation should mean dimensionless molar
concentration, i.e., the molar concentration divided by the concentration unit, 1m, i.e., [CH3COOÀ
(aq)]
actually stands for cCH3COOÀ(aq)/1m, [H3Oþ
(aq)] for cH3Oþ(aq)/1m, [CH3COOH(aq)] for cCH3COOH(aq)/1m, and
[H2O(l)] for cH2O(l)/1m. As already mentioned in Sect. 3, such change in nomenclature within the same
book is confusing; the widely accepted nomenclature is that square brackets indicate molar
concentrations only, i.e., [CH3COOÀ
(aq)] ¼ cCH3COOÀ(aq) , [H3Oþ
(aq)] ¼ cH3Oþ(aq) , etc.
To avoid misunderstandings, it would be much better to use square brackets ([...]) for molar
concentrations only. If one likes to stick on the use of brackets, then one could use, for example, square
brackets with a dot as superscript ([...] .
) for molar concentrations divided by 1m, to emphasize the
difference. With this, Eqn. A1 can then be written as
Kc
CH3COOH ¼
CH3COOÀ
ðaqÞ½ Š

Á H3Oþ
ðaqÞ½ Š

CH3COOHðaqÞ½ Š

Á H2OðlÞ½ Š
 ðA100
Þ
Note that Kc
CH3COOH as defined in Eqn. A1 or A1’’ is different from Ka,CH3COOH defined in Eqn. 8, since
Kc
CH3COOH contains in the denominator an additional [H2O(l)] . ¼ cH2O(l)/1m.
With the certainly correct argument that, in this type of dilute aqueous solutions, the concentration
of water as solvent is constant, Eqn. A1 or A1’’ can be multiplied on both sides with cH2O(l)/1m ¼
[H2O(l)] ., and one obtains
Kc
CH3COOH Á
cH2OðlÞ
1M
¼
cCH3 COOÀ ðaqÞ
1M
Á
cH3 Oþ ðaqÞ
1M
cCH3 COOHðaqÞ
1M
¼ K0c
CH3 COOH ðA2Þ
Kc
CH3COOH Á H2OðlÞ½ Š

¼
CH3COOÀ
ðaqÞ½ Š

Á H3Oþ
ðaqÞ½ Š

CH3COOHðaqÞ½ Š
 ¼ K0c
CH3COOH ðA200
Þ
K’c
CH3COOH is here identical with the thermodynamic constant Ka,CH3COOH , as elaborated for ideal solutions
(gi ¼ 1) (see Eqn. 8 in Sect. 3). Therefore, for 258C, at which the concentration of water is 55.33m, i.e.,
cH2O(l)/1m ¼ [H2O(l)].
¼ 55.33, one obtains Eqns. A3 and A3’’.
Kc
CH3COOHð25 
CÞ Á 55:33 ¼
cCH3 COOÀ ðaqÞ
1M
Á
cH3 Oþ ðaqÞ
1M
cCH3 COOHðaqÞ
1M
¼ K0c
CH3COOHð25 
CÞ ¼ Ka;CH3 COOHð25 
CÞ ðA3Þ
Kc
CH3COOHð25 
CÞ Á 55:33 ¼
CH3COOÀ
ðaqÞ½ Š

Á H3Oþ
ðaqÞ½ Š

CH3COOHðaqÞ½ Š
 ¼ K0c
CH3COOHð25 
CÞ
¼ Ka;CH3COOHð25 
CÞ ðA3’’Þ
Note that both constants, Kc
CH3COOH and K’c
CH3COOH, are dimensionless.
Helvetica Chimica Acta – Vol. 97 (2014)26
If as in Case i, no division of ci by 1m would be considered at all, then the constant defined in analogy
to Kc
CH3COOH , i.e., (cCH3COOÀ(aq)) · (cH3Oþ(aq)) · (cCH3COOH(aq))À1
· (cH2O(l))À1
, would also be dimensionless, while
the constant defined in analogy to K’c
CH3COOH, i.e., (cCH3COOÀaq)) · (cH3Oþ(aq)) · (cCH3COOH(aq))À1
, would have the
unit mol lÀ1
¼ m (and another numerical value). In some general-chemistry textbooks, equilibrium
constants indeed are defined according to Case i, i.e., the acidity constants of BrønstedÀLowry acids
have dimensions [A2 – A7].
The reason for defining equilibrium constants by using only molar concentrations for all chemical
species involved, as in Case i is the early version of the mass action law, as originally proposed in 1864 by
Guldberg and Waage (Fig. A1 [A8 – A10]). This law was formulated before Gibbs published his famous
work on chemical thermodynamics [A11], and well before Lewis introduced the concept of activities
[A12][A13].
It has to be pointed out that the equilibrium constant defined by the Guldberg and Waage version of
the mass action law is not necessarily compatible with the thermodynamic equilibrium constant, since it
does not make reference to the different standard states of solvated species and of the solvent (see the
main text).
Let us return to the equilibrium of the reaction in Eqn. 1. If this equilibrium is considered but as
formulated with Eqn. 1’, the same treatment corresponding to Case ii yields a reaction constant,
abbreviated as K’’c
CH3COOH, which is different from Kc
CH3COOH, as shown in Eqn. A4, although Eqns. 1 and 1’
represent one and the same reaction with one and the same stoichiometry.
K00c
CH3COOH ¼
cCH3 COOÀ ðaqÞ
1M
Á
cHþ ðaqÞ
1M
cCH3 COOHðaqÞ
1M
ðA4Þ
Helvetica Chimica Acta – Vol. 97 (2014) 27
Fig. A1. Carlo Maximilian Guldberg (1836 – 1902; left) and Peter Waage (1833 – 1900; right). Photograph
from http://de.wikipedia.org/wiki/Cato_Maximilian_Guldberg, accessed February 28, 2013.
In this expression, the molar concentration of the solvent water is not considered at all, since it does not
appear explicitly in Eqn. 1’. Since cHþ(aq)/1m ¼ cH3Oþ(aq)/1m, one obtains Eqn. A5, which indicates that
K’’c
CH3COOH ¼ K’c
CH3COOH ¼ Ka,CH3COOH .
K’’c
CH3COOH ¼
cCH3 COOÀ ðaqÞ
1M
Á
cHþ ðaqÞ
1M
cCH3 COOHðaqÞ
1M
¼
cCH3 COOÀ ðaqÞ
1M
Á
cH3 Oþ ðaqÞ
1M
cCH3 COOHðaqÞ
1M
¼ K’c
CH3 COOH ¼ Ka;CH3COOH ðA5Þ
The equilibrium constants K’c
CH3COOH and K’’c
CH3COOH obtained as outlined in this Note have the same
numerical values as the thermodynamical equilibrium constant Ka,CH3COOH elaborated in Sect. 3, and all
constants are dimensionless. Therefore, one may argue that it does not matter whether chemical reaction
equilibria are treated in the strictly thermodynamic way outlined in Sect. 3, or in the way illustrated here,
Case ii. The situation changes, however, if water is a BrønstedÀLowry acid and reacts with water as a
BrønstedÀLowry base in water as solvent, i.e., if the self-ionization of water is considered. The
arguments then become a bit dubious, as outlined in Note A2.
Note A2. On the Molar-Concentration-Based Acidity Constant of Water in Water, K’c
*H2O(258C) and Its
Negative Logarithm, pK’c
*H2O(258C). Considering the reaction in Eqn. 18 and taking for all species, i.e., the
dissolved ions HOÀ
(aq) and H3Oþ
(aq) and the solvent H2O(l), molar dimensionless concentrations, Case
ii mentioned in Note A1, one obtains – again by assuming that all activity coefficients are 1 – Eqn. A6
Kc
H2O ¼
cHOÀ ðaqÞ
1M
Á
cH3 Oþ ðaqÞ
1M
cH2 OðlÞ
1M
Á
cH2 OðlÞ
1M
¼
cHOÀ ðaqÞ
1M
Á
cH3 Oþ ðaqÞ
1M
cH2 OðlÞ
1M
 2 ðA6Þ
Multiplying both sides of Eqn. A6 with the constant term (cH2O(l)/1m)2
gives
Kc
H2 O Á
cH2 OðlÞ
1M
 2
¼
cHOÀ
ðaqÞ
1M
Á
cH3Oþ
ðaqÞ
1M
¼ K’c
H2O ¼ Kw ðA7Þ
At 258C, cHOÀ(aq)/1m ¼ cH3Oþ(aq)/1m ¼ 10À7.0
[A14]. Therefore, one obtains Kc
H2O(258C) ¼ 3.27 · 10À18
and
K’c
H2O ¼ 10À14.0
(see Eqns. A8 and A9.)
Kc
H2 Oð25 
CÞ ¼
Kwð25 
CÞ
cH2 OðlÞ
1M
 2
¼
10À14:0
55:33 mol lÀ1
1M
 2
¼
10À14:0
3061:4
¼ 3:27 Á 10À18
ðA8Þ
K’c
H2Oð25 
CÞ ¼ Kwð25 
CÞ ¼
cHOÀ
ðaqÞ
1M
Á
cH3Oþ
ðaqÞ
1M
¼
10À7:0
M
1M
Á
10À7:0
M
1M
¼ 10À14:0
ðA9Þ
Again, all constants Kc
H2O and K’c
H2O ¼ Kw are dimensionless.
The constant Kc
H2O(258C) ¼ 3.27 · 10À18
and its negative logarithm pKc
H2O(258C) ¼ 17.5 are not of any
great direct use, since they are not compatible with tabulated thermodynamic data, and, therefore can, for
example, not be used for a comparison of acid strengths. On the other hand, K’c
H2O (258C) ¼ Kw (258C) ¼
10À14.0
[A15] and its negative logarithm, pKw(258C) ¼ 14.0, are fully compatible with thermodynamic
data (see Sect. 4). K’c
H2O(258C) ¼ Kw(258C) ¼ 10À14.0
is the self-ionization constant for water at 258C,
which is identical with the thermodynamic acidity constant of water in water at 258C, Ka,H2O(258C), i.e.,
pKa,H2O(258C) ¼ 14.0 (see Sect. 4).
The question we address now is how a pKa value for water at 258C of 15.7 was attained, as
mentioned in essentially all organic-chemistry textbooks. Obviously, pKa ¼ 15.7 means Ka ¼ 10À15.7
, a
value which is obtained if Kw(258C) ¼ 10À14.0
is divided by 55.33 ¼ cH2O/1m. This simple mathematical
Helvetica Chimica Acta – Vol. 97 (2014)28
operation is hard to rationalize from a chemical point of view if one takes into account the
BrønstedÀLowry acidÀbase reactivity of water as formulated the way it is done with Eqn. 18, unless one
argues on the basis of rather dubious considerations; to make it clear, Eqn. 18 is not really compatible
with an equilibrium constant of 10À15.7
, as outlined in the following.
The forward reaction of the chemical equilibrium in Eqn. 18 is the description of the reaction of
water as BrønstedÀLowry acid with water as BrønstedÀLowry base in water as solvent, whereby H2O
simultaneously plays three roles, as BrønstedÀLowry acid, as BrønstedÀLowry base, and as solvent. If
comparison is made with the forward reaction of Eqn. 1, one realizes one fundamental difference. In the
forward reaction of Eqn. 1, water plays only two roles, as BrønstedÀLowry base and as solvent, while
CH3COOH(aq) is the BrønstedÀLowry acid. One may consider a kind of unified view of the two
reactions shown in Eqns. 18 and 1 by replacing Eqn. 2 with Eqn. A10:
H2O(aq) þ H2O(l) > HOÀ
(aq) þ H3Oþ
(aq) (A10)
Eqn. A10 certainly is strange since it implies something which is more hairsplitting than reasonable.
In any case, for Eqn. A10 one obtains – with all the assumptions and conventions mentioned above in this
chapter – for the acidity constant of H2O first Eqn. A11
KcÃ
H2O ¼
cHOÀ ðaqÞ
1M
Á
cH3 Oþ ðaqÞ
1M
cH2 OðaqÞ
1M
Á
cH2 OðlÞ
1M
ðA11Þ
Multiplying on both sides of Eqn. A11 with cH2O(l)/1m yields Eqn. A12
KcÃ
H2 O Á
cH2OðlÞ
1M
 
¼
cHOÀ ðaqÞ
1M
Á
cH3 Oþ ðaqÞ
1M
cH2 OðaqÞ
1M
Á
cH2 OðlÞ
1M
Á
cH2OðlÞ
1M
¼
cHOÀ ðaqÞ
1M
Á
cH3 Oþ ðaqÞ
1M
cH2 OðaqÞ
1M
¼ K’cÃ
H2O ðA12Þ
For 258C, cHOÀ(aq) ¼ cH3Oþ(aq) ¼ 10À7.0
m, and if one uses for cH2O(aq)/1m a value of 55.33, the numerical
value of 10À15.7
is obtained for K’c
*H2O, i.e., as negative logarithm of this constant, pK’c
*H2O(258C), a value of
15.7. Obviously, however, it cannot be that cH2O(aq) ¼ cH2O(l) ¼ 55.33m in one and the same solution, if
H2O(aq) and H2O(l) are considered to be different species as it appears from Eqn. A10. At 258C, the
total amount of water in 1 l is 55.33 mol, not 110.66 mol (see Sect. 3). Therefore, all the above arguments
are dubious and should not be used at all since they are wrong. In contrary, the considerations made
here are useful to explain the students in a hopefully convincing way why the pKa(258C) value of water in
water cannot be 15.7, if pKw(258C) ¼ 14.0.
Two questions remain. Question 1: Why should a division of Kw(258C) by 55.33 be appropriate for
expressing the acidity of water? Question 2: Who was, or who were, the first to define the equilibrium
constant for the reaction of water as BrønstedÀLowry acid with water as BrønstedÀLowry base such that
a value of 10À15.7
at 258C is obtained?
The answer to Question 1 can be given by considering instead of Eqn. 18 Eqn. 18’, or even – to be
somehow consistent with Eqn. 1’ – the following equation:
H2O(aq) > HOÀ
(aq) þ Hþ
(aq) (A10’)
Eqn. A10’ certainly is unusual but it is written here to emphasize the lines of thinking and the ideas of
having a similar formalism as in Eqn. 1’. H2O(aq) is the same as H2O(l), i.e., liquid water.
If one considers the concentration of H2O(aq) at 258C as the same as the concentration of H2O(l) at
258C, although with Eqn. A10 it is assumed that they represent two different species, one obtains
Eqn. A13
KcÃ
H2Oð25 
CÞ ¼
Kwð25 
CÞ
cH2 OðaqÞ
1M
¼
10À14
55:33M
1M
¼
10À14
55:33
¼ 1:81 Á 10À16
¼ 10À15:7
ðA13Þ
Helvetica Chimica Acta – Vol. 97 (2014) 29
and with this value, one obtains Eqn. A14
pKcÃ
H2 Oð25 
CÞ ¼ À log KcÃ
H2Oð25 
CÞ
 
¼ À log 1:81 Á 10À16
À Á
¼ 15:7 ðA14Þ
Note that for the sake of distinguishing the water acidity constant, as obtained with the conventions
just outlined, from the thermodynamic constant Ka,H2O ¼ Kw (see Sect. 4), we use a different abbreviation
that is Kc*H2O (see Eqn. A13).
The true significance of the thermodynamically incorrect equilibrium constant, Kc*H2O, and its negative
logarithm, pKc*H2O, is questionable. It is very doubtful whether pKc*H2O is meaningful for a comparison with
the acidity of organic acids which are almost insoluble in water [A16]. In this latter case, the pKa values
reported for organic acids in organic-chemistry textbooks, e.g., the pKa values of CH4 or benzene are
relative values and cannot be compared with the thermodynamic values determined for conventional
acids that are soluble in aqueous solution. Therefore, these pKa values should not be included in tables
together with thermodynamic pKa values. Alternatively, corrections of listed pKa values for organic
acids should be made, as already mentioned previously [A17]. In any case, the estimated acidity of very
weakly acidic organic molecules are approximate values anyway, although these approximate, relative
acidity values are extremely useful when discussing the reactivity of organic molecules [A18][A19].
If two differently defined pKa values of two acids are compared, then one may draw wrong
conclusions. This is what seems to be the case if the acidity of water is compared with the acidity of
methanol [A20 – A23]. Using for both BrønstedÀLowry acids the same thermodynamic definitions, then
– at 258C – water in water (pKa,H2O(258C) ¼ 14.0) is more acidic than methanol in water
(pKa,CH3OH(258C) ¼ 15.6).
In any case, to avoid confusion, it would be better to use different abbreviations for differently
defined reaction constants, for example Ka,H2O and K*a;H2O, as proposed here.
An answer to Question 2 – who was, or who were, the first stating that pKa,H2O(258C) ¼ 15.7 – cannot
be given with full certainty. They were the ones who did the pioneering work on the quantitative
determination of the relative acidity of weakly acidic organic molecules by non-aqueous competition
experiments [A24 – A27]. They thought – without any experimental data – that the pKa value of water
at 258C has to be 15.7 and the pKa value of H3Oþ
 À 1.8, although the arguments put forward at that time
[A24] were not very convincing. For example, for the reaction of H3Oþ
(aq) with water, the equilibrium
considered was formulated as in Eqn. A15 [A24]:
H3Oþ
þ H2O(solvent) > H3Oþ
þ H2O (A15)
and the argument was that H2O(solvent) is not included in the equilibrium constant, while the
concentration of H2O is taken as 55.33m (or 55.5m as in the original report [A24]). Therefore, an
equilibrium constant K is obtained with a value of 55.33m (or 55.5m, [A24]). The following text is taken
from an organic textbook [A28]: Note that the Ka for water is obtained by dividing Kw by the
concentration of water, 55.5 moles LÀ1
. This change is necessary to put all of the ionizations on the same
scale and in the same units. Recall that the ion product of water, Kw, has units of moles2
LÀ2
or m2
, whereas
Ka values are given in units of moles LÀ1
or m. Obviously, these arguments of the necessity of having
equilibrium constants with uniform units are not compatible at all with the thermodynamic conventions
outlined in Sect. 3 – 6. As repeatedly emphasized in the main text, thermodynamic equilibrium constants
are dimensionless, including Kw; they all have no units.
Final Remarks. There are convincing arguments for introducing and using activities and appropriate
thermodynamic standard states when teaching quantitative aspects of chemical equilibria (main text). If
properly and systematically performed, existing confusions in chemistry textbooks can be eliminated, and
with this, full compatibility of thermodynamic equilibrium constants and tabulated thermodynamic data
can be attained. This is the case not only for the BrønstedÀLowry acidÀbase equilibria discussed in this
review, but also for other chemical equilibria, including the water solubility of sparingly soluble salts with
their characteristic solubility products Ksp(T).
Helvetica Chimica Acta – Vol. 97 (2014)30
REFERENCES
[A1] L. Jones, P. Atkins, Chemistry: Molecules, Matter, and Change, 4th edn., W. H. Freeman, New
York, 2000.
[A2] L. Pauling, Grundlagen der Chemie, Verlag Chemie, Weinheim, 1973.
[A3] R. E. Dickerson, I. Geis, Chemistry, Matter, and the Universe: an Integrated Approach to General
Chemistry, W. A. Benjamin, Menlo Park, 1976.
[A4] C. E. Mortimer, U. Müller, Chemie: das Basiswissen der Chemie, 8. Aufl., Georg Thieme, Stuttgart,
2003.
[A5] E. Riedel, Allgemeine und Anorganische Chemie, 8. Aufl., Walter de Gruyter, Berlin, 2004.
[A6] T. L. Brown, H. E. LeMay, B. E. Bursten, Chemie: Die zentrale Wissenschaft, 10. aktual. Aufl.,
Pearson Deutschland, 2007.
[A7] A. Burrows, J. Holman, A. Parsons, G. Pilling, G. Price, Chemistry3
: Introducing Inorganic, Organic
and Physical Chemistry, Oxford University Press, Oxford, 2009.
[A8] P. Waage, C. Guldberg (transl. by H. I. Abrash, of the original paper published in 1864 in
Norwegian), J. Chem. Educ. 1986, 63, 1044; C. M. Guldberg, P. Waage, Forhandlinger: Vidensk.Selsk.
Christiania 1864, 35.
[A9] K. J. Mysels, J. Chem. Educ. 1956, 33, 178.
[A10] E. A. Guggenheim, J. Chem. Educ. 1956, 33, 544.
[A11] J. W. Gibbs, Trans. Connecticut Acad. Arts Sci. 1875 – 1878, 3, 108 and 343.
[A12] G. N. Lewis, Proc. Am. Acad. Art Sci. 1907 – 1908, 43, 259.
[A13] G. N. Lewis, J. Am. Chem. Soc. 1908, 30, 668.
[A14] N. Bjerrum, A. Unmack, Kgl. Danske Vidensk. Selsk. Math.-fys. Medd. 1929, 9 (1), 1 – 208.
[A15] CRC Handbook of Chemistry and Physics, 93rd edn., 2012 – 2013; http://www.hbcpnetbase.com.
[A16] E. M. Woolley, J. Tomkins, J. G. Hepler, J. Solution Chem. 1972, 1, 341.
[A17] R. A. Cox, Adv. Phys. Org. Chem. 2000, 35, 1.
[A18] M. B. Smith, Organic Chemistry: an AcidÀbase Approach, CRC Press, Taylor & Francis, Boca
Raton, 2011.
[A19] R. D. Rossi, J. Chem. Educ. 2013, 90, 183.
[A20] J. Hine, M. Hine, J. Am. Chem. Soc. 1952, 74, 5266.
[A21] IUPAC, Analytical chemistry division, commission on electroanalytical chemistry, Pure Appl. Chem.
1985, 57, 855.
[A22] D. Gao, J. Chem. Educ. 2009, 86, 864.
[A23] H. I. Abrash, J. Chem. Educ. 2001, 78, 1496.
[A24] E. M. Arnett, Prog. Phys. Org. Chem. 1963, 1, 223.
[A25] D. J. Cram, in Fundamentals of Carbanion Chemistry, Ed. D. J. Cram, Academic Press, London,
1965, Chapt. I, p. 1.
[A26] A. Streitwieser Jr., J. H. Hammons, E. Ciuffarin, J. I. Brauman, J. Am. Chem. Soc. 1967, 89, 59.
[A27] J. B. Hendrickson, D. J. Cram, G. S. Hammond, Organic Chemistry, 3rd edn., McGraw-Hill
Kogakusha, Tokyo, 1970.
[A28] A. Streitwieser, C. H. Heathcock, E. M. Kosower, Introduction to Organic Chemistry, 4th edn.,
Macmillan, New York, 1992.
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