Fig. 26.8. Experimental and theoretical molar conduct AgN03 0.1 0.2 0.3 VW(moI kg"1] 0.4 Comment. The result is not at all bad considering the theoretical difficult. ■• l dealing with a dynamical system of this complexity. Note that the plot c I 1 against «jc is a useful way of making the extrapolation to find A%, Equation (26.1.18) has the same c1/2 concentration-dependence as the empirical Kohlrausch expression, eqn (26.1.3). Furthermore, the slop^ot the curves are predicted to depend on the valence type (z appears in íl c constants A and B). Some comparisons between theory and experiment are shown in Fig. 26.8, and this shows how well the theory account^ W* the observations at low concentrations. The success of the Debye-Hückel-Onsager equation suggests tha ''> t model of ion-ion interactions is substantially correct. A further te r K obtained by investigating what happens when the effect of the iopic atmosphere is eliminated. This can be done in a variety of ways. In unw the conductivities are measured at very high frequencies; then the cental ion is moved backwards and forwards very rapidly, and the retardny effects of the ionic atmosphere ought to average to zero. This is the De11 <■ -Falkenhagen effect, and the predicted increase in mobility at high freq'ien-cies has been observed. The other way of eliminating the effect ol the atmosphere is to move the ions so rapidly that no atmosphere has linuto build up. The Wien effect is the observation of higher mobilities at *«■ ^ high electric fields. (There are two Wien effects. The^rsi Wien effect is i * one just described; the second Wien effect is the enhancement of the dep ^ of ionization of an ionogen, or weak electrolyte, by the applied field i This model of the interactions fails when the concentration bečenu* large, because ions tend to stick together in pairs, and even triples. 'I'''" can be seen quite clearly from X-ray analysis of ionic solution, wh»Ml' peaks of scattering can be interpreted in terms of definite ion-ion distant 20 2 Fundamental aspects of molecular transport i- In this section we begin to draw together the threads of the discussion in this chapter and the last. We do so on the basis of thermodynamic and statistical principles and find that we can make a variety of important and useful connections between properties relating to the motion of molecules and ions in fluids. Diffusion: the thermodynamic view. In Part 1 it was shown that the thermodynamic 1 property that governs the direction of spontaneous change is the thermo- dynamic chemical potential (p. 182). When unit amount of solute is shifted from a region where its chemical potential is ß(l) to one where it is n(2) the work required is w ~ ju(2j—/i(l). Suppose the chemical potential depends on the position x in the system, then the work involved in transferring unit amount of material from x to x + dx is dw = fj.(x + dx)—n(x) = [Ju(x) + (d^/dx)dx]— ß(x) = (d/í/dx)dx. (The derivatives ought strictly to be partial derivatives, and the transfer ought to be carried out under conditions of constant pressure and temperature—see p. 258 for details.) In classical mechanics the work required to shift an object through a distance dx against a force #" is dw — — ^ dx. By comparing the last two equations we see that the gradient of the chemical potential acts like a force. We shall therefore write iMl) ^ ^ -(d/i/dx). There is no real force pushing molecules down the slope of chemical potential, for that is their natural drift as a consequence of the Second Law and the hunt for maximum entropy: nevertheless, thinking in terms of these phantom, effective, thermodynamic forces can be very useful, as we shall see. In a solution where the concentration is c the chemical potential of an ideal solute is (p. 226). A 2.2)° p = n& + R T In (c/mol dm "3). If the concentration depends on position, the thermodynamic force acting is &= -(d/dx)[/ie + i?rin(c/moldm-3)] = -(RT/c)(dc/dx) because /ie is independent of position, and dln//dx = (l/f)df/dx. The form of the last equation lets us derive Fick's Law of Diffusion (that flux is proportional to the concentration gradient, eqn (25.3.1)) from a thermodynamic viewpoint. We suppose that the flux is the response of the molecules to some force. If the force per unit amount is #", the force tlux of material, ./(matter), is proportional to the impressed f< -.^ r /(matter) oc c#". But the effective force is given by eqn (26.2.3), m,; ./(matter) cc ~RT(dc/dx) = ~kT(dJT/áx) and we have the flux as proportional to the concentration gr um.iu , accord with Fick's Law (Jí is the number density of molecules, > ; It is more convenient to develop a different line of argument. tl ij r interpret the flux of particles as the product vJÍ ~ vcL, where ^ rip average velocity and c their concentration. Then Fick's Law, vJÍ = J = -D(dď/áx) = -DL(dc/dx) reads cv= ~D(dc/dx) or, using eqn (26.2.3), v = ~(D/c)(dc/dx) = (D/kT)^. Therefore, in response to a unit force, the molecules diffuse wm'1 • Jru: velocity of magnitude D/kT. We know, however, that the mob:, i ut.-, ion is related to the electrical force on it. Since the mobility i-* iiunc; through 5 - uE, and since the electrical force is ezE, it follows .!■ U s = uE = (u/ze) (ezE) = (u/ze)^, and therefore the drift speed under the influence of unit for« i- in .v The nature of the force is irrelevant; therefore the two drift speeds f /> - /"■ * and (u/ze)^ may be identified: then u/ez = D/kT, and so we arrive at the very important result, known as the Einstein relation: D = ukT/ez , I connecting the diffusion constant and the mobility. The last relation can be developed further in two directions. In [Ik n'i' place it can be used to relate the molar conductivity to the uiiiiKfc* constants of the ions, D+ and Z)_. We write (for 1:1 salts) Am = zF(u++u-) = (z2eF/kT)(D+ +i>_) and so arrive at the Nernst-Einstein relation: Am «= (z2Ez'/RT)(D+ +Z)„). -Í One application of these expressions is to the determination of I (it* ,i*n * diffusion constants from conductivity measurements; the other ^ ,i'il1' calculation of conductivities on the basis of models of ionic diff* *■ ■*" l"*-L below). the mobility to the viscosity. By combining the expressions s = ezEf(>%na and s = uE, the first being the expression for the drift speed defined in eqn (26.1.9), we are able to write u = ez/ÓTtna. Since the Einstein relation is u = ezD/kT, the two may be equated and combined into the 2.6)° I Stokes-Einstein relation: D == kTjénna > which connects the diffusion constant and the viscosity of liquids. An - important feature of this result is that it is independent of the charge of the diffusing species, and therefore it also applies in the limit of vanish-ingly small charge, or neutral molecules. This means that we may use the Stokes-Einstein relation to estimate the diffusion constant from measurements of the viscosity. It must not be forgotten, however, that it is an approximation, being based on the assumption of the validity of the Stokes formula for the viscous drag. Some diffusion coefficients are listed in Table 26.4. 3le 26.4. Diffusion coefficients at 25 °C, J5/10"9 m2 s"1 I2 in hexane 4.05 H2 in CC14 9.75 I2 in benzene 2.13 N2 in CC14 3.42 CCI4 in n-heptane 3.17 02 in CC14 3.82 Glycine in water 1.055 Ar in CC14 3.63 Dextrose in water 0.673 CH4 in CCU 2.89 Sucrose in water 0.521 Water in water 2.26 Methanol in water 1.58 Ethanol in water 1.24 Ions in water: H+ 9.31 OH" 5.30 Li+ 1.03 F- 1.46 Na+ 1.33 cr 2.03 K+ 1.96 Br~ 2.08 r 2.05 Source: American institute of Physics Handbook, McGraw-Hill, and (for the ions) eqn (26.2.4) and Table 26.2. f" »»iple (Objective 12,13,14). Find the diffusion coefficient, the molar conductivity, and the effective hydrodynamic radius of the S04" ion in water at 25 °C. • Method. In Table 26.2 the mobility of the ion is given as 8.29 x 10"4 cm2 s"1 V"1. Relate this to D through eqn (26.2.4). Then use eqn (26.2.5) to relate D (now written D_) to A_ (the anion contribution to AJ. Calculate the effective radius a_ from _(8.29xl0-4cm2s-1V-1)xq.3807xl0-23JK-1)x{298.15K) 2x(1.6022xl(T19C) = 1.065 x 10-5 cm2 s-1. From eqn (26.2.5), _22x(9.6485xl04Cmol~1)2x(1.065xl0-5cm2s~1) (8.3144 JK"1 mol" *) x (298.15 K) = 160.0Q"1cm3mol-1. , From eqn (26-2.6), ü_ = kT/6izt}D- _________(1.3807xlQ-23JK~1)x(298.15K)_______ ""67Tx(1.00xlO"3kgm-1s-1)x(1.065xlO-5cm2s~1) = 2.051 x 10"10 m or 205 pm or 2.05 Á. • Comment. The bond length in SO2." is 144 pm, and so the radius calculated h« (the radius of a sphere representing the molecule) is plausible and compatible wit only a small degree of solvation. Some experimental support for these ideas comes from conductivi measurements, because the empirical Walderis rule is that the prodli A^rj should be approximately constant for the same ions in differei solvents. Since A^, cc u, and u oc l/n, we can see the theoretical basis W this rule. Its applicability is muddied by the role of solvation: different solvents solvate ions to different extents, and so the ions' effective* hydrodynamic radii depend on their nature: both a and n vary with th|, solvent. i1J ? Diffusion as a time-dependent process. We turn now to the discussion of time-dependent diffusion processes, in which some distribution of concentration, or oi temperature, etc., is established at some moment, and then allowed to, disperse without replenishment. One example is a metal bar heated rapidly at one end and then allowed to reach equilibrium, and another is when i layer of solute is spread on the surface of a solvent and the concentration distribution in the solution changes as it dissolves. In order to treat a time-dependent diffusion process we shall concentrate on the diffusion of matter, but the arguments are easily modified to applj to other properties. We fix our attention on a small slab of the systen extending from x to x + Ax, and of cross-sectional area A, Fig. 26.9. Lej the concentration at x be Jr{x,t) at the tíme í, then the increase % concentration inside the slab (of volume AAx) by virtue of the flux from the left is dJT{x,t)/dt = J(x,t)A/AAx = J(x,t)/Ax because JA is the number of particles that enter through a window of area A in each unit time interval. There is also a flow out of the right-hand window; if the flux is J(x + Ax) the concentration inside the slab changes due to this efflux with a rate dJT{x,t)ldt~ -~J(x + Ax,t)A/AAx = -J(x + Ax,t)/Ax, the negative sign appearing because the concentration in the slab decreases when the flow is to the right (J positive). Therefore the total rate of change of concentration is djV{x, t)/dt = J(x, í)/Ax - J(x + Ax, ť)/Ax. The fluxes can now be related to the concentration gradients at each window. Using Fick's Law we can write J(x,t)-J(x + Ax,t) = {-DídjV(x,t)/dxy]}-{-D[djť(x + Ax,t)/dx)~]} = D(ô2jr(x,ť)/dx2)Ax. Substituting this back into the expression for the rate of change of the concentration in the slab leads to the (26.2.7) diffusion equation: {dJŤ(x, ť)/dť) = D(ôKr(x, t)jdx2). [/ This is also sometimes called Fick's Second Law. First, a word about the general form of this equation. We see that the rate of change of the concentration is proportional to the curvature (the second-derivative) of the concentration dependence on the distance. If the concentration changes rapidly from point to point the rate at which the concentration changes with time is correspondingly rapid. If the curvature is zero, the concentration does not change with time. For example, if the concentration falls linearly with distance, the concentration at any point remains constant because the inflow of concentration is balanced by the outflow. The diffusion equation can be regarded as a mathematical eqn (26-2.6). The viscosity of water at 25 °C is 1.00 cp (1.00 x 10"3 kg m""1 s Remember that J = C V and V A"l = a. • Answer. From eqn (26.2.4), Z>_ = u^kTjez __ (8.29 xlQ-4cm2s~1V"1)x (1.3807 x 10~23 JK"I)x(298.15K) 2 x (1.6022 x 10" I9C) = 1.065 xl0~3cm2s_1. From eqn (26.2.5), 22x (9.6485 xlQ4Cmor xf x(1.065 x lO'^m^-1) (8.3144 JK"1 mol"1) x(298.15K) = 160.0n-1cm2mor1. From eqn (26.2.6), a. = kT/6TcrjD- (1.3807 x 10~23 JK-1) x (298.15K) ~ 6re x (1.W x 10"3kgm"^"V(1.065 xl0~scm2s"1) = 2.051 x 10- 10m or 205pm or 2.05Á. • Comment. The bond length in SO2- is 144 pm, and so the radius calculated I.„r. (the radius of a sphere representing the molecule) is plausible and compatible i ii" only a small degree of solvation. Some experimental support for these ideas comes from conducts i1.. measurements, because the empirical Walderfs rule is that the prodi^t A^rj should be approximately constant for the same ions in diffeKW solvents. Since A^ cc h, and u cc 1/n, we can see the theoretical basis '">] this rule. Its applicability is muddied by the role of solvation: diffeicnt solvents solvate ions to different extents, and so the ions' effecnw hydrodynamic radii depend on their nature: both a and t} vary with ilk' solvent. Diffusion as a time-dependent process. We turn now to the discussion of time-depend .nu diffusion processes, in which some distribution of concentration, oi *'t temperature, etc., is established at some moment, and then allowed '^ disperse without replenishment. One example is a metal bar heated rapj.'l. at one end and then allowed to reach equilibrium, and another is when •. layer of solute is spread on the surface of a solvent and the concentrali'"1 distribution in the solution changes as it dissolves. In order to treat a time-dependent diffusion process we shall concenti i * on the diffusion of matter, but the arguments are easily modified to appl* to other properties. We fix our attention on a small slab of the sys\ " extending from x to x + Ax, and of cross-sectional area A, Fig. 26.9. L o" the concentration at x be Jí{x,t) at the time ŕ, then the increase ''' concentration inside the slab (of volume AAx) by virtue of the flux fiiV the left is djf{x,t)jdt = J(x,t)A/AAx = J(x,t)/Ax Fig. 26.9. The diffusion oi material into a region. x+ A.v because JA is the number of particles that enter through a window of area A in each unit time interval. There is also a flow out of the right-hand window; if the flux is J(x + Ax) the concentration inside the slab changes due to this efflux with a rate dJV(x,t)/dt = -J(x + Ax,t)A/AAx = -J(x + AX>t)/Ax, the negative sign appearing because the concentration in the slab decreases when the flow is to the right (J positive). Therefore the total rate of change of concentration is djV{x, t)/dt = J(x, t)/Ax - J(x + Ax,'t)/Ax. The fluxes can now be related to the concentration gradients at each window. Using Pick's Law we can write , J(x,t)-J(x + Ax,t) = {-D[ö^(x,r)/öx)]}-{-D[5^(x + Ax,i)/^)]} ■<^Kh->+ dx Ax = D(ô2jr(x,ť)/ôx2)Ax. Substituting this back into the expression for the rate of change of the concentration in the slab leads to the . diffusion equation: {ô^{x,t)ldt)^D{d2X{xtt)ldx2). This is also sometimes called Fick's Second Law. First, a word about the general form of this equation. We see that the rate of change of the concentration is proportional to the curvature (the second-derivative) of the concentration dependence on the distance. If the concentration changes rapidly from point to point the rate at which the concentration changes with time is correspondingly rapid. If the curvature is zero, the concentration does not change with time. For example, if the concentration falls linearly with distance, the concentration at any point remains constant because the inflow of concentration is balanced by the outflow. The diffusion equation can be regarded as a mathematical _____„v.— u. u.w.1Ui.Uiu*^ iwuuii umi jmiurc nas a natural tendency^ eliminate the wrinkles in a distribution. The diffusion equation is a second-order differential equation in spa<. and first-order in time, and therefore in order to arrive at a solution v„ have to specify two boundary conditions for the spatial dependence arj a single initial condition for the time dependence. This can be illustrate by the specific example of a solvent in which the solute is coated on one surface. At time zero the initial condition is that all the N0 solute particles are concentrated on the yz-plane at x = 0. The boundary conditions aie that the concentration must be everywhere finite, and the total number present must be N0 at all times. The solution of the diffusion equation having these as conditions is (26.2.8) Jf{x,t) = {N0/A(nDt)1/2}exp(~x2/4Dt) as may be verified by direct substitution. The form of the result at different times is shown in Fig. 26.10, and it is clear that the concentration = \ xJT&,t)AdxlNQ = (l/nDt)1'2 \ xQ~xV4Dtdx Jo Jo (26.2.9) = 2(Dt/x)112. The average distance varies as the square root of the time lapse. Tl an important general result which we shall return to later. If we us< Stokes-Einstein relation for the diffusion constant the mean dist covered in a solvent of viscosity n by particles of radius a is (26.2.10) = (2kTßn2r}ä)1/2y/t. The root mean square distance covered is x = -J(x2)>, and its value i (26.2.11) x = V = x2jV(x,t)Adx/N0 = (2Dt)1/2. This is a valuable measure of the spread of the particles when they allowed to migrate in both directions (for then =-0). The value c for molecules having D = 5 x 10~6 cm2 s_1 is shown in Fig. 26.11: you ■ see how long it takes for diffusion to increase x to about 1 cm in an unstir solution. The proportion of particles which remain within a distance ? \Dt=0.05 Fig. 26.10. Diffusion of a solute from a plane surface. the origin is a useful number to have, because the mean might not convey enough information. Since the number in the slab at x is Jr(x, t)A dx, the number in all the slabs up to the one at x is the sum (integral) N(x for then it becomes /•s n/2'n N(x Š x,ť) = {N0/Á(nDty2} dxe-xl'*Dl = (W0M)(2/n1'3) c^dy. It happens that the integral (2/7t1'2)Jge-yidy is a standard mathematical form known as the error function, and written erf z. Standard tables of these are available, Table 25.1, and the value of erf (l/2)l/2, which is what we require, is 0.68. Orlm Fig. 26.11. Root mean square distance covered by particles with D = 5 x 10" cm-1 s z c-l .-s that distance from the origin. The direction of the steps is different or$ each occasion and so the net distance of diffusion must take this into* account. We shall simplify the discussion by allowing the particle to move? only along a straight line, the x-coordinate, but we must not forget thatr,j in a real system a particle is free to move in three dimensions. We shalíj also confine our attention to a model in which the particle can jump withj equal probability through a distance d to the right or d to the left. This is called the one-dimensional random walk; we first met it in Chapter 24. * Our task is to find the probability that a molecule will be found at a-< distance x from the origin at a time ŕ. During that time interval it will ^ have taken t/x steps: we shall write n = ŕ/r. Many of these steps were steps' to the right; many were steps to the left. If nR is the number of steps to ) the right and nL the number to the left, not only can we write the total ' number of steps as n — nR + nL, but we can also write the net distance, travelled as x = nRd—nLd. The probability of being at x after n steps of length d is the probability that of the n steps, nR occurred to the right, nL occurred to the left, and What is the total number of possibilities for left or right steps? Since each step may occur in either of two directions (\eit or right) the total , number of possibilities is 2". How many ways are there of taking nR of the n steps to the right? This is the same as the number of ways of choosing nR objects from n possibilities, irrespective of the order: this is n\/nR\(n — nR)\. We can check this in the case of 4 steps, and ask what is the number of ways of taking 2 right steps. There are 24 possible step sequences: IXLL LLLR LLRR LRRR RRRR LLRL LRLR RLRR LRLL LRRL RRLR RLLL RLLR RRRL RLRl RRLL and clearly there are 6 ways of taking 2 steps to the right and 2 to the left, which tallies with the expression 4!/2!2! = 6. The probability that the particle is at the origin after 4 steps is therefore 6/16. The probability that it is at x = Ad is 1/16 because, in order to be there, all four steps must be towards the right, and there is only one way of organizing that. Returning now to the general case we see that the probability of being at x after n steps, each of length d, is P(x)=n\/nR\(n-nR)\2", with n = nR + nL and x/d = nR~nL. Since _ i Kh + x/d), « - nR - K» - */$> it follows that (26.2.12) P(x) = B!/{Ö(n + s)]![«n-s)3!2-}, '- where s = x/d. This expression does not seem to resemble the Gaussian distribution of probability, such as eqn (26.2.8), and so it looks as though the model of a random walk underlying a diffusion process is quite wrong. This, however, is not the case: the last equation becomes identical to the Gaussian distribution when we examine the limit in which the number of steps becomes very large. The algebraic manipulation of this equation is based on the approximate formula for factorials of large numbers first used in Chapter 20 (p. 668). When N is a large number it is possible to use Stirling's approximation: (26.2.13) lnNlx(N+%)\nN-N + \n(2n)112. This is a more accurate form of the approximation than the one used earlier. Even when N is quite small this expression is quite good. For example, instead of 10! = 3.629 x 106 it gives 10! at 3.60 x 106; when larger numbers are involved we can be very confident indeed about the results it gives. Taking logarithms of eqn (26.2.12), and then allowing n to be large, leads (after quite a lot of algebra) first from lnjP = lnn!-ln{[^n-i-s)3!)-ln([i(n-s)]!)-nln2 to \nP*\n(2/nn)ll2-&n + s+l)\n(\+s/n)-$(n-s+l)\n(l-s/n). If we allow sjn to be a small number (so that x must not be a gi. (. distance from the origin) we can use the approximation ln(l + y) ss y, ai, obtain \nPx\n{2/nn)1/2-s2/2n, or Px(2/nn)1/2Gxp(-s2/2n), which is already of a Gaussian form. Now replace s by x/d and n by t/x. We obtain (26.2.14) P(x,t) = (2x/nt)1/2exp{-xzx/2td2), and this has precisely the form of jV(x,t)/N0 given in eqn (26.2.8) a-solution of the diffusion equation. (The differences of detail arise fr» »i.i allowing the particle to migrate in both directions away from x = 0, i > letting it be found only at discrete points separated by d instead of beingl anywhere on a continuous line.) Therefore we can be confident that the-diffusion can be interpreted as the result of a very large number of small" steps in random directions. This also indicates the region of invalidity oj the diffusion equation: we should not expect it to apply at times so shorť that the particles have had time to take only a few steps. ~< Finally we can make use of the identity of form of the two distributions: to obtain yet another expression for D. Comparison of the two exponents, leads to the identification á 2d2/x~4D, V and therefore we come to the (26.2.15) -" Einsteiň-Smoluchowski relation: D — \d2jx. Example (Objective 17). Suppose that the SO2." ion jumps through about its own diamet every time it makes a move in aqueous solution. How often does it change position? Z • Method. The diffusion coefficient was found in the last Example to be 1.065 x 10" 5 cm2 s"1, and the effective radius was found thereto be 205 pm. Find x Trom eqn (26.2.15). • Answer. From eqn (26.2.15), t = d2/2D = (2x205xlO-12m)2/2x(1.065xlO-9m2s-1) = 7.89xlO-ns. • Comment. The big, heavy SO2." ion jumps through its diameter in about 8 x 10"u s. If the ion were imagined as jumping through a distance equal to tb.% diameter of a water molecule (as 150 pm) the jump time would be about 1.1 X-10" " s. i The Einstein-Smoluchowski relation is a central connection between the microscopic properties of the size (d) and rate (1/x) of a moleculat jump and the macroscopic properties of diffusion constant and viscosity (via the Stokes-Einstein relation eqn (26.2.6)). This also brings the discussion full circle and back to the properties of gases. For if d/x is interpreted as a mean velocity of the molecules undergoing diffusion, and the jump length d is called a mean free path and written X, the Einstein-Smoluchowski equation reduces to D — %Xc, which is the same as that obtained for the diffusion constant from the kinetic theory of gases. This shows that the diffusion of a perfect gas can also be interpreted as a random walk through an average path length X. Mimmary of the general conclusions. The chapter began by examining various aspects of the motion of ions in solution. We saw that the conductivity could be expressed in terms of the mobility of the ions. We also saw that any species could be regarded as moving under the influence of an effective force SF if its chemical potential varied from place to place, and we identified & with — d/Vdx, eqn (26.2.1). The thermodynamic force led to the construction of Fick's First Law of Diffusion. We saw on quite general arguments, that if the particle was subjected to a unit force, it acquired a drift speed D/kT. This led to the Einstein relation between D and the mobility, eqn (26.2.4), and the Nernst-Einstein relation, eqn (26.2.5), between conductivity and D, see Box 26.1. Incorporation of the Stokes frictional force into the argument led to the Stokes-Einstein relation, eqn (26.2.6), between D and the viscosity, valid for molecules of any charge (including zero). We next set up equations for dealing with time-dependent diffusional processes, and derived the basic diffusion equation, eqn (26.2.7). The solutions of this equation could be reproduced, we found, if we modelled the diffusion process as a series of small steps of length d occurring with a frequency Box 26.1 Transport properties in solution Einstein-Smoluchowski relation between jump size ď and jump time t: D = d2l2x. Stokes-Einstein relation between diffusion coefficient D and solution viscosity n: D = kT/Giciia. Einstein relation between diffusion coefficient and ion mobility «+: D± = u+kT/ez = u±RT,'zF. Nernst-Einstein relation between diffusion coefficient and ion conductivity Á+; X±=(z2F2/RT)D±. 1/t; the solutions became the same when %d2/x was identified „if h diffusion constant D: this is the Einstein-Smoluchowski relate» • (26.2.15). With this connection established we can interpret viscosn- ,„ mobility, conductivity, and diffusion processes in general in terrn^.,; .T6 microscopic, dynamical parameters d and x. Appendix: the measurement of transport numbers The following are brief summaries of the three methods used to u„ (Mlr transport numbers of ions and, through them, individual ion condnuivuJj and mobilities. (1) Moving boundary method. Let MX be the salt of interest. Pnur ft? solution of MX into the lower half of a narrow vertical tube. Sekt i i <.$ NX where N is less mobile than M; prepare a solution of NX, and pi u* it on top of the MX solution so that there is a clear boundar ľi^ a current I for a time t. The X" move towards the anode (downward-,) aP(| the M+ and N+ move towards the cathode (upwards). The annuint w* cations transported for this amount (It/F) of electricity is t+(It/z 11 ihjr charge being z+. If they are at a concentration c, the volume s^ip* oui is t+(It/z+cF). But if the cross-section of the tube is S, and the distant moved by the boundary is x, the volume is also equal to xS. 1 krtku-; t+(It/z+cF) = xS, and monitoring the progress of the boundary fc i j ^.m» of times gives i+. Example (Objective 7). The transport numbers of H+ and SO^" were measured in a mo" in^ boundary experiment. The apparatus consisted of a tube of bore 6.40 mm conl 11-ing aqueous sulphuric acid at a concentration of 0.015 mol dm" 3. A steady curu ■ of 1.23 mA was passed, and the boundary advanced as follows: t/s 40 80 120 160 200 x/mm 0.860 1.722 2.586 3.450 4.309 Find t+ and r_. • Method. Use the equations set out above. • Answer. t+ = (ScF/I)(x/t) = Ti x (3.20 x 10"3 m)2 x (0.015 mol dm"3) x (9.6485 x ÍO4 C moľ1) x(l/L23xl