Fig. 26.8. Experimental and theoretical molar conduct
0.2 0.3 VW(mol kg-]
■ Comment. The result is not at all bad considering the theoretical difficult es. of dealing with a dynamical system of this complexity. Note that the plot c I l 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 '( the curves are predicted to depend on the valence type (z appears in 'iiC constants A and B). Some comparisons between theory and experiment are shown in Fig. 26.8, and this shows how well the theory account* f r the observations at low concentrations.
The success of the Debye-Huckel-Onsager equation suggests that the model of ion-ion interactions is substantially correct. A further test is obtained by investigating what happens when the effect of (he ionic atmosphere is eliminated. This can be done in a variety of ways. In one the conductivities are measured at very high frequencies; then the cen ijI ion is moved backwards and forwards very rapidly, and the retarding effects of the ionic atmosphere ought to average to zero. This is the Dehve-Falkenhagen effect, and the predicted increase in mobility at high frequencies has been observed. The other way of eliminating the effect oi the atmosphere is to move the ions so rapidly that no atmosphere has link to build up. The Wien effect is the observation of higher mobilities at \erj high electric fields. (There are two Wien effects. The first Wien effect is the one just described; the second Wien effect is the enhancement of the degree of ionization of an ionogen, or weak electrolyte, by the applied field >
This model of the interactions fails when the concentration bcccnu large, because ions tend to stick together in pairs, and even triples...Th> can be seen quite clearly from X-ray analysis of ionic solution, where peaks of scattering can be interpreted in terms of definite ion-ion distances *
Fundamental aspects of molecular transport
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 property that governs the direction of spontaneous change is the thermodynamic chemical potential (p. 182). When unit amount of solute is shifted from a region where its chemical potential is fi{l) to one where it is (i(2) the work required is w = p(2)—/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 = fx(x + dx)—fi(x) — [fi(x) + (dfi/dx)dx'] — jj.(x) = (d/i/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 ?F is
dw = — ^ Ax.
By comparing the last two equations we see that the gradient of the chemical potential acts like a force. We shall therefore write
(26.2 1)
(26.2.2)°
£6.2.3)°
& = -(dju/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).
H = jxb + R T In (c/mol dm "3).
If the concentration depends on position, the thermodynamic force acting is
#-= -(d/dx)[^e + RTln(c/mol dm-3)]
= -(RT/c){dc/dx)
because /ie is independent of position, and dln//dx = (l//)d//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 IF, the force
-------— ---------»-"v,,m
tlux of material, /(matter), is proportional to the impressed force; traa ./(matter) oc c&. But the effective force is given by eqn (26.2.3), and
J(matter) oc -RT(dc/dx) = ~kT(dJT/dx)
and we have the flux as proportional to the concentration gradient »M accord with Fick's Law (Jf is the number density of molecules, Jf = ',* It is more convenient to develop a different line of argumeni. nvjVw interpret the flux of particles as the product vjV = vcL, where v is thejf average velocity and c their concentration. Then Fick's Law, 1
vJT = J = -D(<\jVldx) = -DUdcjdx) reads 1 cv = — D(dc/dx) . I
or, using eqn (26.2.3), I v = - (D/c) (dc/dx) = (D/k T)&. I
Therefore, in response to a unit force, the molecules diffuse with a dnfra velocity of magnitude D/kT. We know, however, that the mobility oVml ion is related to the electrical force on it. Since the mobility is defined! through s ~ uE, and since the electrical force is ezE, it follows that fj s — uE = (u/ze) (ezE) = (u/ze)^, I
and therefore the drift speed under the influence of unit force is [u/zeM The nature of the force is irrelevant; therefore the two drift speeds (D,'fcT0* and (u/ze)& may be identified: then *
I
and so we arrive at the very important result, known as the S Einstein relation: D'=.uktfez. ™
connecting the diffusion constant and the mobility. j The last relation can be developed further in two directions. In the fua place it can be used to relate the molar conductivity to the diffu«o|| constants of the ions, D+ and D_. We write (for 1:1 salts) M
Am = zF(u++u.) = (z2eF/kT)(D + + D.) J§ and so arrive at the
Nernst-Einstein relation: Am = (z2F2'/RT)(D+ + D-).
One application of these expressions is to the determination of the ioi|| diffusion constants from conductivity measurements; the other is to tS calculation of conductivities on the basis of models of ionic diffusion (s|j below).
the mobility to the viscosity. By combining the expressions
s = ezE/6-ntja   and   s = m£, the first being the expression for the drift speed defined in eqn (26.1.9), we are able to write
u ■=■ ez/6nna.
Since the Einstein relation is u = ezD/kT, the two may be equated and combined into the
Stokes-Einstein:relati6n::D. ^kTl&faja—z-
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.
i ihle 26.4. Diffusion coefficients at 25 °C, D/W
9 m2 s"
I2 in hexane	4.05	H2 in CC14	9.75
I2 in benzene	2.13	N2 in CCU	3.42
CC14 in n-heptane	3.17	02 in CCU	3.82
Glycine in water	1.055	Ar in CCU	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	
Sourcfc American institute of Physics Handbook, McGraw-Hill, and (for the ions) eqn (26.2.4) and Table 26.2.
example (Objective 12,13,14). Find the diffusion coefficient, the molar conductivity, and the effective hydrodynamic radius of the SO2,- ion in water at 25 °C. • Method. In Table 26.2 the mobility ofthe ion is given as 8.29xl0"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 X- (the anion contribution to Am). Calculate the effective radius a_ from
_ (8.29 x 10->cmas"1 V') x (1.3807 x 1CT 23 J K"1) x (298.15 K)
2 x (1.6022 x 10_I9C) = 1.065 x 10"5 cm2 s_1. From eqn (26.2.5),
_22x(9.6485x 104Cmor1)2x(1.065x lQ-5cm2s"')
(8.3144 J K~1 mol- J) x (298.15 K) = 160.0 a-1 cm2 mol-1. From eqn (26.2.6), a_ = kTj&Kt}D-
(1.3807 x 10~23JK~l)x (298.15 K)
6ttx(I.00x 10~3kgm-1s_1)x (1.065 x lO-'cnv's-1) = 2.051 x 10"10 m  or  205 pra  or  2.05 A.
• Comment, The bond length in SO+" is 144 pm, and so the radius calculated 1 (the radius of a sphere representing the molecule) is plausible and compatibles only a small degree of solvation.
Some experimental support for these ideas comes from conductivity measurements, because the empirical Walder?s rule is that the prodtt| A%fl should be approximately constant for the same ions in differeS solvents. Since /f° cc u, and ucclfy, we can see the theoretical basis c this rule. Its applicability is muddied by the role of solvation: differenl solvents solvate ions to different extents, and so the ions' effective hydrodynamic radii depend on their nature: both a and t} vary with the solvent. *
Diffusion as a time-dependent process. We turn now to the discussion of time-dependeri| diffusion processes, in which some distribution of concentration, or otj temperature, etc., is established at some moment, and then allowed tor] disperse without replenishment. One example is a metal bar heated rapidll at one end and then allowed to reach equilibrium, and another is when 4 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 apply to other properties. We fix our attention on a small slab of the system;! extending from jc to x + Ax, and of cross-sectional area A, Fig. 26.9. Lrf the concentration at x be Jf(x,t) at the time t, then the increase in concentration inside the slab (of volume AAx) by virtue of the flux from the left is
dJf{x,t)ldt = J(x,t)A/AAx = J(x,t)/Ax
(26.2.7)
x+ Ax
because J A 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
dJr{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
djr(x, i)/dt = J(x, t)/Ax -J(x+Ax, t)/Ax.
The fluxes can now be related to the concentration gradients at each window. Using Fick's Law we can write
J(x, 0- J(x + Ax, t) = {- DtdJT{x, t)/dxj}} - {- D[d>(x + Ax, t)/5x)]}
= —D
dx
+ D-^J^(x,t)+
8J^{x,t) dx
Ax
= D(d2jV(x,t)/dx2)Ax. Substituting this back into the expression for the rate of change of the concentration in the slab leads to the
" diffusion equation: (d.V(x, r)/3t) = D(d2A'(x,t))dx2).
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 Remember that J = C V and V A"1 => £1.
Answer. From eqn (26.2.4), Z)_ = u^kT/ez
_ (8.29 x 10"4cm2s~1V"*)x (1.3807 x 10'23 JK"') x(298.15K)
2 x (1.6022 x 10"19 C)
= 1.065 xl0-5cm2s-'. From eqn (26.2.5),
_ 22 x (9.6485 x 104Cmo1"1)2 x(1.065 x 10"5cm2s"') (8.3144 J K"1 mol^x (298.15 K)
From eqn (26.2.6), a_ = kT/6nriD-
=_(1,3807 x 10~23JK~1)x (298.15 K)_
_ 6k x (1.00 x 10"3 kgm"1 s"l) x (1.065 x 10-5 cm2 s"*) = 2.051 x 10" 10m  or  205pm  or 2.05A.
• Comment. The bond length in SO2." is 144 pm, and so the radius calculated (the radius of a sphere representing the molecule) is plausible and compatible only a small degree of solvation.
here with
Some experimental support for these ideas comes from conducts ity measurements, because the empirical Walderts rule is that the product A^tj should be approximately constant for the same ions in difici.-m solvents. Since A^ cc u, and uocl/n, we can see the theoretical basis of 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 t\ vary with the solvent.
Diffusion as a time-dependent process. We turn now to the discussion of time-dependent diffusion processes, in which some distribution of concentration, or of temperature, etc., is established at some moment, and then allowed to disperse without replenishment. One example is a metal bar heated rapidh at one end and then allowed to reach equilibrium, and another is when a 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 appl} to other properties. We fix our attention on a small slab of the system extending from x to x + Ax, and of cross-sectional area A, Fig. 26.9. Let the concentration at x be J^{x,t) at the time t, then the increase in concentration inside the slab (of volume AAx) by virtue of the flux from the left is
dJf(x, t)/dt = J(x, t)A/AAx = J(x, t)/Ax
Fig. 26.9. The diffusion of material into a region.
because J A 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
dJf{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
dJT{x, t)/dt = J(x, i)/Ax - J(x+Ax;t)!Ax.
The fluxes can now be related to the concentration gradients at each window. Using Fick's Law we can write
. J(x,i)~J{x + Ax,t) = {-DldJS(.x,t)/dx)-]}-{-D[dJ>~(x + Ax,t)/dx)]}
dx
2H
= D(d2^V(x,t)/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'eqmtion:(8J^(xjydr)=='D(82vr'(x,t)/dx2).
This is also sometimes called Ficfe'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^iuuuuTb iiuuun mai nature nas a naiurai tendency
eliminate the wrinkles in a distribution.
The diffusion equation is a second-order differential equation in space and first-order in time, and therefore in order to arrive at a solution we have to specify two boundary conditions for the spatial dependence and a single initial condition for the time dependence. This can be illustrated 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 j>z-plane at x = 0. The boundary conditions are that the concentration must be everywhere finite, and the total number present must be JV0 at all times. The solution of the diffusion equation having these as conditions is
(26.2.8)       Jf(x, t) = {N0/A(7iDt)112} 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 of particles spreads through the material as time advances. The use of the diffusion equation is that, with its aid, the concentration can be predicted, at any point in the system at any time.
A number of important features of the diffusion process can be explained on the basis of the diffusion equation, and in particular with the help of the simple solution just quoted. For example, we can ask what is the mean distance through which the solute has spread after a time t. The number of molecules in thejslab of thickness dx at x is >"(x, t)A dx, and so tne||; probability that any of the N0 molecules is there is JT(x,t)Adx/N0. II Iht | molecule is there it has travelled a distance x from the origin; therefore -the mean distance travelled is
|£>f=0.05
(26.2.9)
<x>
= 2(Dt/n)112.
■JTfy, t)A dx/N0 = (1/nDt)^2     x e~x2/4Dtdx
The average distance varies as the square root of the time lapse. Tlu>> i> an important general result which we shall return to later. If we use the Stokes-Einstein relation for the diffusion constant the mean distance covered in a solvent of viscosity n by particles of radius a is
(26.2.10) <x> = (2fcT/37tV)1/Vf-
The root mean square distance covered is x = ^/<x2), and its value is
(26.2.11) 5c = V<x2> =        x2JT(x, t)A dx/N0 = (2Dt)112.
This is a valuable measure of the spread of the particles when they are, allowed to migrate in both directions (for then <x> =-0). The value of x for molecules having D = 5 x 10"6 cm2 s ~1 is shown in Fig. 26.11: you can see how long it takes for diffusion to increase 3c to about 1 cm in an unstirred solution. The proportion of particles which remain within a distance x of
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 JT(x,t)Adx, the number in all the slabs up to the one at x is the sum (integral)
N(x < x
:,*) =
Jo
jr(x,f)Adx = (0.6S...)(N0/A),
where x has been replaced by (2Dt)l/2, and the integral evaluated numerically.* It follows that the proportion of molecules inside the range 0 < x < x is 0.68. Therefore, over two-thirds of the molecules are still clustered around the origin, and only 32 per cent have escaped beyond x (but do not forget that the value of x grows with time).
The diffusion equation can be solved for more complicated arrangements, for example, when ions are continuously generated at a plane electrode dipped in the solution, or when ions are deposited on an electrode and withdrawn from the solution. Calculations like these play an important part in the discussion of rates of reactions at electrodes (Chapter 30).
Diffusion: the statistical view. An intuitive picture of the mechanism of diffusion is one in which particles move in a series of small steps, and gradually migrate away from their original position. We shall build a model of diffusion on the basis that the particles can jump through a distance d, and do so in a time t. This means that the distance covered by a molecule in a time (is (t/x)d. This does not mean that the particle will be found at
* The integral can be simplified by substituting y = x/2(Dt)1/2, for then it becomes
fx l*l/2"J
N{x ^x,i) = {NjAfyDtf2}    Axe-*2/40' - WOC/it1'3) t^'dy.
Jo Jo It happens that the integral (2/7t',2)J*e">Jdy 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)1/2, which is what we require, is 0.68.
Fig. 26.11. Root mean square distance covered by particles with D = 5x 10"6 cm2 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 I account. We shall simplify the discussion by allowing the particle to move?j only along a straight line, the x-coordinate, but we must not forget thai: in a real system a particle is free to move in three dimensions. We shallh also confine our attention to a model in which the particle can jump withl 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. j Our task is to find the probability that a molecule will be found at aj distance x from the origin at a time t. During that time interval it wifla have taken t/x steps: we shall write n — t/x. Many of these steps were steps! to the right; many were steps to the left. If nR is the number of steps toll 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 1 travelled as x = n$d—nLd. "1 The probability of being at x after n steps of length d is the probability 1 that of the n steps, nR occurred to the right, nL occurred to the left, and 1
»R-«L = X/d. !
What is the total number of possibilities for left or right steps? Since .1 each step may occur in either of two directions (left or right) the totalj number of possibilities is 2". Jj 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 'i case of 4 steps, and ask what is the number of ways of taking 2 right steps. There are 24 possible step sequences: 1
LLLL   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) = ».ynr!(b-mR)!2", with n = nR+nL and x/d = nR — nL. Since =       x/d),     n-nR =%n-x/d), it follows that (26.2.12)      P(x) = «!/{Ci(»+s)]!Ci(»-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 A1" is a large number it is possible to use Stirling's approximation:
(26.2.13) hxNl
(JV+£> In iV-iV +In (27r)1/2.
This is a more accurate form of the approximation than the one used earlier. Even when JV is quite small this expression is quite good. For example, instead of 10! = 3.629 x 106 it gives 10! as 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 InP = ln»!-ln(ri(n+s)]!)-ln([i(ii-s)]!)-»ln2
to
In P * In (2/nn)112      + 5 +1) In (1 + s/n) - fa - s +1) in (1 - s/n).
If we allow sfn to be a small number (so that x must not be a greajS distance from the origin) we can use the approximation ln(l +y)fvy, arid obtain
lni'«ln(2/7tn)1/2-s2/2n,
or
P ss (2/7t»)1/2exp( —s2/2n),
which is already of a Gaussian form. Now replace s by x/d and n by tj%. We obtain
(26.2.14)      P(x, t) = (2z/nt)112 exp(-x2t/2td2),
and this has precisely the form of J^(x, t)/N0 given in eqn (26.2.8) as> a solution of the diffusion equation. (The differences of detail arise from allowing the particle to migrate in both directions away from x = 0, aad letting it be found only at discrete points separated by d instead of beings 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 of/ the diffusion equation: we should not expect it to apply at times so short 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
2da/t»4J>,
and therefore we come to the
(26.2.15)   "'" Einstein-Smoluchowski relation: D = ?d2/v.
Example (Objective 17). Suppose that the SO%' ion jumps through about its own diametefij every time it makes a move in aqueous solution. How often does it change position?
• Method. The diffusion coefficient was found in the last Example to be l.OiiS < 10"5 cm2 s~ \ and the effective radius was found there to be 205 pm. Find t from-eqn (26.2.15).
• Answer. From eqn (26.2.15),
z = d2/2D
= (2 x 205 x 10"12 m)2/2 x(1.065 x 10-9 m2 s_J) = 7.89 x 10"11 s.
• Comment. The big, heavy SO2.- ion jumps through its diameter in about 8 10"11 s. If the ion were imagined as jumping through a distance equal to thfr diameter of a water molecule (as 150 pm) the jump time would be about l.l -lO'^s. i
r
The Einstein-Smoluchowski relation is a central connection between the microscopic properties of the size (d) and rate (1/r) of a molecular'
1
*~r 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/z is interpreted as a mean velocity of the molecules undergoing diffusion, and the jump length d is called a mean free path and written I, the Einstein-Smoluchowski equation reduces to D = \lc, 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,
Summary 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 & if its chemical potential varied from place to place, and we identified !F with — d/i/dx, eqn (26.2.1). The thermodynamic force led to the construction of Fields 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" sfee" d' and tiuhip ' time t: ' -
D = d2/2T.      ;'■ ,; ' ;:
Stokes-Einstein relation between diffusion coefficient D and solution1 ■ viscosity n: ■
D = kT/6nria. ' , .    '][''''     " f
Einstein relation between diffusion coefficient and ion mobility «±: ■
t)± - u±kT/ez ± u±RTfzF.        ' '■
Nernst-Einstein relation between diffusion coefficient and ion^con-. , ductivity X±;. ' . ; '     ." ' ■
'   l±=(z2F2/RT)D±. . '    - "i'.'vJ
1/t; the solutions became the same when \d2j% was identified w„ diffusion constant D: this is the Einstein-Smoluchowski relation (26.2.15). With this connection established we can interpret viscosit mobility, conductivity, and diffusion processes in general in terms of microscopic, dynamical parameters d and x, *
Appendix: the measurement of transport numbers
The following are brief summaries of the three methods used to meas M
transport numbers of ions and, through them, individual ion condi
and mobilities. %%.
(1) Moving boundary method. Let MX be the salt of interest. Pour {jaj solution of MX into the lower half of a narrow vertical tube. Select a a!)? NX where N is less mobile than M; prepare a solution of NX, and potjg it on top of the MX solution so that there is a clear boundary Passar current / for a time t. The X" move towards the anode (downwards) and? the M+ and N+ move towards the cathode (upwards). The ann imf.f cations transported for this amount (It/F) of electricity is t+(It/z^ F), theirs charge being z+. If they are at a concentration c, the volume s^lp* .ui-is t+Qt/z+cF). But if the cross-section of the tube is S, and the distance1 moved by the boundary is x, the volume is also equal to xS. Therefore* t+Qt/z+cF) = xS, and monitoring the progress of the boundary fo i j  11 of times gives t+.
Example (Objective 7). The transport numbers of H+ and SO*" were measured in a mo\ing."
boundary experiment. The apparatus consisted of a tube of bore 6.40 mm containing aqueous sulphuric acid at a concentration of 0.015 raol dm"3. A steady current 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 (_.
• Method. Use the equations set out above.
• Answer. t+ = (ScF/l)(x/t)
= x x(3.20 x 10"3 m)2 x(0.015 mol dm"3) x (9.6485 x 104 C mol"'l
x(l/1.23xl0"3 A)x(x/t) = 3.785 x 10* m"1 s (x/t) = 37.85 [(x/mm)/(t/s)]. Draw up the following table:
t/s 40        80        120       160 200
104(x/mm)/(r/s)     215     215.3     215.5     215.6 215.5
Average: 0.02154 mm s'1. Therefore £+(H+) = 0.815; and so t±(SOi") = 1-0.815 = 0.185.
• Comment. These results can be used to recover the mobilities and the individual ionic conductivities, and are the basis of the data in the last Example.
(2) The Hittorf method. A cell is divided into three compartments and
an amount It of electricity is passed. An amount It/z+F of cations are discharged at the cathode, but an amount t+(It/z+F) of cations move into the cathode region. The net change in the amount of cations near the cathode is ■ -(It/z+F)+t+(It/z+f)= -(l-t+)(ft/z+F)= -r_(Ir/z+F). Therefore, measuring the change in composition in the cathode compartment gives f_, the anion transport number. Likewise, the change in composition of anions at the anode is — £+(if/j2_|F).
(3) E.m.f. measurements. The e.ra,f. of a cell with transference having an electrode reversible with respect to anions is related to the e.m.f. of the cell with the same net reaction but without transference by Et = 2t+E (the argument is similar to that used for the Hittorf method). Therefore set up two cells and compare their e.mi.s.
Further reading
Conductimetry. T. Shedlovsky and L. Shedlovsky; in Techniques of chemistry (A. Weissberger and B. W. Rossiter, eds.), Vol. IIA, 163, Wiley-Interscience, New York, 1971.
Determination of transference numbers. M. Spiro; in Techniques of chemistry (A. Weissberger and B. W. Rossiter, eds.), Vol. IIA, 205, Wiley-Interscience, New York, 1971.
The principles of electrochemistry. D. A. Maclnnes; Dover, New York, 1961. The physical chemistry of electrolytic solutions. H. S. Harned and B. B. Owen;
Reinhold, New York, 1958. Ionic solution theory. H. L. Friedman; Wiley-Interscience, New York, 1962. '', Electrolyte solutions. R. A. Robinson and R. H. Stokes; Academic Press, New York,
1959.
Electrolytic conductance. R. M. Fuoss and F. Accascina; Wiley-Interscience, New York, 1959.
Treatise on electrochemistry. G. Kortum; Elsevier, Amsterdam.
Experimental methods for studying diffusion in liquids, gases, and solids. P. J. Dunlop, B. J. Steel, and J. E, Lane; in Techniques of chemistry (A. Weissberger and B. W. Rossiter, eds.), Vol. IV, 205, Wiley-Interscience, New York, 1972.
Diffusion in solids, liquids, and gases. W. Jost; Academic Press, New York, 1960.
Problems
26.1. Conductivities are often measured by comparing the resistance of a cell filled with the sample to its resistance when filled with some standard solution, such as aqueous potassium chloride. The conductivity of water is 7.6 x 10"* Q"1 cm-1 at 25 °C and the conductivity of 0.1 mol dm"3 aqueous KC1 is 1.1639 xlO-2 H"1 cm"1. A cell had a resistance of 33.21 £1 when filled with 0.1 mol dm"3 KC1 solution and 300 Q when filled with 0.1 mol dm"3 acetic acid. What is the molar conductivity of acetic acid at that concentration and temperature?
26.2. In conductivity measurements it is common to write k - C/R, where R is the measured resistance of the sample in the cell and C, the cell constant, is a characteristic of the particular cell in use. Since k = l/RA we see that C = l/A. Both / and A may be difficult to determine directly, and so it is normal to calibrate the cell with a sample of known conductivity. In several of the Problems that follow we shall refer to 'the cell'. We shall always mean the cell of this Problem,