12-7 Activities and Activity Coefficients of Electrolytes
A. Introductory Comments
An electrolyte, like any other solute, tends to give nonideal solutions, approaching Henry's law behavior at infinite dilution. We know that at high dilution the positive and negative ions act independently. The colligative property effects report, for example, the number of particles expected from the complete dissociation of the electrolyte. On the other hand, it is not possible to vary a single ion concentration, keeping everything else constant. An attempt to do so would immediately result in the solution acquiring an enormous electrostatic charge.
An excess of even I O- 10 mole l i t e r-1 of one kind of ion over another would result in a static charging of the solution to a potential of about ΙΟ6 V! In other words, we cannot prepare a solution containing only one kind of ion and therefore cannot determine individual ion activities or activity coefficients; we can only observe a mean value for the positive and negative ions present.
The situation is illuminated if we consider the case of a solubility product
TABLE 12-6. Equivalent Conductivities of Aqueous Ions at 25° Ca
Cation aS e e D. Maclnnes, "Principles of Electrochemistry." Van Nostrand-Reinhold, Princeton, New Jersey, 1939.
b To convert to SI units, multiply by 1 0- 4. Thus A(Na+) = 50.1 cm2 equiv"1
o h m-1 = 50.1 x 1 0 "4 m2 e q u i v-1 o h m- 1.
is characteristic of the isolated ion free of long-range interionic attraction effects.
A number of such values are given in Table 12-6. These are, of course, parallel to the mobilities in Table 12-4, being related by Faraday's number. The same general comments apply here as were made in Section 12-5C.
- \η αι = ^ Π [Eq. (10-20)]
equilibrium. We write for a slightly soluble salt M X the solubility equilibrium
MXO) - MX(solution) = M+ + X". (12-46)
The thermodynamic criterion for equilibrium is satisfied if we write
H<MX(S) = /xMx(solution)
as required by Eq. (9-44). We can introduce the activity of the electrolyte species:
^Mx(solution) = /xM X(solution) + RTln aMX . (12-47) The activity aMX corresponds to the solute activity a2 in the equations of Chapter 10.
It can be obtained, for example, by application of the Gibbs-Duhem integration procedure to solvent activities as determined from colligative property measure
ments. As with solutes generally, we use a Henry's law standard state, usually the one based on molality as a concentration unit (see Section 10-6).
The treatment up to this point is rather unsatisfactory, however. It does not tell us that the electrolyte is dissociated or how aMX is apt to be influenced by the presence of a common ion in the solution. Returning to the solubility equilibrium, the normal way of writing the equilibrium constant for process (12-46) is in terms of a solubility product:
#sp = (M+XX-). (12-48) We know that the constant KSÎ> i s well-behave d i n dilut e solutions . Fo r example ,
while w e canno t avoi d havin g essentiall y equa l number s o f positiv e an d negativ e ions presen t i n an y solution , w e can , b y usin g mixe d electrolytes , var y th e concentrations o f specifi c kind s o f ions , suc h a s M + o r X~ , independently . I f S denotes th e solubilit y o f MX(i) , the n i n pur e wate r w e hav e
Ksï> = S2. (12-49 )
If adde d X - io n i s present , a s i n th e for m o f a concentratio n C o f NaX , the n Eq. (12-48 ) become s
#sp = (S)(S + C) . (12-50 )
We no w hav e th e commo n io n effec t whereb y adde d X ~ io n depresse s th e solubilit y of MX(i) . Sinc e th e solutio n i s stil l i n equilibriu m wit h th e solid , aMX mus t no t have changed , eve n thoug h th e individua l value s o f ( M+) an d ( X-) ar e no w quit e different fro m before . Thu s observatio n tell s u s tha t i n dilut e solutio n
aux = (Μ+ΧΧ-). (12-51)
Use of Eq. (12-51) allows a more realistic treatment of colligative effects in dilute solution. The Gibbs-Duhem integration gives In aMX , and for a single electrolyte solute MX, (M+) = (X~) = m. We then have In aMX = In m2 = 2 In m, and the experimentally observed factor of 2 has now appeared.
Consider, for example, the osmotic pressure effect. The basic equation is
12-7 ACTIVITIES AND ACTIVITY COEFFICIENTS OF ELECTROLYTES 455
and this is to be used with the Gibbs-Duhem relation
x1 d(ln + x2 d(ln a2) = 0 [Eq. (9-79)],
where x1 and x2 must denote the mole fraction of components 1 (solvent) and 2 (salt MX). It is simpler for the present illustration to use mole fraction rather than molality for the salt concentration, and so we write aMX = ( X M+X * X ~ ) = * 22 (rather than aMX = m2). This amounts to using the first of the Henry's law conven
tions of Section 10-6. Since d(ln aMX) = 2 d(ln x2), combination of Eqs. (10-20) and (9-79) gives
2x2 d(\n x2) = - x1 d(\n = ^~χχάΠ (12-52) or
The integral of dx2/x1 is — ln(l — x2) and since the solution is to be dilute, this becomes just x2. The final result is
- ^ / 7 = 2 x2. (12-53)
Thus the osmotic pressure is predicted to be twice the value expected just from the mole fraction of the salt, or the van't Hoff/factor comes out equal to 2, as observed.
β . Defining Equations for Activity and Activity Coefficient
The preceding analysis was presented to show that the activity of an electrolyte is equal to the product of the ion concentrations in very dilute solutions. For the more general case of a nonideal solution we therefore write
a2 = a+a_ , (12-54)
where a2 denotes the electrolyte activity and a+ and a_ the individual ion activities, which become equal to m+ and m_ , respectively, in the limit of infinite dilution.
We then further define the activity coefficients γ+ and y_ :
a+ = γ+ιη+ and a_ = yjm_. (12-55)
Since a2 involves the square of a concentration, it is convenient to define a new activity a± as the square root of a2 :
a2 = a±2. (12-56)
The activity a± is known as the mean activity. Similarly, γ± is called the mean activity coefficient:
y±2 = y+y _ , (12-57)
and m± the mean molality:
m±2 = m+m_ . (12-58)
Û2 (12-62) where ν = v+ + v_ , (12-63)
a+ a_ = y_w_ [Eq. (12-55)],
Y± = y + y _ , (12-64)
m± = mv+m"-, (12-65)
and, as before,
a± = y±m± [Eq. (12-59)].
The complications introduced by these definitions are regrettable. They develop naturally, however, when we deal with nonideal electrolyte solutions.
C. Activity Coefficients from Solubility Measurements
The preceding material allows us to write the thermodynamic equilibrium constant for the solubility equilibrium of an electrolyte. Thus for AgCl we have
AgCIO) - Ag+ + Cl-, (12-66) Kth = aAg+acl. = (Ag+)(C1-) yA g +yc l- , (12-67)
or
Kit, = Ks p r ±2, (12-68)
where Kth is the thermodynamic solubility product and Ks$ is the usual form in which concentrations are used. The solubility of AgCl in water is about 1 0-5 m and it seems likely that y± will be unity for so low a concentration. We can investigate the situation by adding some neutral or noncommon-ion electrolyte.
We find experimentally that on doing so the solubility of AgCl increases, and hence Then
a± = y±m± . (12-59) This set of définitions is for the specific case of a 1-1 electrolyte, that is, for an
electrolyte which produces 1 mole of ions of each kind per formula weight. The general treatment is suggested by consideration of the solubility product for a salt Mv+X„- :
Mv+Xv_(s) = v+M*+ + v_X- Aep = (Mz+Y+(Xz-)\ (12-60) where z+ and z_ are the respective ion charges and v+ and v_ are the numbers of
ions of each type. Electroneutrality requires that
v+z+ = v_z_ . (12-61)
We want to define the activity of this general electrolyte in terms of individual ion activities such that in the limit of infinite dilution we obtain the expression on the right-hand side of Eq. (12-60). The definitions are then
12-7 ACTIVITIES AND ACTIVITY COEFFICIENTS OF ELECTROLYTES 457
so does its KSO value. Evidently y± is varying with the concentration of added neutral electrolyte, since Kth must remain a constant.
A useful way of graphing such data follows if we write Eq. (12-68) in the form
log KSO = log - 2 log y t . (12-69)
We then plot log Ksp against some function of concentration. Figure 12-7 gives the results of measurements that show the effect of added K N 03 and M g S 04 on the solubility and hence the ATsp for AgCl. As with equivalent conductivity, the experimental observation is that a plot is nearly linear if the square root of the added electrolyte concentration is used. The theoretical explanation is given in the next section, as is the definition of ionic strength, /.
Since we are using the Henry's law convention for activities and activity coeffi
cients, y± approaches unity at infinite dilution. The intercept of Fig. 12-7 therefore gives l o g ^ t h = —9.790, or Kth = 1.62 χ 10~1 0. We may then calculate y± for the ions A g+ and Cl~ in the presence of added K N 03 by inserting Km and the measured Ksp into Eq. (12-69). Thus log Ksv = —9.645 at Vm = 0.175 or m = 0.0306; then log y± = [(-9.790) - (-9.645)]/2 = - 0 . 0 7 2 5 , whence y± = 0.846.
The data allow a tabulation of y± versus concentration of added K N 03 (strictly speaking, m includes the contribution of the dissolved AgCl to the total salt concentration).
D. The Ionic Strength Principle
Studies such as the preceding were carried out for a number of systems, especially by V. LaMer and co-workers, and these confirmed an earlier observation by G. Lewis that the activity coefficient of an electrolyte depends mainly on the total concentration of ions and only secondarily on their specific chemical natures.
Thus for concentrations below about 0.05 m, the data of Fig. 12-7 would be essentially the same if N a N 03, K C 1 04, and so on were used instead of K N 03. The
results, incidentally, would also have been the same if NaCl were the added electrolyte. Although there is now a common-ion effect, depressing the solubility of the AgCl, the experimental Ksv> values will still show the same variation with concentration of total electrolyte present.
Empirical observation showed, however, that if other than uni-univalent salts were used, there was an increased effect on activity coefficients. It was found that all types of electrolytes can be put on a common basis by expression of the ionic concentration in terms of a quantity called the ionic strength I, where
In the case of a uni-univalent electrolyte, I = m, but for, say K2S 04, / =
\{2m + 4m) = 3m, while for M g S 04, / = \{Am + Am) = Am. The increase in solubility of a salt in the presence of any electrolyte is nearly the same if the results are plotted in terms of / rather than m. The circles in Fig. 12-7 are the points obtained with M g S 04 as the neutral salt and with the abscissa reading V 7 rather than Vm.
The discovery of the ionic strength principle was a major step toward the under
standing of nonideality effects in dilute electrolyte solutions. The principle constituted strong evidence that it was the charges on ions and not their particular chemical natures that determined activity coefficients. It paved the way for the development of the interionic attraction theory (described in Section 12-8).
£. Methods of Determining Activity Coefficients
The change in solubility of a salt with ionic strength provides one means for determination of the activity coefficient of the salt in the presence of some other electrolyte. The method is not applicable to soluble salts, and, moreover, necessarily gives activity coefficients in mixed electrolyte solutions. The activity coefficient of an electrolyte in solutions containing no other ionic species is of more general importance.
As indicated in Section 12-7A, a2, the solute activity, may be obtained from colligative property measurements through use of the Gibbs-Duhem equation.
Equation (12-52) would be written with d(\na2) rather than d(\nx2), and then
ι = \ Σ mizi2- (12-70)
12-7 ACTIVITIES AND ACTIVITY COEFFICIENTS OF ELECTROLYTES 459
0.01 0.02 0.03
FIG. 12-8. Variation of Kd for acetic acid with concentration at 25° C.
manipulated into the form of Eq. (9-80). The detailed procedures for simplifying the handling of this integration may be found in more specialized texts [for example, Lewis and Randall (1961)]. Much of our activity coefficient data comes from measurements of the emf of electrochemical cells. This approach is discussed in detail in Chapter 13 and will not be reviewed here.
Activity coefficients may also be obtained from a study of ionic equilibrium. As a specific example, Table 12-7 gives the concentration equilibrium constant for the dissociation of acetic acid as determined from conductivity measurements at various concentrations. The concentration equilibrium constant is
HAc = H+ + Ac-, Κ = - ^ = - (12-71) 1 — oc
and oc in the table has been corrected for the ion atmosphere effect οηΛ, as described in Section 12-3C. The residual variation in Κ is attributed to nonideality, and Eq. (12-71) is written
Kth = ^ V H + T A C - = .7) 2 ( 1 2
y H A c
The solutions are dilute enough that, as a nonelectrolyte, HAc is probably in the Henry's law region of behavior, so yH Ac = 1. Therefore Ky = y±2, and Eq. (12-72) may be plotted according to the form
log Κ = log Kth - 2 log y± . (12-73) The ionic strength derives entirely from the H+ and Ac~ ions, whose concen
tration is aC, and the plot of log Κ versus ( a C )1 /2 is shown in Fig. 12-8. Extra
polation to zero ocC gives log(105^th) = 0.2440, or Kth = 1.754 χ I O- 5. Insertion of this value into Eq. (12-72) allows Ky , and hence y± , to be calculated for each concentration. Thus for 0.050 m HAc, y±2 = 1.754 χ 10"5/1.849 χ 10~5 = 0.949, or y± = 0.97.
The same approach would be used in the case of an equilibrium not involving
ionic species. Thus for the esterification reaction in aqueous solution, CH3COOH + C2H5OH = CH3COOC2H5 + H20 , one again writes
( C H3C O O C2H5) ( H20 )
y C H3C O O C2H5y H20 R.R.
( C H3C O O H ) ( C2H5O H ) yCH 3C O O H y c2H5O H
and extrapolates the measured values of Κ to infinite dilution. Pure water is taken to be in its standard state so that yH 2o approaches unity at infinité dilution of the other species, and the extrapolated value of ATis then equal to Κχκ . We are primarily concerned with rather dilute solutions in this chapter, and activity coefficients for nonionic species are close to unity. It is only in the case of ions that the long-range Coulomb interactions lead to nonideal behavior even in quite dilute solutions (C < 0.1m).
F. Activity Coefficients of Electrolytes
As noted earlier, a number of methods may be used for the determination of the activity coefficient of an electrolyte. The results have been collected and tables of standard activity coefficients are available. A selection of such results is given in Table 12-8 and the mean activity coefficients of some typical electrolytes are plotted against concentration in Fig. 12-9.
The points to notice are the following. First, of course, even 0.1 m solutions can be drastically nonideal in their behavior. Second, electrolytes of a given charge type tend to show similar variations of γ± with concentration at first but deviate from each other at higher concentrations. Third, the higher the product z+z_ , the earlier the deviations from ideality set in. The differences within a given family
TABLE 12-8. Mean Activity Coefficients for Aqueous Electrolytes at 25°Ca
Mean molal activity coefficient
Electrolyte 0.001 m 0.005 m 0.01 m 0.05 m 0.1 m 0.5 m 1.0 m 2.0 m 4.0 m HC1 0.965 0.928 0.904 0.830 0.796 0.757 0.809 1.009 1.762 NaCl 0.966 0.929 0.904 0.823 0.778 0.682 0.658 0.671 — KCI 0.965 0.927 0.901 0.815 0.769 0.650 0.605 0.575 0.582 HNOa 0.965 0.927 0.902 0.823 0.785 0.715 0.720 0.783 0.982
KNOa — —
—
— 0.733 0.542 0.548 0.481 —A g N 03 — 0.92 0.90 0.79 0.72 0.51 0.40 0.28 —
NaOH — — — 0.82 — 0.69 0.68 0.70 0.89
H2S 04 0.830 0.639 0.544 0.340 0.265 0.154 0.130 0.124 0.171
K2S O4 0.89 0.78 0.71 0.52 0.43 — — — —
BaCl2 0.88 0.77 0.72 0.56 0.49 0.39 0.39 — —
CuS04 0.74 0.53 0.41 0.21 0.16 0.068 0.047 — —
K4Fe(CN)6 — — — 0.19 0.14 0.067 — — —
a Adapted from W. M. Latimer, "The Oxidation States of the Elements and Their Potentials in Aqueous Solutions," 2nd ed. Prentice-Hall, Englewood Cliffs, New Jersey, 1952, and H. S.
Harned and Β. B. Owen, "The Physical Chemistry of Electrolytic Solutions," 3rd ed. Van Nostrand-Reinhold, Princeton, New Jersey, 1958.