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 1 0- 1 0 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
T A B L E 12-6. Equivalent Conductivities of Aqueous Ions at 25°Ca
Cation -Reinhold, Princeton, N e w Jersey, 1939.
b T o convert to SI units, multiply by 1 0- 4. T h u s A ( N a+) = 50.1 c m2 e q u i v "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.
- I n ^ = A / 7 [Eq. (10-20)]
equilibrium. We write for a slightly soluble salt M X the solubility equilibrium
M X ( i ) = MX(solution) = M+ + X~. (12-46)
The thermodynamic criterion for equilibrium is satisfied if we write
^ M X C O = /^Mx(solution)
as required by Eq. (9-44). We can introduce the activity of the electrolyte species:
/xM X(solution) = / ^ ( s o l u t i o n ) + RT In 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:
* s p = (M+XX-). (12-48) We know that the constant ATsp is well-behaved in dilute solutions. For example,
while we cannot avoid having essentially equal numbers of positive and negative ions present in any solution, we can, by using mixed electrolytes, vary the concentrations of specific kinds of ions, such as M+ or X~, independently. If S denotes the solubility of MX(i), then in pure water we have
* s p = S2. (12-49)
If added X~ ion is present, as in the form of a concentration C of NaX, then Eq. (12-48) becomes
tfsp = (S)(S + C). (12-50)
We now have the common ion effect whereby added X~ ion depresses the solubility of MX(i). Since the solution is still in equilibrium with the solid, aMX must not have changed, even though the individual values of ( M+) and (X~) are now quite different from before. Thus observation tells us that in dilute solution
αΜχ = (M+XX-). (12-51)
Use of Eq. (12-51) allows a more realistic treatment of colligative effects in dilute solution. The Gibbs-Duhem integration gives ln aMX , and for a single electrolyte solute M X , ( M+) = (X~) = m. We then have ln aMX = ln m2 = 2 ln 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
xx d(\r\ + 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 = ( * Μ+) ( * Χ ~ ) = * 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(ln x2) = -X l d(ln ax) = ^±χ1άΠ (12-52) or
^ n = l f ^ ( l n *2) = 2 f ^ .
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
^Π=2χ2. (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+ = y+m+ and a_ = y_m_. (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±\ (12-56)
The activity a± is known as the mean activity. Similarly, γ± is called the mean activity coefficient:
r ± 2 = r +y _ , (12-57)
and m± the mean molality:
m±2 = m+m_ . (12-58)
a2 = av+avs, (12-62) where ν = v+ + v_ , (12-63) a+ = y+m+, a_ = yjm. [Eq. (12-55)],
Y± = y++y: (12-64)
m±" = m"+mv-, (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
AgClC*) = Ag+ + C1-, (12-66)
*tn = aAg+acl. = (Ag+)(C1-) yA g +yc l- , (12-67) or
Atn = Ks py± 2, (12-68)
where Km is the thermodynamic solubility product and K8p 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 definitions 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 Μν +Χν- :
MV +X, _ ( 5 ) = V +M-+ + v_x*-, Κ8Ό = (M'+y+tX2-)"-, (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 K8p value. Evidently γ± 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 Κ8Ϊ> = log Kth - 2 log y t . (12-69) We then plot log KSTt 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 K8p 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, γ± 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 γ± for the ions Ag+ and C I- in the presence of added K N 03 by inserting Kth and the measured Kav into Eq. (12-69). Thus log ATSP = —9.645 at Vm = 0.175 or m = 0.0306; then log γ± = [(-9.790) - (-9.645)]/2 = - 0 . 0 7 2 5 , whence γ± = 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 Q3, K C 1 04, and so on were used instead of K N Q3. The
-9.600 h
, -9.700 h
-9.790 -9.800
T h e o r e t i c a l s l o p e —ν
Ο = M g S 04
— = K N 03
0.04 0.08 0.12 0.16 yfm~ (for KNO3)
V T ( f o r M g S 04)
FIG. 12-7. Variation of KBP for A g C l at 25°C with increasing K N OS concentration (plotted as Vm) and increasing M g S 04 concentration (circles, plotted as v7).
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 KSp 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 /, where
In the case of a uni-univalent electrolyte, I = m, but for, say K2S 04, / =
\(2m + Am) = 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 V7 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 o f 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 (12-70)
T A B L E 12-7. Dissociation Constant of Acetic Acid at 25">Ca
° Adapted from S. Glasstone, "Textbook of Physical Chemistry," p. 955. Van N o s t r a n d -Reinhold, Princeton, N e w Jersey, 1946.
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