Paradox of the Activity Coefficient 7, Ei-lchiro Ochiai Junlata College, Huntingdon, PA 16652
The Paradox
Most modem physical chemistry textbooks deal with the activities and activity coefficients of an ionic salt (M,+Z+X,-2-) solution in the following way:
+
where n = n+ n-. It appears, therefore, that y+ represents a kind of geometric mean of y+ and y-. Let us use the simplest Debye-Huckel equation, which is derived in accord with the notations given ahove, to calculate y values:
where A = - d l . "P'in this eauation is the ionic strength . of the solution. Now let us do some calculations. Case 1:Let us add 0.1 mol of CuSO4 to a 0.2 m NaCl(0.2 mol in 1000 g water). We are interested in estimating y for Cu2+in the solution, but the textbooks suggest that we should calculate a+ or y* instead; so we do the following: Since y+2= yc,yso,, it may be assumed that: YC* = Y* = YSO,
(4)
Then:
(Here NaCl is assumed, as usual, to he providing an ionic environment). Case 2: Next let us suppose that we add 0.1 mol of CuClz to a 0.1 m NanS04 solution. A simple calculation will tell you that the ionic strength of this solution is the same as that of the first solution above. As a matter of fact, these two solutions are chemically the same. But let us move on. We calculate y+' (for CuC12) as follows: I ~ , appear (to an uninitiatNow, because y+'3 = ~ C , , ~ itCmay ed) that: y,, = y,' = yc, (becausethis Leads to ycuyc? = Y*'~) (7) Then: In y,? = A(2+)(1-) = 2A or yc, = 7,' = exp @A)
(8)
Here lies the paradox. We have come up with two different y+ values, which are supposed to represent yc, in the two chemically identical solutions.
lhs Solutlon and Suggestions Fortunately or unfortunately, the kinds of examples and exercises usually found in the textbooks do not encounter this difficulty. Obviously, some assumptions or suppositions are not valid in the argument above. But I suspect that an average student would have a hard time solving this mystery because no clue is given in the description above nor is any good clue given in the textbooks. The resolution of this paradox can be uncovered as follows: the supposition represented by eq 7, which may be derived from the general implicit assumption made in the first paragraph of this article, is invalid. Admittedly, no textbook states that such a supposition is valid; in fact none of them makes any comment on the relationship between y+ and the individual y+ or y- values (other than relevant to that of eq 1,that is.). I t can be shown that yc, = yso, = y+ in case 1is correct (within the Debye-Huckel approximation), whereas in case 2 yc, = y+I2and y a = .\/y&'. Therefore, yc, = y,(= exp (4A)) = yhf2(=exp (4A));thus, they are reconciled. The clue lies in the derivation of the DehytHuckel equation. Equation 2 is derived from the more fundamental equation:
where Z; is the electric charge of an ionic species i. The derivation (from eq 9) of the final limiting law for y+ (eq 2) is rather circuitous, though legitimate. Equation 9, therefore, occurs only in passing, and no significance seems to be attached to it in most textbooks. The problem, it seems, stems from the statements (made in the textbooks) that (1)activities or activity coefficients for individual ions cannot be measured because an individual ion cannot exist independently in a solution, and (2) therefore.. r+ .- is the onlv meanineful measure of activitv coefficient. The activitv and. hence, the activitv coefficient of an individual ion can, however, he measured-at least approximately by, for example, an ion-selective electrode. The classical glass electrode is an example. Therefore, statement 1does not seem to be warranted, and the corollary 2 may need to be abandoned, though it is still true that no solution containing only one kind of ion is possible. A logical solution to these problems would be to assume that we should he able to calculate (at least conceptually)the activity coefficient for an individual ion using eq 9 (or the modifications thereof). A treatment of ionic activity based on this equation would he, pedagogically, less confusing and more crisp and straightforward. If one chooses, one can still use the notation of y+ in comparing the experimental and theoretical y values in appropriate cases; but now this can be explicitly related to those of the individual ions.
Volume 67
Number 6
June 1990
489
