COMMUNICATIONS TO THE EDITOR
1653
Logarithmic Term in Conductivity Equation for Dilute Solutions of Strong Electrolytes
Sir: The well-known Fuoss-Onsager equation1r2 for extrapolation of conductivities of (1: 1) electrolytes to infinite dilution is
A = A0
- S d i + EClog c + JC
in which X gives the limiting law. The Ec log c term makes a small but significant contribution in the accurate extrapolation to Ao, when a plot of (A -I- 8 4 ; Ec log c) us. c is applied, using an approximate value for A,. The form of E’ is (Ed,, - E 2 ) ,in which both El and Ez depend only upon solvent properties and temperature. Recently, Fernhidez-Prini and Prue, in a comparison of equations derived by Fuoss and by Pitts,4 state that both give the same value of E. This is certainly true for El, the dominant term, which is not surprising, since a common model is treated by closely similar mathematical methods. It is also true that Ez is the hame in both cases, but it has in fact a completely different origin in the two theories. Both derivations are based upon the Onsager continuity equation, for which the terminology of Fuoss and Accascina2 will be used. For conductance, with a static applied field, E , and the solution at rest, the continuity equation can be written
+
VzfJl(r)uJ1(rl,r) V ~ f l J ( - ~ ~ v-r) l ~ (=r ~0, which, as Vz = -V1 to
=
V and fJl(r) = fll(-r), reduces
~ f , , ( r{vJ1(r1,r) ) - vLJ(rz,-r)] =
o
I n this, v,, is the velocity of an i ion, situated a t distance
rz from an arbitrary origin, in the vicinity of j ion, situated at rl, v,,, rl, and rz all being vector quantities. As indicated, vJ1depends upon the position of the j ion at rl and the distance of the i ion from the j ion, r = rz - rl. Correspondingly, v13is the velocity of a j ion in the vicinity of an i ion. The dominant part of vJl is the velocity obtained from the product of its mobility, a,,and the total local force acting upon the i ion. The first approximation to this part alone is used in calculating the first approximation to the asymmetric potential, +,’, about a j ion. The calculations involved are fully presented by Fuoss and Accascina2 and by Pitts4 for their respective theories and are not relevant to the present note. By using a second approximation to this part of vJI,both theories introduce a complex group of exponential integrals into the second approximation to $J’; and, when these have been expanded and reduced for use at small concentrations, both theories lead to the same logarithmic term, EIAoclog c , for A. I n the second approximation, however, it is necessary also to introduce an additional term into vl,,
namely, vl(rz),the velocity of the fluid medium at the site of the i ion. Since the i ion has a finite size, this is really a fiction, and is taken as the fluid velocity which would exist at the center of the i ion if it were a point. I n the Fuoss theory, this is assumed to be determined by the velocity field in the fluid which is created by motion of the j ion. If the fluid velocity at a distance r from this ion is denoted by vj(r), then this can be evaluated and is put equal to v,(rz). Solving we find that further for the effect of v,(r) upon exponential integrals are introduced and expansion of these leads to the logarithmic term -Ezc log c in A. Pitts4 placed a completely different interpretation upon vl(rz) in that he has put it equal to the electrophoretic velocity produced at the site of the i ion by the action of the external field upon the whole ionic atmosphere about the i ion. The introduction of such a term is open to question, but, in any case, if it is used, it should be additional to the v,(r) term used by FUOSS, and, most important for the present note, it does not lead to any exponential integrals and thus gives rise to no term in c log c. On the other hand, in assessing the second approximation as it affects A, Pitts solved the hydrodynamic equation for the effect of the first approximation to the asymmetric distribution in the ionic atmosphere of the j ion upon its velocity. This introduces a term containing exponential integrals and, as pointed out by Fernhndez-Prini and Prue,3 these lead to the same contribution -Ezc log c upon expansion. To clarify the difference in approach, if the relaxation correction to be applied field E is denoted by AEIE, and the electrophoretic correction to A for the unperturbed ionic atmosphere is AS, the Fuoss expression for A is +J’,
A
= (A0
- &)(I + U / E )
Tvhile the Pitts expression is A = Ao(1
+ AE/E) - A,
- he’
Here, A,’ is the additional electrophoretic effect due to asymmetry and replaces A, X AEIE. In applying the Onsager continuity equation, the evaluation of vl(r2) by Fuoss as equal to v,(r) is more correct than that of Pitts, while the calculation of A,’ is more rigorous than A, X AE/E. On this basis, if the value of Ez is left unchanged, E should really be given by (Edo- 2Ez). As an example, for a (1:l) electrolyte in water at 25”, E1 = 0.5276 and E2 = 20.33, whence, for A, = 150, E = 38.48 instead of 58.81. The proportionate change in E varies considerably for other cases, e.g., (1) R. M. Fuoss, J . A m e r . Chem. Soc., 81, 2659 (1959). (2) R. hf. Fuoss and F. Accascina, “Electrolytic Conductance,” Interscience Publishers, New York, N. Y., 1959. (3) R. FernAndee-Prini and J. E. Prue, Z. P h y s i k . Chem. (Leipeig), 228, 373 (1965). (4) E. Pitts, Proc. Roy. Soc., A217, 43 (1953).
V o l u m e 7.4, N u m b e r 7
A p r i l 8, 1970
1654
CORII\IUKICATIOSS TO THE
it would be greater for lithium halides in water, due to the smaller values of A. and much less for halogen acids in water, due to the very high values of A,. Since the Ec log c term is small, the effect of the extrapolation of (A S d i - Ec log c ) vs. c upon A ~ i not s serious; but the gradient of the line obtained, giving J , can be seriously affected. Since J depends upon the distance of closest approach between ions, a, and is used to calculate a, the significance of such calculations becomes open to queBtion. Actually, the true relationship between J and a presents a difficult problem. As pointed out by FernBndez-Prini and P r ~ e the , ~ Fuoss theory and the Pitts theory give different expressions, since they employ different terms in the continuity equation, as just discussed, different mathematical methods of approximation, and different boundary conditions. A further discussion of these differences appears in a recent paper by Pitts, et aL6 Still other expressions arise if a synthesis of the Fuoss and Pitts
+
The Journal of Physical Chemistry
EDITOR
theories is attempted, along the lines presented for the E'c log c term, but a discussion of this requires a lengthy analysis which lies beyond the scope of this note. The sole point it is desired to make here is that the Fuoss and Pitt theories need not be mutually exclusive, since they are based upon the same model. The fact that, as they stand, both yield the same values of E is accidental. On theoretical grounds, the contributions of both to E seem to be sound and, if this is accepted, the correct value of E is (E& - 2E2) instead of (Elno- E2). (5) E. Pitts, B. E. Tabor, and
J. Daly, Trans. Faraday SOC.,65,
849 (1969).
NATIONAL CHEMICAL RESEARCH LABORATORYP. SOUTHAFRICANCOUNCIL FOR SCIEKTIFIC AND INDUSTRIAL RESEARCH PRETORIA, SOUTHAFRICA RECEIVED NOVEMBER 13, 1969
c. CARA1.W
