T H E L I X I T I N G LAW FOR TRANSFEREKCE XCMBERS B Y MALCOLM DOLE
Scatchard' and Jones and Dole2 have recently published empirical equations connecting the transference numbers of strong electrolytes with the concentration, but as yet, apparently, no one has attempted to give these equations any theoretical basis. Indeed, on the basis of the old Kohlrausch conception of constant ion mobility the transference numbers mould be constant and independent of the concentration,-an idea which would discourage any effort to determine the relationship between transference numbers and the concent rat ion. The more modern views regarding the change of ion mobility with the concentration as brought to final mathematical perfection in the work of Debye and Hucke13 and of Onsager' offer a possible source of explanation for the transference number variation with the concentration since according to the "theory of complete dissociation" the ion mobilities are not independent of the concentration but depend very definit,ely upon the ionic constitution of the solution. At great dilution the ion mobilities, or on the assumption of complete dissociation the ion conductances are given by the Debye equation &\i
=
i(m
- aid;
(1)
In equation ( I j A i is the ionic conductance of the ith ion at the concentration c, AT is the ionic conductance at infinite dilution and ai is a constant which may be calculated according to the equation of Onsager or which may be estimated from the experimental data. The transference number, t , of the positive ion of the salt PN, for example, is given by the equation (by definition)
Substituting the values of the ionic conductances as given by the Debye equation (I) into equation ( 2 ) we obtain an equation between the transference number and the concentration as follows:
In equation (3) -1,D and B are all constants, independent of the concentration. They are given by the following equations:
1
G. Scatchard: J. Am. Chem. SOC.,47, 696 f19zj). G. Jones and 11. Dole: J. Am. Chem. SOC.,51, 1073 (1929). P. Debye and E. Huckel: Physik Z., 24, r8j, 3 0 j (1923). L. Onsager: Physik Z., 28, 277 (1927).
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MALCOLM DOLE
B=+-
a,
ap
+ a,
Equation (3) is the limiting law for transference numbers. Scatchard's equation for the transference numbers is
t,
=
tz
+k f i
(7)
Equation (3) may be expressed in a series as follows:
By dropping off all terms beyond __ Scatchard's equation D* inasmuch as
( j ) is
obtained
This analysis indicates that Scatchard's equation is an approximate form of the theoretical limiting law. Jones and Dole's transference equation which they somewhat accidentally discovered is
'
t "dZ/c+N
+R
By referring to equation (3), it is readily seen that Jones and Dole's equation has exactly the same form as the Zimzting law. This seems rather remarkable inasmuch as the Jones and Dole equation is valid for all salts as yet studied up to the high concentration of one normal whereas the limiting law is theoretically valid only for very dilute solutions. However an examination of the constants shows that in the two cases they are quite different; for example, the limiting law gives for B the value 0.5, the analogous constant, R, in the Jones and Dole equation has the value -I. At very small concentrations the two functions for the transference numbers will coincide since a t zero concentration, M/N R must equal B - AID
+
+
In conclusion we may remark that the Debye theory gives by far the most satisfactory explanation of the transference number variation with the concentration. Chemical Laboratory, Yorthwestern LJniverdy L vanston, Illinois.
