A Note on the Limiting Equivalent Conductance of Electrolytes PIERRE VAN RYSSELBERGHE University of Oregon, Eugene, Oregon TUDENTS of physical chemistry invariably marked diicultV in trvinz to obtain a clear understanding of the concept of limitinc equivalent conductance or equivalent co'nductance at infinite dilution. Current texts of physical chemistry fail to give them anv effective heln because thev all omit the mathe,~ matical reasoninp'necessarv for ;he exact definition of the limiting equivalent conductance. Yet, this reasoning does not require more than the mere rudiments of calculus. That the specific conductance L of the electrolyte decreases with concentration and vanishes a t zero ,concentration when allowance is made for the conductivity of the solvent is clear to everyone. That, a t all finite concentrations, there is an equivalent conductance connected with the specific conductance and the equivalent concentration C (in equivalents per liter) or the dilution V (in liters per equivalent) by the relations ~
-
* = lWOL c = 1000 LV
(1)
is usually acceptable to everyone. The existence of a finite limit Ao for A a t infinite dilution is, however, frequently misunderstood as long as purely intuitive arguments are used. The following procedure, on the other hand, removes the difficulty. As C tends toward zero, L also tends toward zero, and the expression for A becomes of the form 0/0. In other words, & is an indeterminate form. The indeterminacy is removed by taking the ratio of the derivatives of L and C with respect to C (L'Hospital's rule), C being regarded as the independent variable:
lim A =
= 1000 lim
JL JC
(3)
since the experimental curve describing the variation L with c is smooth, continuous, starts at zero for c = 0, and since its slope is also finite and continuous 2r
a t all values of C, it is clear that the derivative
has a
finite limit a t C = 0. Among the intuitive arguments which can be used to explain the meaning of Lo,we have found the following one to be the most satisfactory: The equivalent conductance a t any finite coucentration can be regarded as the sum of the velocities of the ions per unit of potential gradient multiplied by the charge of an equivalent (96, 494 coulombs) and multiplied by the degree of dissociation. These velocities increase with dilution, since the interference due to the ionic atmospheres gradually vanishes. The degree of dissociation is equal to unity a t all small concentrations, or, in the case of weak electrolytes, it tends toward unity a t infinite dilution. The ionic velocities a t infinite dilution are those which would actually be exhibited in a solution containing one single dissociated molecule of the electrolyte in any finite volume, say, one milliliter. The sum of these velocities is multiplied by the charge of one equivalent in order to obtain the conductance corresponding to 6.023 X loa3(Avogadro's number) pairs of elementary plus and minus charges (i. e., 6.023 X loa3K+ and 6.023 X loa3C1- ions in the case of KC1, '/a X 6.023 X Mg++ and X 6.023 X loz3 SO4-- ions in the case of MgS04, X 6.023 X Al+++ and 6.023 X loz3C1- ions in the case of AlCls, etc.). The true concentration of such an "infinitely" dilute solution would be 1000/ mole per liter. 6.023 X lozaor 1.660 X
