Sept., 1934
THETEMPERATURIC COEFFICIENT OF CONDUCTANCE
Summary The dissociation constants of some hydrocarbon derivatives of boric acid have be,en determined. These constants have been coin-
[CONTRIBUTION FROM THE CHEMICAL
1857
pared and discussed on the basis of the resonances and negativities of the various groups involved. BERKELEY, CALIF.
RECEIVED MAY1, 1934
LABORATORY OF BROWNUNIVERSITY ]
Properties of Electrolytic Solutions. XI. The Temperature Coefficient of Conductance BY RAYMOND M. Fuoss
I. Introduction It has been shown’ that the conductance of many electrolytes up to concentrations cormsponding to the minimum in conductance can he described in terms of the hypothesis that free ions, ion pairs and ion triples are present. The constants describing the equilibria have been calculated as functions of ion size,2 dielectric constant and temperature. It should therefore be possible to calculate the temperature coefficient of conductance in the above range of concentration. In this paper we shall derive for the case of weakly dissociated electrolytes an explicit expression for the following function
where ii = equivalent conductance, T = temperature, a = ion size, D = dielectric constant, c == concentration and q = solvent viscosity.
11. Calculation of the Temperature Coefficient For the case of binary electrolytes in solvents of dielectric constant under 10, the conductance over a considerable concentration range is given by the following limiting form of the general conductance equation Ag(c) = Ao
-/d + (Xo/ /lo.'@ The general formula for k in o w fourth paper" contains an e m r which, fortunately, does not affect the resilt's: the upper limit of integratioxi for 8 should not extend to T for all values of the distance r . The limits are 0 and T for r > 2a and 0 and 2 a / 3 fbr r = a. For 2a>rba, the upper limit is arc cos(r/&). Integrating from ( 2 r / 3 ) to T contributes a very small amount to' the integral, because the-integrarl&becomes unity for r = a, 8 = 2k/3 and approaches z m ertponekti tially for larger values. Piacticalfy' tbe entire value of the integra1 for large (i. e., b 10) values of b comes from values of the integrand in the neighborhood of r = a: 8 = 0. If distances are measured'in multiples of a, we have then
b
(1 =
+ xe + 2% cos 8
V )
sin 0 d0
(13)
2r~~3~(b)/1000
In (13), K-' is represented as the volume under a surface in the x-8 plane. As pointed out in the previous paragraph, this volume has an expodential peak near the point (1,O): we therefore look (10) For simplicity, w e shall drop the subscript on bt throughout this section (11) Fuoss and Kraus, THISJOURNAL, SS, 2387 (10331
1859
6 9 e - 3 W 4 d6
whikh gives
&mbining,the above results, we'obtain and 64irNa3 b -1nk~In+i.-2lnb+ 3000
whfchi is &e rewlt used above- itt dkriting' ('7). The &eive et?@ in usirlg (173'insteadl of. the pkdWter*ni&d;to eud6bte the 16(4b&hrn Of I% isd'thk!&der df &**, bemuse the lek&g, tern of (l?)is b/2 add &e e m f is in1terms cstit.airiing b2
irl the?dWiflh&tW. SUXllmilry 1. A theoretical derivation df the temperature coefficieht of the conductance of elcctrol~tes~ in solventaof low dielectric constant iwpresentkd! 2 . This c o d c i e n t is shown, tu con'tain, three terms: a viscosity term and two tkms arising from the shift of the simpler ionic equilibria h t h temperature. 3. The dependence of the coefficient on teknperature and concentration is discilswd. 4. An asymptotic expansion for the calculation of the triple ion constant, valid.fot solvezits of dielectric constant under 10, is obtained2 PROVIDENCE, R. I. RECEIVED &TAY 16, 1934 (12) Fuosn, ibrd., 16, 1030 (1934)
