2876
ERNESTO J. DEL ROSARIO AND JOHNE. LIXD,JR.
Temperature Dependence of Electrolytic Conductance: Tetrabutylammonium Fluoroborate in Phenylacetonitrile1v2
by Ernesto J. del Rosario3and John E. Lind, Jr. Department of Chemistry, Cornell Unioersity, Ithaca, !Yew York
(Received March $, 1968)
The electrolytic conductance of dilute solutions of tetrabutylammonium fluoroborate in phenylacetonitrile has been measured between 25 and 176", and the theory of Fuoss and Onsager has been used to correlate the data. The variations of the physical constants, characterizing the solutions for the sphere-in-continuum model, largely compensate for one another so that the data present more of a test of the model than of the theoretical approximation of the model. The model appears adequate, although the association constant goes through a slight maximum rather than rising continuously with the inverse of the product of dielectric constant and temperature.
Recently, Fuoss and Onsager4 reexamined their theory for ionic conductance in dilute solutions and have obtained a more exact solution for the model of spheres in a continuum. They have shown the adequacy of the theory primarily by applying it to systems where the electrical properties and the viscosity of the solvent were varied by using mixed solvents a t 25". Data are presented here to show the adequacy of the theory and the model when the electrical properties and viscosity are changed by changing the temperac ture of a pure solvent. The measurements mere made on tetrabutylammonium fluoroborate in phenylacetonitrile over the range 25-175". This system is especially useful for showing whether there are inadequacies in the model rather than in the mathematical approximation of the model. The fractional change of the equivalent conductance with concentration is predicted by the model to be about the same at all temperatures for this system if Walden's rule applies. This means that all discrepancies caused by mathematical approximations should contribute proportionally the same a t all temperatures and not cause changes in the values of the arbitrary parameters needed to fit the equation to the data. The reasons for these statements are as follows. Since the thermal motion is always attempting to destroy the effects of the electrical forces, the electrostatic energy always appears divided by the thermal energy, and thus the dielectric constant is always T h e Journal of Physical Chemistry
multiplied by the temperature raised to the same power. Because the dielectric constant decreases with increasing temperature, their product for phenylacetonitrile is decreased by only 8% while the temperature changes 50%. So far as the electrostatic theory for the spherein-continuum model is concerned, little change is noticed in the association of the ions during this large change of the temperature. The relaxation-type terms in the equation for the equivalent conductance depend only upon the reciprocal of this product of temperature and dielectric constant and upon the limiting equivalent conductance at infinite dilution. Thus these terms are just scaled by the limiting conductance or by the inverse of the viscosity if the Walden product is constant. The electrophoretictype terms not only depend on the inverse of the product of dielectric constant and temperature but also upon the reciprocal of the viscosity. The result is that the whole equation is just scaled by either the limiting conductance or the reciprocal of the viscosity. Now if there are effects such as significant van der (1) This study was aided by a grant from the Office of Saline Water, U. S. Department of the Interior. (2) This paper is based on a thesis presented by E. J. del Rosario to the Graduate School of Cornell University in partial fulfillment of the requirements for the M.S. degree, Jan 1966. (3) Ford Foundation Scholarship, University of the PhilippinesCornell University Graduate Education Program. (4) R. M. Fuoss, L. Onsager, and J. F. Skinner, J . P h y s . Chem., 69, 2581 (1965).
TEMPERATURE DEPENDENCE OF ELECTROLYTIC CONDUCTANCE
Waals forces between ions or between ions and solvent molecules, these should cause marked changes in the parameters of the equation since these forces are not directly related to the dielectric constant and the viscosity.
Table I: Dielectric Constant of Solvent OC
Dielectric constant
24.9 58.9 78.6 99.0 117.7 138.9 162.8 185.4 201.4
18.77 16.57 15.46 14.45 13.53 12.69 11.86 11.14 10.72
Temp,
Experimental Section Tetra-n-butylammonium fluoroborate was prepared by titrating a methanolic solution of tetrabutylammonium hydroxide with an aqueous solution of fluoroboric acid. The salt was recrystallized three times from 1 : l methanol-water mixtures and dried at 70" under vacuum to constant weight. The melting point was 161". Fluorine analyses yielded 22.99 and 22.86% compared to the theoretical 23.08%. The phenylacetonitrile was dried over molecular sieves and carefully fractionated. The specific conductance was less than 0.3 X lo-' mho/cm and the refractive index was n Z 51.5208. ~ The dielectric constants of the solvent were measured on a General Radio 716-C Schering bridge equipped with a 716-P4 guard circuit. To reduce the effects of the higher solvent conductances at high temperature, measurements were made at 263 kHz and checked at 100 kHz. The capacitance cell consisted of an inner high potential electrode 11/8 in. in diameter and 5g/16 in. long which was suspended in a deep cup leaving an annulus of "16 in. The cup was the grounded electrode and the top inch of it mas a guard ring. All metal parts were of stainless steel and the insulation was Supramica. The capacitance cell was calibrated at 25" with nitrobenzene and ethylene dichloride whose dielectric constants are 34.82 and 10.36, respe~tively.~ The nitrobenzene was purified by the methode used previously, yielding a specific conductance of less than 7 X mho/cni. The Eastnian Spectrograde ethylene dichloride was redistilled and had a specific conductance of much less than 2 x low9mho/cm. The cell constant was 0.04213 which was independent of temperature within the precision of the measurements. The results in Table I agree with the approximate value of Grimm and Patrick5!' and are about 1-2% above those of Walden.* The viscosities were measured under argon in Ubbelohde viscometers. The densities were measured in a Sprengle-Iype pycnometer on which the two arms were interconnected and the whole pycnometer was completely submerged in the thermostat. The small vapor space required a maximum correction for the vapor volume of 0.03y0 at 200". The conduci ance bridge, the method of calibration, and the technique for the conductance measurements were similar to those used by one of the authors pre-
2877
viously.6 The cells used were two erlenmeyer cells whose cell constants were 0.15591 f 0.00002 and 2.0714 f 0.0003 cm-'. Both cells were calibrated directly with KC1. A small cell with a volume of about 20 ml was also used for the high-temperature runs and the cell constant was 2.8278 0.0003. The temperatures were measured by a platinum resistance thermometer and the variation was no greater than +0.03" at the highest temperature. All runs were perfornied with the solutions under an atmosphere of argon. The runs at 25 and 100" were performed by diluting the solutions in the erlenmeyer cell. The run at 75" was made in the same cell by diluting the solution successively in a flask at rooin temperature and then transferring a portion of the solution by a syringe to the cell. This could be done easily because the vapor pressure of the solvent at room temperature was so low that there was no significant aniount of solvent lost through evaporation. This technique had to be used for the runs at 150 and 175" since decomposition was noticeable and thus a single sample could not be diluted for measurement at five different compositions. The small cell was used for these two hightemperature runs and the vapor space at equilibrium was only about 1 ml, so that no correction for the amount of solvent in the vapor space was needed. The decomposition at 150" caused a resistance change of 0.1% in 30 min and at 175" the change was 1% in the same period. For these two runs the resistance was measured only at 10 kHz since the runs at lower teniperature indicated that the value extrapolated to infinite frequency differed by only about 0.01%.
*
~~~
~~
( 5 ) A. A. Maryott and E. R. Smith, "Tables of Dielectric Constants of Pure Liquids," National Bureau of Standards Cirrular 514, U. S.
Government Printing Office, Washington, D. C., 1951. (6) J. E. Lind, Jr., and R. M.Fuoss, J . P h y s . Chem., 65, 999 (1961). (7) F. V. Grimm and W. A. Patrick, J . Am. Chem. Soc., 45, 2794 (1923).
(8) P. Walden, 2. P h y s i k . Chem., 70, 669 (1910).
V o l u m e 7 0 , iyumber 9
September 1966
ERNESTO J. DEL ROSARIO AND JOHN E. LIND,JR.
2878
The resistance a t 10 kHz was then extrapolated back to zero time, ignoring the initial period when the temperature had not reached equilibrium. A run was made a t 200* but discarded because resistances changed of the order of 10% in 0.5 hr. The physical constants are summarized in Table I1 and the conductance measurements in Table 111. The values of the dielectric constant in Table I1 are interpolated from Table I.
Table 11: Solvent Properties
Dielectric constant
Density,
OC
g/ml
Viscosity, poise
Specific conductance, 107 mho/cm
25.00 75.00 100.00 150.00 175.00 200.00
18.77 15.66 14.38 12.31 11.45 10.67
1.0125 0.9719 0.9506 0.9079 0.8866 0.8637
0.01971 0.00949 0.00716 0.00473 0.00396 0.00334
0.13 0.60 0.51 0.84 0.75 2.0
Temp,
Table I11 : Conductance of Tetrabutylammonium Fluoroborate ir Phenylacetonitrile 104c
t 31.004 24.720 18.432 12.450 6.290
t 28.929 23.194 17.393 11.695 5,876
AA
A
104c
A
AA
t = 75.00’
= 25,OOO 21.882 -0.008 22.664 0.011 23.635 0.002 24.858 -0.012 26.711 0.004
18.664 14.873 11.072 7.401 3.976
100.00” 57.284 0.001 59.427 -0.011 62.218 0.027 65.772 -0.021 71.120 0.006
17.726 13.886 10.304 6.898 4.502
48.137 49.697 51.628 54.045 57.197
-0.006 0.008 0.002 -0.010 0.004
t = 150.00°
=
90.06 93.74 98.03 103.00 107.59
0.00 -0.04 0.05 -0.04 0.01
t = 175.00”
16.037 9.642 6.479 4.095
105.84 114.46 120.19 125.78
-0.00 0.02 -0.02 0.01
Discussion Fuoss and Onsager obtain eq 1 when they retained the Boltzmann factor in the exponential form during the integration of the equation of continuity A = .lo- Scl”
+ E’c In r 2 + Lc - AAocf2
(1)
All terms depend upon the adjustable parameter .io and the last two also depend upon the ion-size The Journal of Physical Chemistry
parameter, which represents the distance of closest approach of cation and anion. The A term has the form of the Bjerrum approximation to the association constant. FUOSS, Onsager, and Skinner have shown that this equation is applicable where “association” is very low. For systems where association is high and especially for association arising in part from forces not accounted in the model such as van der Waals forces, the law of mass action can be introduced by replacing the last term by the analogous expression given by the law of mass action. The conductance equation becomes
where the association constant, K A , becomes the third independent parameter. To either equation an additional term was proposed to account for termination of power series arising out of lower order effects. This B term is [r3(21)- 3)Ao/b2] and is substracted from the right-hand side of the equation. It is of the three-halves power of the concentration and depends upon only the same arbitrary parameters, .ioand a, upon which the L term depends. In Table IV are tabulated the parameters and the terms required to fit the data with eq 2 both with and without the B term. The precision of three of the five sets of data is increased appreciably by the addition of the B term, while the other two are not affected. At the same time, the association constants K A increase 60% but do not change significantly their dependence upon temperature and dielectric constant of the solvent. The values of the ion-size parameter, a&, arise from the L term and the B term, and they become larger and more reasonable when the B term is used. The values aL both with and without the additional term decrease at high temperature. It is important to remember that the L term and the B term vary the most rapidly of any terms with concentration. Thus they are small terms which are most sensitive to the final fit and will gather into the parameter uL all the inadequacies of the theory and the model. Because the Walden product is nearly constant, the coefficients of the conductance equation with the B terms are scaled by the limiting conductance and are given in Table V. As predicted, the scaled values vary slowly with temperature. The one exception is the L term at high temperature which drops to half its lowtemperature value. This result is reflected in the slight decrease of K Aat high temperatures. To account for this variation, consider the electrostatic model modified with a “solvation” energyg (shown in eq. 3)
2879
TEMPERATURE DEPENDENCE OF ELECTROLYTIC CONDUCTANCE
Table IV:
Constants in the Conductance Equation
oc
L
KA
c
Aa
25.00f0.01 75.00 & 0 . 0 2 100.00&0.02 150.00 & 0.03 175.00 f 0.03
0.01 0.01 0.03 0.05 0.02
31.16 4IO.05 65.20i0.04 8 3 . 8 i0 . 1 125.0 i 0 . 3 145.3 f 0 . 1
Without B Term 202 f 34 711 f 63 283 f 68 -645 f 327 -985 f 171
94 f 7 145 f 4 117 & 5 119 f 13 114 f 6
4.1 & 0.1 4.56 f 0.09 3 . 9 5 & 0.07 3.4 & 0.2 3.40 =t 0.08
5 . 5 f0 . 2 5.00 f 0.05 5.6 & 0.1 5 9 =t 0 . 3 6 . 2 =k 0 . 1
25.00 f O . O 1 7 5 . 0 0 2 ~0.02 100.00 1 0 . 0 2 150.00f0.03 175.00 4 ~ 0 . 0 3
0.002 0.002 0.04 0.06 0.007
31.444i0.007 65.44f0.01 84.7 f0.2 125.5 =t 0 . 3 145.72 i 0 . 0 4
With B Term 1256 f 6 2758 4I 17 3463 f 157 2912 f 413 3017 f 57
175.7 f 0 . 9 207 i 1 214 i 11 178 rk 15 167 & 2
7.75 f 0.02 7.86 4I 0.03 7 . 6 Z!C 0 . 2 5 . 7 f0 . 3 5.48 f 0.03
4.486 =tO0.0O7 4.510=tO0.O06 4 . 6 0 f 0.06 5 . 1 410.1 5.38 f 0.02
t,
4irNf.X~~ K a = ____ exp(e2/aDkT - E,/kT) 3000
(3)
where the first term in the exponential is the coulomb energy and E , is the remainder of the energy difference between the free ions and the ion pair. As can be seen in Figure 1, where the two lines represent eq 3 with E, of zero and a K of 5.5 and 7.5 A, no single value of a K will fit the equation to the data. The several values needed to fit the data are given in Table IV. A single value of E, other than zero does not represent the data. I n fact, for these two values of a K , E, has values favoring ion pair formation which go through a maximum of 400-800 cal/mole, respectively, and decrease to a few hundred at higher temperatures. Such an effect could be caused by van der Waals forces. These forces might decrease with temperature because the motion of the chains of the cation increases with temperature resulting in fewer of the atoms of the chains interacting with the anion. The accuracy of the data and the questions arising in the definition of an ion pair, which primarily affect the preexponential term in eq 3, do not warrant a more careful analysis of the association constants. Table V : Scaled Constants for the Conductance Eq 2 with the B Term o c
S/Ao
E’/&
Li/Aoa
L/Ao
Aov
--25
3.736 3.836 3.943 4.107 4.227
14.51 15.81 16.56 18.12 18.93
49.43 53.76 56.33 61.62 64.42
39.95 42.2 40.8 23.2 20.7
0.620 0.621 0.607 0.594 0.577
t,
10
100 150 175 a
L1is the part of L which is independent of the parameter a.
0
0.01
0.02
loo/( DT). Figure 1. Comparison of the experimental values of the association constant with eq 3 for E, = 0.
I n conclusion, the model of spheres in a continuum of dielectric fluid seems to account well for the electrolytic conductance. However, further investigation should pursue the finer detail of the small deviations of the association constants in such systems where any limitations of the theory are minimized and the inadequacies of the model are more readily seen. (9) H. Sadek, E. Hirsch, and R. M. Fuoss, “Electrolytes,” Pergamon Press Ltd., Oxford, 1962,p 134.
Volume 70, ,Vumber 9 September 1966
