J. L. DYEAND F. H. SPEDDING
888
From these data, it is, of course, impossible to say how far the Debye-Hiickel limiting law involving perchlorates and nitrates will hold for activity coefficients. However, in light of the fact that a constant value of d results in agreement
[CONTRIBUTION No. 279
PROM THE
Vol. 76
of experiment and theory for the chlorides and bromides up to 0.1 N , it is reasonable to assume that the limiting law would hold equally well for the perchlorates and nitrates. AMES,IOWA
INSTITUTEFOR ATOMICRESEARCHAND DEPARTMENT OF CHEMISTRY, IOWA STATE COLLEGE1]
The Application of Onsager’s Theory of Conductance to the Conductances and Transference Numbers of Unsymmetrical Electrolytes BY J. L. DYEAND F. H. SPEDDING RECEIVED JULY 27, 1953 Consideration of the transference number behavior of unsymmetrical electrolytes in aqueous solution led t o an examination of the mathematical development of Onsager’s theory of conductance. A mathematical treatment of Onsager’s theory of conductance is described, which employs graphical methods t o evaluate integrals which were only approximately evaluated by Onsager. The approximate methods, while satisfactory for 1-1 electrolytes, are unable t o explain the transference number behavior of unsymmetrical electrolytes. The new treatment of the theory was applied to the conductances and transference numbers of CaClz, XdCla and ErC13. The agreement between theory and experiment was greatly improved for both transference numbers and conductances of these unsymmetrical electrolytes.
Introduction expansion and subsequent neglect of high order In a theoretical study of the processes of diffusion terms in the usual treatment of the electrophoretic and electrical conduction in electrolytic solutions, effect, is invalid even a t quite low concentrations. Onsager2 developed equations which quantitatively In attempting to include the complete integral, describe the changes in conductance with con- however, the equation could not be put into an centration in sufficiently dilute solutions. Because analytical form, and certain integrals were evaluthe theory was developed for the individual ionic ated graphically. The results are compared with species in the solution, the results were also applic- experimental values of the transference numbers able to the transference numbers of the constituent anti conductances. ions. The theory predicted the transference numDevelopment of the Method ber to be a linear function of the square root of the concentration in dilute solutions. This has actuFrom the treatment of the electrophoretic effect ally been found to be the case for most electrolytes, upon conductance given by Onsager2 one may oband the slope of the resultant straight line agrees tain the electrophoretic correction to the conductquite well with the predicted slope for 1-1 elec- ance to be trolytes. I t has been pointed out, h ~ w e v e r , ~ - ~ that unsymmetrical electrolytes for which the transference numbers have been measured exhibit slopes which differ by as much as a factor of onein which n,i is the concentration of the i t h kind of fifth from the slope predicted by the Onsager equaion a t a distance r from a n y j ion and is given by tion. Because of this dependence upon charge type, the Boltzmann distribution to be approximately i t was deemed advisable to re-examine the mathematical development of the conductance equation. The change in the conductance of an electrolytic in which $! is the electrostatic potential a t a dissolution with concentration may be said to be due to two factors, which are called the “time of relaxa- tance r from the j t h ion and was found by Debye tion effect” and the “electrophoretic effect.” and Hiickel to be The electrophoretic effect normallv makes the larger contribution to the conductance. If the fundamental assutnptions of Onsager’s treatment T h e quantities in these equations are defined by of conductance are assumed to be correct, it can be 11 15 t h e viscosity of the solvent shown that [or icms of high charge ari exponential c , is the charge O I L the it11 ion 31id i~icludesthe slgil of the ( I ) Work \*as prrfurmed iu the I\rnrs Laboratury of t h e Atomic Energy Commission. (2) (a) L. Onsager, Physik %.. 28, 277 (10!27), (b) L. Onsager and R . M. Fuoss, J . Phys. Chem., 36, ?I389 (1932). (3) H. S. Harned and B. B. Owen, “The Physical Chemistry of Electrolytic Solutions,” Second Edition, Reinhold Publ. Corp., h-ew York, N. Y . , 1950, p. 165. (4) L. G. Longsworth a n d D. A. blaclnnes, THISJOUKNAL. 60, 3070 (1038). ( 5 ) F. H. Spedding, P. E:. Pixttlr a n d J 51. %‘right, i h r d . . 74, 2778 (1952).
charge on the ion k is the Boltzmanu constant D is the dielectric constant of the solvent a, is the mean distance of closest approach of the ions t o the j ion
the total number of kind$ of ions in the solution, :inti
s
ii
e
is t h e charge on the electron
Feb. 5 , 1954
UNSYMMETRICAL ELECTROLYTES : ONSAGER'S THEORY APPLIEDTO CONDUCTANCES S89
Combining 1, 2 and 3 gives for the electrophoretic correction to the conductance
and for this system
I Ahjl in which N is the Avogadro number (5)
For an electrolyte which dissociates into v ions by the scheme M"+d y- --+ v+ JI'+ + v.-.1*and having a mean distance of closest approach of the ions, a , equation 5 may be specialiied to
and
in which = Kr 96,500Dk T Jf = 1800re7(/2+ 1 12p
+
C , is the concentration of the salt in moles per liter
We may let the integrand of equations 6 and 7 be represented by G+ and G-, respectively. The electrophoretic correction can then be determined at any specified concentration and a value, by a graphical integration of a plot of G+ against p, and G- against p, from K a to infinity. It is to be noted that M is dependent upon the charge type of the electrolyte, and P is dependent also on the value of a, the mean distance of closest approach, and the concentration of the solution. Typical plots of G+ against p and G- against p are shown for 0.00333 M neodymium chloride ( a = 5.49 X lo-* cm.) in Figs. 1 and 2 . For comparison purposes, plots of the integrands of the usual Onsager equation also are included. These plots of the usual equation include the complete first term of the expansion of the integrand not neglecting 8.
I)
AN ION
-COMPLETE
INTEGRAND --FIRST TERM OF EXPANSION OF INTEGRAND
CATION COMPLETE INTEGRAND - - - -- - - FIRST TERM OF EXPANSION OF INTEGRAND
6-
t 4
z 0 c 0 K
a
0 0
0
Ew
2
Q
K 0 I
t
4-
8
8 K
'!
0
IW
4a!
z 0
W
5-
z
2K
1 1
b W
A W
a 3-
0
c P
K
z a
K
0 W I-
r
0
Fig. l.-Comparison of the integrands of the ordinary arid extended equations for the electrophoretic correction on the conductance of 0.00333 M NdCla a t 25".
Fig. 2.-Comparison of the integrands of the ordinary and extended equations for the electrophoretic correction on the conductance of 0.00333 M NdC13 a t 25".
One car1 obtain the area uiider ffit: curve from a large value of p to infinity analytically instead of graphically because e - p / e becomes very small for
H . SPEDDING CATION
PLUS ANION 0.48 COMPLETE INTEGRAND FIRST TERM OF EXPANSION OF INTEGRAND
---__-
12-
4 z
1 I
Vol. 76
-
__ CALCULATED BY THE SIMPLE ONSAGER EQUATION
____-ONSAGER
LIMITING SLOPE CALCULATED BY THE EXTENDED EQUATION
0
2z 10044
Y K
K
s p
8-
ti
I
Q
t
5
6-
W
K
‘!
e 0
3
4-
O0 L
I
P.
2
a32
3
Fig 3.-Comparisoii of the integrands of the ordinary and extended equations for the electrophoretic correction on the conductance of 0.00333 Jd XdCla a t 25 ’.
large values of p . Let F, be the area under the curve Gj from a large value of p, say B , to infinity. Then F,is given by IF,/
=
+ Iz,s,I]e-B
;ZdP[Iz,/*
(8)
This value of the integral from B to infinity is added to the integral from Ka to B which is determined graphically. It is not necessary to perform a new graphical integration for each value of a used in this treatment. To determine the electrophoretic correction a t a value of a different from that used in obtaining the graph, one uses the first graphical integration but with a different lower limit and a t a slightly different concentration. This concentration is determined from K in the expression for P. For the case of 3-1 electrolytes P = 7.135 X lo-* ( e K @ / ( l K a ) ) (9) The same value of P that was used in the original graphical integration is used to solve for K by successive approximations, using the new value of a, and equation 9. The integral from the old lower limit K’a’ to the new lower limit Ka,may be determined graphically. The expressions for AA+ and AA- due to the electrophoretic effect may be combined with the “time of relaxation” corrections to obtain the individual ionic conductances, A+ and A-, from
+
I
I
I
Feb. 5 , 1954
UNSYMMETRICAL ELECTROLYTES : ONSAGER'S THEORY APPLIEDTO CONUUCT~INCES 891
I -CALCULATED BY THE SIMPLE ONSAGER EQUATDN ---_ONSAGER
15C
LIMITING SLOPE
BY THE - -CALCULATED EXTENDED EQUATION _.___EXPERIMENTAL
.. ,
(LCNGSWORTH)
140
., .... . .. EXPERIMENTAL (ALLGOOD,
______
4
CALCULATED BY THE SIMPLE ONSAGER EQUATION.
'\
LEROY AND GORDON)
CALCULATED BY THE EXTENDED EOUATION EX PER IMENTAL
a I30 W
V
z
2 V
a n 120 z 0
0 I-
z
W -1
2
110
3 W 0
Ioc
9c I I 0.10 0.15 (NORMALITY)~.
I
0.05 (NORMALITY) k.
0.2
Fig. 5.--Cornparison of theoretical and observed transference numbers of calcium chloride at 25 '.
Fig. 6.-Comparison of theoretical and observed equivalent conductances of erbium chloride a t 25".
ordinarily applied. The smoothed experimental values are also listed as well as the percentage deviations of the calculated values from experiment. The table for calcium chloride transference numbers gives the average of the values obtained by Allgood, LeRoy and Gordon,8 and Longs-
worth.g The transference numbers as calculated and measured for ErC13 and CaClz are plotted in Figs. 4 and 5. The calculated and observed conductances of ErC13 are plotted in Fig. 6.
ENCE TABLE I COMPARISON OF THEORETICAL AND OBSERVED CONDUCTAXES OF NEODYMIUM CHLORIDE,ERBIUMCHLORIDEASD CALSalt CIUM CHLORIDE AT 25'
Salt NdCla
ErCla
CaClz
Normality
0.00009294 .0003340 ,003507 .01002 ,03506 ,07520 .00009291 .0009993 ,003489 ,009945 ,03423 .07257 .0001000 .0009000 .002114 .01000 .02000 .04000
Measured conductance
143.1 137.1 130.4 122.1 109.9 102.1 139.7 134.0 127.5 119.3 108.2 99.6 134.1 130.6 128.0 120.3 115.6 110.3
NdCla
Calculated conductance ExOnsager tended
142.91 136.88 129.14 117.72 93.35 69.06 139.64 134.22 126.03 114.84 91.37 68.15 134.06 130.53 127.71 118.16 110.85 100.50
143.05 137.37 130.53 121.05 103.87 89.50 139.77 133.59 127.72 118.76 102.73 89.15 134.08 130.76 128.09 119.38 113.49 105.47
Deviation, % OnsaExger tended
0.14 0.03 0.16 .20 1.00 .lo 3.6 .80 15.0 5.5 32.3 12.3 0.04 0.05 0.33 .l6 1.2 .17 3.8 .42 15.5 1.1 31.6 10.2 0.03 0.01 .05 .l2 .24 .07 1.7 .76 4.1 1.7 8.9 4.3
TABLE I1 THEORETICAL AND OBSERVED TRANSFERNUMBERS O F h'EODYMIUM CHLORIDE, ERBIUM CHLORIDE AND CALCIUM CHLORIDE AT 25' T + calculated Deviation, Yo T+
COMPARISON O F
ErCla
CaClz
Normality
measured
Onsager
Ex-
tended
Onsager
0,00009294 . , , . 0,4731 0,4738 , , . ,0003340 ..,. ,4655 ,4695 , .. ,003607 ..,. ,4551 ,4657 , , , ,01002 0.4650 ,4370 ,4592 4.8 ,03506 ,4542 ,3833 ,4469 14.2 ,07520 ,4440 ,2924 ,4339 32.6 .00009291 . ., . ,4605 ,4610 ., , ,0009993 ., , ,4524 ,4563 .,, 003189 .... ,4411 ,4522 ,.. ,009946 ,4471 ,4217 ,4451 R.7 ,03423 ,4366 ,3658 ,4318 16.2 ,07257 ,4262 ,2723 ,4182 36.17 .0001000 .... ,4361 ,4364 , .. .0009000 . .. ,4322 ,4329 .,, .002114 .... ,4289 .4313 ._. .01000 ,4271 ,4167 ,4245 2.4 .02000 ,4226 ,4060 ,4206 3.9 ,04000 ,4167 ,3850 ,4120 7.6
.
.
Ex-
tended
.. ..
.. 1.2 1.6 2.3
..
.. .. 0 45 1.1
1.9
.. ..
.. 0.60 0.50
1.1
Discussion of Results It may seem inconsistent to retain the complete exponential in this treatment, while using the value
J. L. DYEAND F. H. SPEDDINC for the electrostatic potential which was obtained by neglecting higher terms in an expansion of the same exponential when solving the PoissonBoltzmann differential equation. The PoissonBoltzmann equation in its complete form is known to be inconsistent, however, and one can only hope that the potential function obtained by a solution of the consistent part of the equation is close to the true answer. The close fit of activity coeffkient data to the theory of Debye and Huckel, derived on the basis of this assumption, seems to substantiate it as a good first approximation. The true distribution of charge is given by S
S
ein,e-Wi/kT
n,iei = %=I
i=l
where nji is the concentration of i ions in the neighborhood of a j ion. Here TVi is the true extra energy an i ion obtains a t the point in space considered. In the treatment of Debye and Hiickel and also in this paper the approximation is made that W i = ei+,o Because of the close fit of activity coefficient data t o the theory of Debye and Hiickel, this approximation is believed t o be more valid than it appears to be from observation of the Poisson-Boltzmann equation, and so eiJ/jo is used in the treatment of the electrophoretic effect as if it were the same as Tt’i. The agreement of the final result with both conductance and transference data seems to add weight to this premise. Figures 4, 5, 6 and Tables I and IT show that the new calculated values for the transference numbers agree with experiment much better than those calculated using the Onsager equation, while the fit of conductance data is also improved. The individual ionic conductances using this method result in quite different values than are obtained using the abbreviated equation, even at low concentrations where the Onsager equation begins to ,:rive the correct total conductance. The reason for this may he seen by comparing Figs. 1, 2 and 3.
Vol. 76
Figure 3 is a plot of the integrand G of the sum of the ionic conductances, compared with the integrand of the ordinary Onsager equation, while Figs. 1 and 2 are similar plots for the cation and anion conductances only. The areas under the two curves for the total conductance are approximately equal because the shaded areas almost cancel each other. It is seen from Figs. 1 and 2 that no such complete cancellation occurs for the individual ionic constituents. I t is of interest to note t h a t the calculation of d for the rare earth nitrates from conductance data by Spedding and Jaffe,” using the extended equgtion described above, gives a value of about 4.5 A . for 8, while the usual Onsager equation including B gives a value of about 2 A. The value of 4.5 A. for 8. is in accord with the physical meaning of d . The original Onsager equation explains the transference number behavior of 1-1 electrolytes quite well. This is because the low charges on the ions result in small exponents in the integrands of equations 6 and 7 , and also because, for symrnetrical electrolytes, both the simple and the extended equation demand equal contributions to the conductance from the cation and anion. The calculations described in this paper are of value because they show t h a t the underlying assumptions made by Onsager are valid at higher concentrations than was previously thought t o be the case. I t will be necessary t o know the extent of validity of the present physical assumptions before an extension of the present theories of conductance can be made. It would be desirable to develop mathematical methods t o obtain the exact solutions of the Debye-Huckel and basic Onsager equations to determine the concentrations a t which the present assumptions break down. Acknowledgments.--The authors wish to tharik Dr. J. AI. Keller for his helpful advice during the course o f this work, and Mr. Gordon dtkinson for his assistance in making calculations and carrying out nunierous graphical integrations. AMMES, IOWA - ____---
(11)
F,H Speddini: and S. Jaffe, THISJ O U R N A L . 7 6 , 884 (1954).
