TRANSFERENCE IN BINARY MOLTENSALTSOLUTIONS
April, 1957
485
TRANSFERENCE I N BINARY MOLTEN SALT SOLUTIONS BY BENSONRoss SUNDHEIM Department of Chemistry, Washington Square College, New York University, New York, N . Y. Receiued November 18, 1966
The coupling between diffusion and electrical conduction is treated for a solution of two molten salts having a n ion in common. The quantities of transport are related to transference numbers, diffusion potential and the steady state in electrolysis.
It has been observedl that the quantity ordinarily termed transference number in a pure molten salt has a value which is determined by the conditions of measurement and by the reference point chosen for the velocities. An investigation is made here of the significance of transference numbers and junction potentials in electrochemical cells of certain molten salt solutions. The diffusion process has been studied repeatedly from the point of view of the thermodynamics of irreversible processes.2 Ordinarily the motioh of the individual ionic components is studied, leading to the use of single ion chemical or electrochemical potentials. In the derivation presented below the introduction of such unmeasurable quantities is avoided. In addition, the usual treatment of the diffusion potential uses Xi+) for the force on a the expression grad ( p i single ion. This immediately implies that the Nernst-Einstein equation applies, relating the diffusion coefficient and mobility of the ion. No such special assumption is imbedded in the present treatment. The Phenomenological Equations.-Consider a cell containing a mixture of two molten salts, the local concentrations being designated as cl, c2 moles/cc. We restrict ourselves to the particularly simple case where the salts have an ion in common since the important results are seen most clearly here. The general multi-component case will be treated elsewhere. The first component is A,B, and the second C,B, and the cell also contains a pair of electrodes reversible to one of the ionic species. The frictional drag of cell walls and electrodes is assumed to be negligible. A porous plug may be inserted in the cell to establish a reference point for measurements. (This arrangement corresponds to that of various experimental cells used to measure transference numbers or junction potentials in molten salts.) Designate by
+
A
A
&
L
JB =
(I - Z A J A
A
- ZCJC)/Z€i
(d)
A
_L
A
[I - ~ z c J ~pZC(dfl/d~Z)Jl]/ZB
=
c2
CAZA
=
(1 -
(e)
ClDl)/OZ
+ CBZB + cczc = 0
(f)
It is seen readily t,hat all the measurable fluxes are expressed in terms of J1 and I and that only the A
d
concentration c1 is needed to specify the composition of the solution a t any point. A
A
The forces corresponding to the fluxes J i and I are,2 respectively, - grad pi and - grad 4 where pi is the chemical potential and 4 is the electrostatic potential. In these terms the phenomenological equations express a linear relation between the forces and fluxes A
-LIIgrad p1 - Llz grad pz - grad 4 JZ = - LZIgrad p1 - Lzzgrad wz - L23 grad 4 I= grad pi grad pz - La3 grad 4 JI =
L13
(2)
L32
-L31
Certain of the coefficients are readily identified3
L33
=
K
Here K is the isothermal specific electrical conductance measured in faradays, yi the activity coefficient of the ith component and Dik the interdiffusion coefficient of the ith and kth components. The Onsager relations then state that L i k = L k i . Designate the quantity L13 = L31by K ~ I ,L32 by Ktz. The quantity grad p 2 can be eliminated from (2) by use of the Gibbs-Duhem equation, (& grad /Ji)T,P
=o
-
4
Jl(J2) the flux (moles/cm.2 sec.) of (neutral) salt 1
J1
d
= ( -Lu
- LIzcI/cz) grad p1 -
Kt
grad 6
(4)
A
(2) and by I the electrical current density (Faradays/cm.2 see.), all measured with respect to the mass average velocity of the solution. Let 61, f l 2 be the respective partial molar volumes, M1, M 2 the appropriate formula weights and XA, ZB, xc the ionic valences. I n view of the essential incompressibility of liquids, of the requirements of electroneutrality and of the fact that momentum is conserved in such systems’ the set of interrelations occurs A
MlJl A
Jz =
+ AdzJz = 0 A
(a)
&
-(Mi/Mz)Ji
(1) B. R. Sundheim, THISJOURNAL, 6 0 , 1381 (1956). ( 2 ) S. R. de Groot, ”Thermodynamics of Irreversible Processes,” Interscience Publishers, Ino., New York, N. Y., 1952.
JZ = - ( L z ~- LZZCI/CZ) grad A
I =
-(ti
-
/ZCI/CZ)K
grad
p1
p1
- K&
-
K
grad 6
grad 6
Using (la) and designating [LII - LIZCI/CZ] by C, (4) becomes d
J1 =
- 6: grad p, - K t l
grad 4
d
JZ = + ( l M ~ / f i f z ) Ggrad A
+
p1
+ ( M I / M z ) Kgrad ~~+
- K grad 9 ( 5 ) I = -[I ( ~ l C I ) / M z C z ) ] K h grad Only one diffusion coefficient, 2, occurs in accord(3) (a) J. G. Kirkwood in “Ion Transport Across Membranes,” edited by R. T. Clarke, Academic Press, Ino., New York, N. Y., 1954, P. 119 ff; (b) H. Fujita and L. J. Costing, J. Am. Ckem. Soc., 7 8 , 1090 (1956).
CHEVESWALLING,E. E. RUFFAND J. L. THORNTON, JR.
486
ance with the fact that only one composition variable is required and the fact that the fluxes are coupled through (la). The following quantities are of particular interest. -
+
A
Ed.
+ M d ( M 2 c d 1 grad AEd = +(a) - d o ) -Joatlll + M ~ c ~ / ( M z c grad d l rldz Ed =
-
= td-M,/Mz)
12
(6)
nzAll [I - nzAtl f p(fill/MZ)zAtl] = -p(Ml/Mz)zctl
fA = tB IC
4
A. ( J 1 I I ) p I = &.-Thus tl is seen to be the amount of component one convected along with a unit current in a uniform fluid. We shall call it the electrochemical transport coefficient of component one. B. [grad d/grad ~ 1 1 1 - 0 = -ttl[1 Mlcd (M~Jl.-Here grad 4 is the diffusion potential, &
Vol. 61
fil
tl[1
2
It is clear that there is only one significant transport quantity describing this system and it may be chosen to be any of the above as well as tl itself. The numerical value of the transport quantities depends upon the point of reference used in defining the velocities. If a new reference point, moving with a velocity 210 with respect to the former one (the mass average velocity) is chosen, then the new fluxes, identified with primes become A
=
With the use of this expression the equation for the current becomes I
=
-KEd
-K A
grad
@
= Ed
4
A
I‘
= Z(Vi
A
+ = I + vozc,zi = I
-
U O ) C ~= V0)CiZi
A
A
( 4 ) P. Drossback, 2. Elektrochem., I S , 66 (1954).
(7)
-
-
ti’ = Ji’zi/I =
- I/K
[grad fillgrad d]J,-const. = &/d: - [Jd(d: grad .$)].-Grad ,ul here is the gradient of the chemical potential, from which the gradient of the concentration can be obtained, during the steady stat.e of electrolysis of the s ~ l u t i o n . ~Since 2, tl and K are functions of the concentration, integration of this expression involves solution of the diffusion equation and is ordinarily a formidable task. The Electrochemical Transport Coefficient.With the aid of expressions 1the transference numbers of the individual ions may be expressed in terms of the electrochemical transport coefficient of one of the components.
VOC~
+ VOCl/I 2
-
stating that the potential difference across dx is equal to the diffusion potential minus the I R drop.
A
Ji
A
d
tl‘ = tl
grad 9
c.
+ +
The transport coefficients are then changed to
&
A
&
Ji’ = (vi
(8)
+ voci~i/I -
ti
A
In this way the transport coefficients may be referred to any desired reference point. One reference point seems to be particularly convenient. A porous plug inserted in the liquid will have the mass average velocity of the liquid provided that viscous drag along the walls is negligible and provided that the plug is free of electrophoretic effects. Diffusion potentials and transference numbers in molten salts are ordinarily measured with reference to such a porous plug. I n any event it is important that the transference number occurring in the expression for the diffusion potential be expressed in terms of the same reference velocity as is used in expressing the gradients of concentration and potential. It is a pleasure to acknowledge assistance to this work from the United States Atomic Energy Commission (Contract AT 30-1 1837).
THE ADSORPTION OF CATIONS BY ANIONIC FOAMS] BY CHEVES WALLING,~ EDGAR E. RUFFAND JAMES L. THORNTON, JR. Contribution from the Research, and Development Division of Lever Brothers Co., Edgewater, N . J . Received November 23, 1966
A study of the relative adsorption of calcium and sodium ion by N-palmitoyl methyl taurine foams indicates a strong preferential adsorption of calcium ion by the anionic surface layer. Quantitative interpretation of the phenomenon is however complicated by a competitive preferential adsorption of calcium ion by micelles in the solution being foamed. Similar preferential adsorption of a number of other cations by anionic foams has also been demonstrated, polyvalent ions being, in general, the most strongly held, and the relation of this absorption t o foam properties is discussed.
The layer of an anionic surfactant adsorbed a t an air-water interface must have associated with it an equivalent quantity of cations in order to neutralize its electric charge. In general such ions, bound by a colloid phase are termed “gegenions.” The fact that insoluble surface films such as stearic acid, show very strong preferential adsorption of certain (1) Presented in part before the Colloid Division a t the meeting of the American Chemical Society, Buffalo, New York, March, 1952. (2) Department of Chemistry, Columbia University, New York 27, N. Y.
ions was strikingly demonstrated many years ago by Langmuir and S h ~ e f e r and , ~ the subject has reaently been investigated further by H a ~ i n g a . ~ The possibility that a similar preferential adsorption might take place with soluble films occurred to us, and has now been investigated, using the apparatus for the study of foam compositions and foam drainage properties which we have recently de(3) I. Langmuir and V. J. Schaefer, J. A m . Chem. Soc., 69, 2400 (1937). (4) E. Havingr, Rec. trav. chim., 71, 7 2 (1952).
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