2385
SOMEINTERACTIONS IN PERMSELECTIVE MENBRANES
Evaluation of Some Interactions in Permselective Membranes
by N. Lakshminarayanaiah Department of Pharmacology, University of Pennsylvania, Philadelphia, Pennsylvania
19104
(Received August 27, 1969)
The number of moles of water (f)accompanying ion transport, following passage of 1 faraday of current across three cation selective membranes, one of high water content and the other two of low water content, has been measured at 25” using 0.01 N solutions of NaCI, RbC1, or CsCl. Also, specific conductance, counterion selfdiffusion coefficient, and hydrodynamic permeability of the membranes have been determined. These data and other equilibrium parameters like water con_tent_andexchange-capacity have been used to derive values for the Stefan-Maxwell diffusivity coefficients, D1_*1,OM,0 1 3 , and D a h , from a few relations given by Scattergood and Lightfoot. The diffusivity coefficient Dl*l was the lowest (i,e,, the interaction between species 1* and 1 was the highest) in the case of membranes of low water content in which f showed little dependence on the current density used in its measurement, In the case of high water content membrane, t3was dependent on the current density employed during electrolysis. At low current densities, Z3 was greater than the quantity of water associated with 1 equivalent of counterion in the membrane phase. With this value of f,D34 assumed negative values. On the other hand, when the limiting value of t3 observed at high cur-rent dsnsities-wasused in the equations, values obtained for the diffusivity coefficients increased in the o_rder0 1 4 < Ill*, < 0 1 3 < 0 3 4 . The order of these coefficientsfor membranes of low water content was Dl*l < 0 1 4 < 1513 < 1534.
Introduction A variety of transport phenomena arise across a membrane when it is subject to different driving forces.’V2 Some of these phenomena such as ion migration, electroosmosis, self-diffusion, hydrodynamic permeability, etc., occurring across ionic membranes have been described by Spiegler3 applying the principles of nonequilibrium thermodynamics and by Lightfoot and coworker^^-^ using the generalized Stefan-Maxwell equations. Under simplifying conditions, vix., co-ions absent from the membrane phase (e2 = 0 and counterion transport number tl = l), Spiegler3 derived the following set of equations =
a
=
(aE
- 1)/B
(RT/D1) =
+
(1) x 1 4
E = (E/C1F2)
(2)
(3)
radioactive isotope 1” employed to measure D1 with its abundant species, the counterion itself. The importance of this isotope interaction has been pointed out by Laity,’ and Caramazea, et al.,s have questioned the omission of this term and furthermore have given an expression for b1 including this interaction term. The effects of this interaction term on measurements of unidirectional fluxes using radioactive isotopes have been considered by some w o r k e r ~ , ~ -and l ~ Curran, et a1.,13 have examined the available experimental data and came to the conclusion that the tracer flow could be used to predict the flow of the bulk substance. Recently, Scattergood and Lightfootj6 for the first time, have evaluated the diffusivity coefficient D1*1 . This followed from the fourth measurement of hydrodynamic permeability (L,) of the membrane under the condition of zero current. They obtained the following relations, again for the condition, = 0 and 11 = 1.
B = (Et3/C3F2) =
+ c3x34(x13 + xl4) 1
(clxl8)/ [clxl8xl4
(4)
where subscripts 1, 3, and 4 represent counterion, water, and membrane including the ionized groups fixed to the membrane matrix, respectively, ci)s are the concentrations, xij’s are the friction coefficients between the components i and j , E is the specific conductance, 6 is the transport number of water, and b1is the selfdiffusion coefficient. R , T, and F have their usual significance. Overbars refer the parameters to the membrane phase. With the help of these equations values for the friction coefficients could be derived provided reliable measurements of 1 3 , bl, and E are made for the membrane in its proper ionic form. Equations 1-4 do not contain the term X1*1 which indicates the interaction of the
(1) N . Lakshminarayanaiah, Chem. Rev., 65, 493 (1965). (2) N. Lakshminarayanaiah, “Transport Phenomena in Membranes,” Academic Press, Inc., N e w York, N. Y., 1969, p 6. (3) K . S. Spiegler, Trans. Faraday Soc., 54, 1408 (1958). (4) E. N. Lightfoot, E. L. Cussler, Jr., and R. L. Rettig, A.1.Ch.E. J., 8, 708 (1962).
(5) E. N. Lightfoot and E. M. Scattergood, {bid., 11, 175 (1965). (6) E . M. Scattergood and E . N. Lightfoot, Trans. Faraday Soc., 64, 1135 (1968). (7) R . W. Laity, J . Phys. Chem., 63, 80 (1959). (8) R. Caramazza, W. Dorst, A. J. C. Hoeve, and A. J. Staverman, Trans. Faraday Soc., 59, 2415 (1963). (9) L. F. Nims, Yale J . B b l . Med., 31, 373 (1959). (10) L. F. Nims, Science, 137, 130 (1962). (11) 0. Kedem and A. Essig, J. Gen. Physiol., 48, 1047 (1965). (12) A. Essig, J . Theor. Biol., 13, 63 (1966). (13) P. F. Curran, A. E. Taylor, and A. K. Solomon, Biophys. J., 7, 879 (1967).
T h e Journal of Physical Chemistry, Vol. 74, N o . 11, 1070
2386
N. LAKSHMINARAYANAIAH
Lp
(f3/&)/
=
=
(53CV3*)/[{(2i/&.)
= (C1F2/fiT/[(23/D13)
(1/Dd
= [(2i f
+ (24/b34) ] + (Z4/Da)]RTd]
[ (21/-&3)
t3
f (Z4/D14)
51*)/151*1I
+ [Z,/bi3]
-
(t3%/&)]
f [24/D14]
(5) (6)
(7) (8)
where %'s are mole fractions, Dl,'s are Stefan-Maxwell diffusivity coefficients, C is the total concentration, ?3 is the partial molar volume of water, d is the thickness of the membrane, and * indicates the radioactive counterion. The two treatments mentioned above are essentially equivalent in that the friction coefficients of eq 1-4 are related to the diffusivity coefficients of eq 5-8 by the relation&
If,,
=
(RT/b13)Z3
(9)
and
D,,= D,,
Z,If,,
= 2,1fJi
(10)
These relations indicate that small values for the diffusivity coefficients blJ's mean large values for the interaction between species i and j , i e . , the values for the friction coefficients Xl,'s are high, and vice versa. The four measurements (i3, b,, 8, and Lp) including the membrane equilibrium data and the results of evaluation of the four diff usivity coefficients, b1*1,b14, b13, and &, for three strong acid cation-exchange membranes and three ions, Na+, Rb+, and Cs+ are presented in this paper.
Experimental Section Cation-ExchangeMembranes. Two membranes, AMF C-103 and AILIF C-104, were supplied by the American Machine and Foundry Co. They were polyethylenestyrene graft copolymer type containing sulfonic acid groups. The third was the cross-linked phenolsulfonic acid (PS4) membrane which was prepared following the procedure described elsewhere.l4 The membranes were conditioned in 1 N HC1 solution, washed thoroughly with deionized water, and equilibrated with the required 0.01 N solution (NaC1, RbC1, or CsC1) which was changed a number of times for complete conversion of the membrane into the proper ionic form. Measurement of Different Equilibrium Properties. Pieces of equilibrated membrane were blotted with filter paper, placed in a weighing bottle, and weighed. They were then dried to constant weight at 103" to determine their water content which was estimated to have an error of 5 4 % in the case of PSA membrane and 5 3% in the case of other membranes. A piece of equilibrated membrane of accurately known area, surface dried between filter papers, was held in a micrometer caliper for measurement of its thickness (error = 520j0). This piece of known volume was weighed for the determination of its density which was accurate to 50.3OjO0. Next it was equilibrated with stirring overnight with a known excess of 0.1 N "08 The Journal of Physical Chemistry, Vol. 74, N o . 11, 1970
solution. The acid used up by ion exchange only was estimated in the usual way14 with an error of kl%. The chloride ion (co-ion) content of HNOI solution and washings was negligible (
