Transport of electrolytes across cation-exchange membranes

Journal ofMembrane Science, 78 (1993) 155-162 Elsevier Science Publishers B.V., Amsterdam

155

Transport of electrolytes across cation-exchange membranes. Test of Qnsager reciprocity in zero-current processes Stanislaw Koter Institute of Chemistry, N. Copernicus University,

Gagarin Street, PL-87100, Toruri (Poland)

(Received June 2,1992; accepted in revised form November 11,1992)

Abstract In the framework of a series layer model of a membrane a test of the Onsager reciprocal relations (ORR) for a cation exchange membrane in aqueous and nonaqueous electrolyte solutions was performed. It was found that for zero-current transport processes the ORR and the linearity of the fluxes to forces relation are not fulfilled. Keywords:

Onsager reciprocity; series layer model; cation-exchange membranes

cients of eqns. (1) at non-zero differences of concentration and pressure.

Introduction The symmetry of coefficients of linear transport equations had been known long before Onsager proved it on the base of statistical mechanics [ 11. Although the symmetry of coefficients (denoted as ORR - Onsager reciprocal relation) was experimentally proved for many transport phenomena [ 11, there are still only few papers considering this problem for transport of electrolyte solutions through ionexchange membranes [ 2,3]. Foley and Meares [ 21, who investigated the transport of NaBr across a cation-exchange membrane, have found that ORR are obeyed by the differential conductance coefficients, i.e. the coefficients at vanishing forces. Using a complicated apparatus and mathematical procedure, Chu et al. [3] were able to determine simultaneously all transport coeffi-

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In eqn. (1) Ji is a flux,and A/Q is the difference of chemical potential of species i; subscripts s,O denote solute and solvent, respectively. They proved that for an anion-exchange membrane under the investigated conditions (NaCl, solutions, E,= 0.15-0.4 mol/dm3, AC= 0.1-0.2 mol/ dm3, Ap=O.O7-0.4 atm) ORR is fulfilled only in those experiments, where the logarithmic mean of the concentrations of the solutions surrounding a membrane was nearly constant. Using simple arguments Kedem and Katchalsky [4] pointed out that for a membrane composed as a series array of layers with transport coefficients depending on the concentra-

0 1993 Elsevier Science Publishers B.V. All rights reserved.

156

Stanidaw

KoterjJ. Membr ne Sci. 78 (1993) 155-162

TABLE 1 The example of A, X, B of eqn. (9) for a membrane divided into 4 slices

fL$.,0 + =%,LPo LlG.,O+ %,lPo

CL%1

Pl %P PZ CEE.3

0 0 0 0

P3

J. Jv

TABLE 2 Difference of concentrations and of osmotic pressure, AC, Al7, and the mean concentration defined by eqn. (3), E,, corresponding to different Y/c’ for logarithmic mean c,=O.l mol/dn?; 25°C C/CT

c’

C"

AC

AI7” (atm)

EBb (mol/dm3)

0.01 0.069 0.16

0.44 3.2 7.6

0.1003 0.1003 0.1002

(mol/dm3) 1.1 2 5

0.095 0.069 0.040

0.105 0.138 0.20

*Within 4% the same for all solutions. bFor NaCl/H*O solutions.

tion ORR should not be expected for such a system. Since the transport coefficients of ionexchange membranes do depend on the concentration, it is justified to approximate the membrane by a series array of slices with the

-L&s*4 - =%,4P4 -4&,4 - %,A

transport coefficients determined by the intermediate concentrations (such a model was applied by McCallum and Meares [5] for calculation of concentration, pressure, and electric potential profiles in the membrane). Thus, the aim of this paper is the examination of ORR in zero-current processes based on the slice model of the membrane and on the assumption that in each slice ORR is fulfilled. The investigated system will be a Nafion 417 membrane in aqueous and methanolic solutions of sodium chloride and sulfuric acid. Method of calculation of Likcoefficients The transport equations (1) were transformed according to general rules in [ 61 to the equations

TABLE 3 Differential coefficients, Y&, and coupling coefficient, q, for Nafion 417 in different electrolyte solutions at c,=O.l mol/ dm3 and 25°C Solution

2-x 108 (mo12/Nm3-set)

-4p,x 10’2 (m3/N-see)

sp.”x 1O’O (mol/N-set)

9

NaCI/H,O NaCI/MeOH H,SOJH,O H,SO,/MeOH

0.391 1.28 0.867 4.33

0.328 1.13 0.966 4.11

0.299 0.989 0.839 4.01

0.835 0.822 0.917 0.951

Stanidaw

157

Koter/J. Membrane Sci. 78 (1993) 155-162 2.0

4.0 Nofion

417

25°C

I MeOH

Nafion

417

(c) O.sl c,

25°C

, 0.10 ( mol/dm3)

I

, MaOH

I

0.20

Here An, A& are the differences in osmotic pressure and the concentration part of the chemical potential of a solute of solutions separated by a membrane. Changing one of the forces, e.g. Ap to (1 + E) Ap, and assuming that in that range the transport coefficients are constant it is possible to calculate Las coefficients

I

t I

c,

Fig. 1. Concentration concentration range rithmic mean E,=O.l % (i’,), (c) =%/6p,

0.10 ( mol/dm3)

(da)

58

= Js - LsvAWs Ap-AII

(4b)

=dJv w CAP

(4c)

‘” - CAP

04 o~8.!0

-3

L

1

L

0.20

dependence of y&s/9& (i’,) in the corresponding to &‘/CL=5 for logamol/dm3; (a) 2’??/~3&(Q, (b) A!&/ (6).

L

L

_

“8

J,=L,,A~/E,+L,,(Ap-417)

(2)

Jv=Lv~lI/Zs+Lw(Ap-AII) with the mean concentration, c”,,defined as [ 71 (3)

-

Jv

-

LvAlTI~, Ap-AL’

(4d)

The choice of Ap as the force to be varied seems to be more justified than AC, because a change of Ap does not change the concentrations of solutions on both sides of the membrane. The fluxes J, and J, in stationary state caused

Stanislaw Koter/J. Membrane

158

Nafion

2.0

c.=O.

417 25°C

1

p’=l

-

NaCl

Nafion

2.0

417 25°C

-

Sci. 78 (1993) 155-162

NaCl

c.=O. 1 mal/dm’

mol/dmJ

oh

p’=l

p”(otm)=

1.5

T;;

3

atm

1.5

0 X.-.X .$

Jl.0

rn -I

>

1.0

A -.I

p”(atm)= OQQW AA&IA

0.5

0.5

(4

(4

I I

0.0

2

4

6

8

,

0.0

1

T

o.,

2

4

I

8,0

1

0.1

1

C”/C’ Nafion

2.0

c.=O. p’=l

417 25’C

1

-

1.1 3

10

C/c’ p”(otm)=

lizSO,

I’

mol/dm’

A’

otm

2.0