CATIONEXCHANGE ACROSS ION-EXCHANGE MEMBRANES
435
Cation Exchange across Ion-Exchange Membranes
by A. S. Tombalakian,’ C. Y. Yeh, and W. F. Graydon Department of Chemical Engineering and Applied Chemistry, University of Toronto, Toronto, Ontario, Canada (Received July 69, 1966)
A study of the exchange of inorganic cations of various valence for hydrogen ion has been made using polystyrenesulfonic acid ion-exchange membranes. The results indicate that the preference of the ion-exchange membrane for the various inorganic counterions increases with increasing valence and decreasing hydrated ionic size. A marked dependency of the selectivity coefficient on external solution concentration is noted only in the case of multivalent ions exchanging for hydrogen ion. From the ion-exchange equilibria data and cation-interchange fluxes obtained in counterdiff usion experiments across the same membrane, single-ion diffusion coefficients for the interdiffusing ion species in the membrane have been estimated using the Nernst-Planck equation. The dependence of the membrane interdiffusion coefficient on the ionic composition of the membrane for various ion pairs has been determined.
Introduction Equilibria between synthetic ion exchangers and solutions of cations of various valence have been the subject of numerous investigations.*-lO A complete listing of these studies is given by Helfferich” in his comprehensive treatise on ion exchange. These studies of exchange equilibria with synthetic ion-exchange resins in granular form have revealed that the exchantngjng properties of a resin depend upon its macromolecular structure. The homogeneity of some of the ionexchange resins used in these earlier investigations has been a matter of some doubt. Typical ion-exchange resins prepared by sulfonation of cross-linked resins, for example, contain sulfonate groups the environment of which differ markedly from that of other sulfonate groups. Ion-exchange resins of the polystyrenesulfonic acid type of improved homogeneity may be prepared by direct copolymerization of styrene, divinylbenzene, and the ester of p-styrenesulfonic acid. The ion-exchange behavior of such homogeneous ionexchange membranes in solutions of cations of various valence is thus of interest. The results of measurements of the selectivity of polystyrenesulfonic acid ion-exchange membranes for the exchange of inorganic cations of various valence for hydrogen ion are given in this report. The results are consistent with theoretical considerations of ionexchange equilibria for synthetic ion-exchange resins.
The dependence of the ion diffusivities in the membrane on membrane properties and the interdiffusing ion species has been determined.
Experimental Section ( A ) Membranes. Two polystyrenesulfonic acid ionexchange membranes were prepared by the bulk copolymerization of the n-propyl ester of p-styrenesulfonic acid with styrene, divinylbenzene, and benzoyl peroxide as catalyst and subsequent hydrolysis in 5% caustic soda solution following the procedure described p r e v i ~ u s l y . ~ ~Membranes -~~ one and two had a (1) To whom correspondence should be addressed at Department of Chemistry and Engineering, Laurentian University, Sudbury,
Ontario, Canada. (2) T. R. E. Kressman and J. A. Kitchener, J. Chem. SOC.,1190 (1949). (3) T.R. E. Kressman and J. A. Kitchener, ibid., 1201 (1949). (4) D.Reichenberg, K. W. Pepper, and D. J. McCadey, ibid., 493 (1951). (5) H. P. Gregor, J . Am. Chem. Soc., 73, 642 (1951). (6) H. P. Gregor, 0. R. Abolafia, and M. H. Gottlieb, J . Phys. Chem., 58, 984 (1954). (7) D.Reichenberg and D. J. McCauley, J . Chem. SOC.,2741 (1955). (8) G. E. Myers and G. E. Boyd, J. Phys. Chem., 60, 521 (1956). (9) 0.D.Bonner and L. L. Smith, ibid., 61, 326 (1957). (10) H. C. Subba Rao and M. M. David, A.I.Ch.E. J., 3, 187 (1957). (11) F. Helfferich, “Ion Exchange,” McGraw-Hill Book Co., Inc., Toronto, 1962.
Volume 71, Number 6 January 1967
436
A. S. TOMBALAKIAN, C. Y. YEH,AND W. F. GRAYDON
Table I: Separation Factors and Selectivity Coefficients for the Exchange of Various Cations for H+ M emCounterion
brane no.
Moisture content at 100% R H at 2 5 O , moles of HzO/equiv
Equiv ionic fraction of counterion in resin
Separation factor,
Selectivity coeff,
aBA
KBA
H
18.8 19.8
Li
19.0 20.0
0.431 0.421
0.760 0.729
Na
17.3 18.7
0.509 0.507
1.04 1.02
K
1 2
16.2 17.5
0.554 0.549
1.21 1.20
cs
1 2
13.6 15.0
0.564 0.560
1.26 1.25
Mi3
1 2
12.0 16.6
0.878
7.22
0.825
Ca
11.3 14.8
0.892
8.26
1.03
Sr
10.3 12.9
0.910
10.1
1.31
Ba
9.2 11.2
0.926
12.5
1.71
AI
11.5 14.5
0.936
14.6
3.03
nominal cross linking of 4 mole % of DVB, an exchange capacity of 2.01 and 2.61 mequiv/g of dry resin H form, and a thickness of 0.0545 and 0.0670 f 0.0002 em, respectively. ( B ) Ezchange Capacity. Samples of the membranes in the hydrogen form were added to 50-ml portions of 0.1 N sodium chloride (or the salt of a different counterion) solution and the solution was titrated with 0.1 N sodium hydroxide using brom cresol green. ( C ) Ion-Exchange Equilibrium Determination. Weighed samples of the membranes in the hydrogen form were equilibrated at 25" in solution containing hydrochloric acid and the salt of the counterion (LiCI, NaC1, KC1, CsC1, MgC12, Ca(NO&, SrC12, BaC12, and AlC4). After equilibration the membrane sample was removed from the solution and the solution analyzed for hydrochloric acid and chloride ion content by titration. The difference between these titration results gave the counterion content of the solution. The membrane sample was thoroughly rinsed with conductivity water and the hydrogen ion on the membrane determined as in the capacity determinations. The counterion on the membrane was calculated from the difference between the hydrogen ion on the membrane The Journal of Physical Chemistry
and the capacity. There was no change in the capacity of a membrane sample after each ion-exchange equilibrium determination for the different ion-pair exchange systems investigated.
Results and Discussion The equilibrium data obtained for the exchange of various univalent and divalent inorganic cations for hydrogen ion across two typical polystyrenesulfonic acid ion-exchange membranes are given in Table I. These measurements were made with solutions (0.1 N total solution concentration) containing the same number of equivalents of the two cation species. The separation factors, wA, and molal selectivity coefficients, KgA,for the membranes were calculated from the experimental data using eq 1 and 2 , respectively. Cy**
=
RAXB 7
XBXA
(1)
(12) W. F. Graydon and R. J. Stewart, J. Phys. Chem., 59, 86 (1955). (13) A. 5. Tombalakian, H. J. Barton, and W.F. Graydon, ibid., 66, 1006 (1962). (14) P. Rosenblum, A. S. Tombalakian, and W. F. Graydon, J. PoEumer Sci., A l , 1703 (1966).
CATION EXCHANGE ACROSS ION-EXCHANGE MEMBRANES
K~~
=
(2)
where X and X are the equivalent ionic fractions of the ion species in the membrane and external solution, respectively; i V and M are the molalities of the ion species in the membrane and external solution, respectively; and 2 is the valence of the ion species. For the exchange of univalent ions the above equations yield the same numerical value for the separation factor and the selectivity coefficient. The data in Table I show that the preference of the ion-exchange membrane for the various inorganic counterions increases with increasing valence and also iR the order lithium, sodium, potassium, cesium, magnesium, calcium, strontium, barium, and aluminum. This latter effect is consistent with the variation in the effective radii of these ions of the same valence in solution in their hydrated forms. These results may be compared, for example, with data for sulfonated styrene-divinylbenzene copolymer beads of comparable capacity and cross linking which showed higher value^.^^^^^ This variation may be due to differences in detail structure between the polystyrenesulfonic acid ion-exchange membranes and the sulfonated styrenedivinylbenzene copolymer beads. The effects of variation of the compositjon and concentration of the external solution on the exchange properties of the membranes also were determined. No significant variation in the selectivity coefficient (for membrane one, 1.06 f 0.03; membrane two, 1.04 f 0.03) for the exchange Na+-Hf was observed using solutions of varied composition (0.1 N total solution c~ncentration).'~The results obtained using solutions of different concentration are given in Table 11. It can be seen that although both univalent and divalent counterions exhibit an increase in the selectivity
437
of the membrane with dilution of the external solution, this effect is much greater for divalent than monovalent counterions. In counterdiffusion experiments of two cation species across an ion-exchange membrane, the cations exchange in equivalent quantities. If the cation-interchange flux is described by a special form of Fick's first law, we may write
(3) The flux in a diaphragm diffusion cell having solutions of equal volume and normality is
J
dc = d(ACA)
1
1
Exchange system
Na+-H+
0.001 0.01 0.1 1.0
0.523 0.515 0.510 0,505
1.08 1.04 1.03 1.02
Mg'+-H+
0.001 0.01 0.1 1.0
0.988 0.952 0.878 0.661
0.927 0.860 0.825 0.644
(ACA)t=O ~
e*)
d(AcA)
(5)
where J is the interchange flux, equiv/cm2 see; DAB is the membrane interdiffusion coefficient, cm2/sec; Ki is the over-all ion-interchange mass-transfer coefficient ( = ( V / 2 A t ) In (ACo/ACf)),16,17 cm/sec; L is the membrane thickness, cm; ACA is the difference in concentration of the counterion between half-cells, Table 111: Membrane Interdiffusion Coefficients
System Selectivity coeff
=
(A t =0 Substitution of eq 4 and 5 in eq 3 and integration lead to the relationship
Nor-
normality of ext soln
Tot.
Membrane no.
(4)
Using the approximation
Table 11: Selectivity Coefficients for the Exchange of Na+ and Mg2+ for H + Equiv ionic fraction of counterion in resin
= Vdc/2dt
MgClrHCl
mality of ext soln
0.01 0.1 1.0
Interchange coeff (Ki), DABcalcd by eq 6, -cm sec-1 X 1 0 k om2 sec-1 X 106 MemhlemMemMembrane brane brane brane 1
2
1
2
38.6 13.2 4.58
61.0 20.9 7.32
1.07 1.22 1.36
2.88 3.20 3.35
__
Av 1.26
~~~
__
3.14
~
(15) I. H. Spinner, J. Ciric, and W. F. Graydon, Can. J . Chem., 32, 143 (1954). (16) A. S. Tombalakian, C. Y. Yeh, and W. F. Graydon, Can. J . Chem. Eng., 42, 61 (1964). (17) M. Worsley, A. S. Tombalakian, and W. F. Graydon, J . Phys. Chem., 69, 883 (1965).
Volume 71, Number B January 1967
438
A. S. TOMBALAKIAN, C. Y. YEH, AND W. F. GRAYDON
~~~~
~
Table IV : Membrane Interdiffusion Coefficients Ion-pair exchange system
Membrane no.
CS+-H
1
Normality of ext soln
N a +-H + Ba2+-H + Sr2+-H+
0.1 0.1 0.1 0.1
Ca2+-H + ?*lgz+-H
0.1 0.1
+
+
equiv/ml; A c A is the difference in concentration of the counterion in the membrane between the membrane faces (=2J;iBCi at zero time), equiv/ml; €i is the membrane internal ion concentration, equiv/ml; and (€A)= ~ ~XACi equiv/ml. Adherence of the experimental data to eq 6 is illustrated by the close agreement in the values of the membrane interdiffusion coefficient for the interchange of magnesium with hydrogen over a 100-fold increase in external solution concentration gradient as shown in Table 111. The results indicate an increase of about 20% in the value of the interdiffusion coefficient for the Mg2+-H+ exchange as the external solution concentration is increased from 0.01 to 1.0 n;. It is of interest to determine the dependence of the membrane interdiffusion coefficient for the various exchange systems on the ionic composition of the membrane. The equilibrium data (Table I) together with cation-interchange fluxes measured previouslylB in mass-transfer experiments across the same membrane equilibrated with isotonic solutions were used in eq 6 to obtain membrane interdiffusion coefficients for our experimental conditions given in Table IV. The membrane interdiffusion coefficients obtained by the above method were used in eq 7 together with the assumption that limiting ionic mobility ratios in water apply in the membrane solution to estimate single-ion d 8 u sion coefficients for hydrogen and the various counterions.
where b is the single-ion diffusion coefficient of the exchanging ion species in the membrane, cm2/sec. Equation 7 is an evaluation of the membrane interdiffusion coefficient (DAB) from the Nernst-Planck equation'* for the exchange of two cations. The limiting evalua0 and tions of DABfrom eq 7 are DAB= DA for €A DAB = DB for CB 0. These single-ion diffusion COefficients of the two exchanging cations in the membrane
-
Exptl DAB calcd by eq 6, cmz sec -1 x 106
1.42 1.24 2.69 2.53 2.18 1.22
Single-ion diffusion coeff calcd by eq 6 and 7, cm3 aec-1 x 106
Cs+, 0.798 Na+, 0.697 Bas+, 0.533 Sr2+,0.481 Ca2+, 0.416 Mgz+, 0.219
H+, 3.62 H+, 4.88 H+, 2.93 H+, 2.84 H+, 2.45 H+, 1.45
1
A Bd'-H' 6
s~"-H'
0
Go''-"
0
Mf-H'
I ~
05 1.0 0 0.5 EQUIVALENT IONIC FRACTION OF COUNTERION IN MEMBRANE
1.0
Figure 1. The variation of the membrane interdiffusion coefficient with variation of the equivalent ionic fraction of counterion in the membrane for the exchange of various univalent and divalent inorganic counterions with hydrogen ion.
represent the limiting evaluations of the membrane interdiffusion coefficient. Samples of such values of single-ion diffusion coefficients (limiting evaluations of DAB) calculated by combination of eq 6 and 7 for the exchange of various univalent and divalent inorganic cations with hydrogen ion are also given in Table IV. These single-ion diffusion coefficients serve to correlate interdiffusion
N
The Journal of Physical Chemistry
(18) F. Helfferich and M. S. Plesset, J. Chem. Phya., 28, 418 (1968).
SELF-DIFFUSION IN SIMPLE FLUIDS
behavior for various ion pairs and also describe the range of membrane interdiffusion coefficients predicted from the experimental data by the Nernst-Planck equation, eq 7. The observed dependence of the interdiffusion coefficient on the ionic composition of the
439
membrane for the various exchange systems across membrane one is shown in Figure 1. Acknowledgment. The authors are indebted to the National Research Council, Ottawa, Ontario, Canada, for financial support,.
Self-Diffusion in Simple Fluids
by John A. Palyvos and H. Ted Davis Department of Chemical Engineering, University of Minnesota, Minneapolis, Minnesota (Received September $8, 1966)
Application of the formula for the friction coefficient derived independently by Helfand and by Rice and Allnatt yields generalized charts comparing hard-core interaction contributions to the friction constant to contributions arising from soft interactions as predicted by the linear trajectory approximation. Numerical calculations based on the theoretical pair correlation functions of Kirkwood, et al. , are presented for liquid argon, krypton, and xenon. On the basis of these calculations it is concluded that the use of the linear trajectory approximation in the Rice-Allnatt theory yields fairly reliable predictions (to within 10-40% over the entire liquid range) for the self-diffusion coefficients of simple liquids.
Introduction Rice and Allnatt’ have developed a theory of transport in dense fluids based on the assumptions that (1) the interaction potential can be split into a hard-core repulsive part and a longer range soft part and (2) the dissipative effects of the hard and soft parts are additive. According to their model, a molecule moving through a dense fluid will undergo a motion in which it experiences a hard-core collision followed by a “Brownian motion” caused by soft interactions with the potential field of the neighboring molecules. Under these assumptions Rice and Allnatt obtained a modified Boltzmann equation in which the hard-core collisions were treated as in Enskog’s theory2 of a dense rigidsphere fluid while the soft interactions were handled by Ross’s weak coupling theory3 which leads to the Fokker-Planck approximation for these interactions. In order to predict values of transport coefficients from the formulas of Rice and Allnatt, one must have,
in addition to the parameters of the interaction potential, values for the equilibrium pair correlation function. Several numerical comparisons, discussed in detail in the text by Rice and Gray,4 between experimental and theoretical viscosities and thermal conductivities of the noble liquids indicate that the Rice-Allnatt theory gives a quantitative description of transport in simple liquids. In obtaining the theoretical predictions, the LennardJones potential was used as the soft part of the potential. The theoretical pair correlation function for this potential model has been computed by Kirkwood, Lewinson, and Alder using the superposition approximation on the Born-Green-Yvon integral e q ~ a t i o n . ~ (1) S. A. Rice and A. R. Allnatt, J . Chem. Phys., 34, 2145 (1961). (2) 8. Chapman and T. G. Cowling, “The Mathematical Theory of Nonuniform Gases,” Cambridge University Press, Cambridge, 1964. (3) J. Ross, J . Chem. Phys., 24, 375 (1956). (4)S. A. Rice and P. Gray, “The Statistical Mechanics of Simple Liquids,” John Wiley and Sons, Inc., New York, N. Y., 1964.
Volume 71, Number 2 January 1967
