J. Phys. Chem. 1981, 85, 635-642
635
higher free-ion yields in water16 and some hydrocarbons.40 Furthermore, measurement of the optical absorption by anions formed by the scavenging of electrons by aromatic hydrocarbons indicates that there is an isotope effect in ammonia, the value of G(anion) being 3.2 in NH3 and 3.6 in ND3.41 The greater electron scavenging efficiencies in perdeuterio water42and hydrocarbons?O relative to those in the perprotio compounds, are due to the greater thermalization ranges of the electrons in the former. The large free-ion yield in liquid ammonia is attributed mainly t o the inefficiency of the neutralization e; + NH4,s+,43 and the isotope effect upon this reaction is not known. We cannot offer a reason for the apparent discrepancy between the optical results from the ammonias in the presence and the absence of an electron scavenger. The oscillator strength of the band may be expressed as
E (eV). The value of the integral was estimated as follows.
where €(E) is the decadic extinction coefficient at energy
Acknowledgment. We thank the Natural Sciences and Engineering Research Council for financial assistance and the staff of the Radiation Research Center for aid with the electronics.
(40) Wang, H. Y.;Willard, J. E. J. Chem. Phys. 1978, 69, 2964. (41) Seddon, W. A.;Fletcher, J. W.; Sopchyshyn, F. C.; Jevcak, J. Can. J. Chem. 1974, 52, 3269. (42) Stradowski, Cz.; Hamill, W. H. J. Phys. Chem. 1976, 80, 1054. (43) Belloni, J.;Saito, E. In “Electrons in Fluids”;Jortner, J.; Kestner, N. R., Eds.;Springer: Berlin, 1973; 461.
The low-energy side that follows the Gaussian shape function was integrated analytically. The high-energy tail beyond 1.5 eV was integrated according t o t a ES. The middle portion was summed graphically. The total area is shown in Table I1 as W 1 , J where (3) The value of I is 1.30 f 0.01, independent of temperature and isotopic substitution (Table I). The value of I in water is 1.35 f 0.01, but WtI2and hence W1121are 5% smaller in D20 than in H20. Using E,, = 4.8 X lo4 M-l cm-’ (ref 2 and 8) in both ammonias, one obtains F = 0.89 for the band in both solvents. However there is a 20-30% uncertainty in the value off obtained from eq 1,44so it is not known whether the observed band accounts for all of the oscillator strength.
(44) Maccoll, A. Q.Rev. Chem. SOC.London 1947, I, 16.
Transport Properties of the Barium Form of a Poly(styrenesu1fonic acid) Cation-Exchange Membrane Ellchl Kumamoto”+and Hldeo Klmlzuka Department of Chemisty, Faculty of Science, Kyushu University, Fukuoka 812, Japan (Received: April 3, 1980; I n Final Form: October 3 1, 1980)
A detailed study of the transport properties of the barium form of a poly(styrenesu1fonic acid) cation-exchange
membrane was completed over a wide range of external solution concentrations (10”-1 M) of BaC12. The electric conductance, the ionic and water transport numbers, and the hydrodynamic permeability coefficient were measured with the membrane in contact with two external solutions of the same concentration. Moreover, data were obtained on the salt and volume fluxes across the membrane when the two external solution concentrations are different. Equations were derived for determining the six independent phenomenological coefficients (li,) that were required to characterize a system of a single salt and water through the membrane. The li; coefficients were calculated from the six independent experimentalquantities described above as a function of the external solution concentration. The tendencies for the variation of the 1ij coefficients with concentration were attributed to the concentrations of the individual species. Frictional and coupling coefficients (fii and q i j ) were also evaluated from the phenomenological coefficients, ri;, which are defined in the inverse description of the li;_formalism. From the ratios of f i j (i # j ) to rii times the concentration of species i in the membrane, Le., fij/(Cirii), or qij coefficients, it was suggested that there is a strong interaction between the counterion and the fixed charge and that the interactions between the co-ion and the other species are very small.
Introduction When an ion-exchange membrane is subjected to various driving forces such as the gradients of electric potential, hydrostatic pressure, and concentration, etc., a variety of transport phenomena take place across the membrane. Many papers have appeared on the application of none+ Department of Physiology, Saga Medical School, Saga, 840-01, Japan.
0022-3654/8 1/2085-0635$01.25/0
quilibrium thermodynamics to the transport phenomena across the membranes since the original papers of Staverman and Kirkwood in the 1950s.’ In particular, major contributions have been made by Kedem and Katchalsky? me are^,^ Paters0n,4.~and others. (1) P. Meares, J. F. Thain, and D. G . Dawson in “Membranes”, Vol. I, G . Eisenman, Ed., Marcel Dekker, New York, 1972, Chapter 2. (2) 0. Kedem and A. Katchalsky, Trans. Faraday SOC., 59, 1918 (1963).
0 1981 Amerlcan Chemical Society
636
The Journal of Physical Chemistry, Vol. 85,
No. 6, 1981
This paper is concerned with an analysis of an isothermal system consisting of a cation-exchange membrane, water, and a single salt dissociating into two kinds of ions, on the basis of nonequilibrium thermodynamics. Even in this simple case, there are six independent phenomenological coefficients, and hence six independent experiments are required to elucidate the transport properties. Meares and his c o - ~ o r k e r shave ~ . ~ evaluated the six independent phenomenological coefficients of a membrane with high water content, i.e., Zeo-Karb 315 membrane (0.67 g of H 2 0 per gram of wet resin in the sodium form), by performing six independent experiments. On the other hand, Paterson and his c o - ~ o r k e r have s ~ ~ ~attempted to evaluate the phenomenological coefficients from five experimental quantities other than the experiment with an applied pressure, either by ignoring certain coefficients or by assuming nonthermodynamic relations between them, since the volume flux under the applied pressure was considered to be very small because of the low water content of the membrane which they used, i.e., an AMF C60 membrane (0.35 g of H 2 0 per gram of wet resin in the sodium form). In this paper, on a poly(su1fonic acid) cation-exchange membrane with low water content, the barium form was studied. There are two reasons for this choice. The first is that Ba2+might be strongly bound to the fixed charges? and the second is that it is of importance to evaluate also the phenomenological coefficients for the membranes with low water content without any assumption about these coefficients. Since the experiment using a high hydrostatic pressure difference entails many difficulties, as pointed out by Kramer and me are^,^ the osmotic pressure of raffinose was applied as a pressure force.1° The experiments were conducted by selecting as the driving forces the gradients of electric potential, hydrostatic pressure, and concentration. That is to say, the electric conductance and the ionic and water transport numbers were measured under the electrical force alone, and the hydrodynamic permeability coefficient was measured under the gradient of hydrostatic pressure alone, when the membrane is in contact with solutions of the same concentration on its two sides. In addition, the fluxes of the salt and water were measured across the membrane where there is a difference in the concentrations of the solutions on its opposite sides. The phenomenological coefficients (Zij) are functions of temperature, pressure, and the composition of the membrane system. Although the 1, coefficients are fundamental to the characterization of the transport phenomena across membranes, few investigations have been reported of the concentration dependence of the 1, coefficients. Therefore, we examined how the I;, coefficients depend on the concentrations of the individual species in the membrane which were obtained from the experimental data of the (3) P. Meares in “Charged Gels and Membranes-Part I”, E. SBlBgny, Ed., Reidel, Dordrecht, The Netherlands, 1976. (4) R. Paterson and C. R. Gardner, J . Chem. SOC.A, 2254 (1971).
( 5 ) R. Paterson, R. G . Cameron, and I. S. Burke in “Charged Gels and Membranes-Part I”, E. SBlBgny, Ed., Reidel, Dordrecht, The Netherlands. 1976. (6) T. Foley, J. Klinowski, and P. Meares, Proc. R.SOC.London, Ser. A , 336, 327 (1974). ( 7 ) C. R. Gardner and R. Paterson, J. Chem. SOC., Faraday Trans. 1, 68, 2030 (1972). (8) E. Kumamoto and N. Yoshida, J. Phys. Chem., 83, 2169 (1979). (9) H. Kramer and P. Meares, Biophys. J.,9, 1006 (1969). (10) Demarty and SBlBgny” have measured the coefficients for NaCl in a cation-exchange membrane by using the osmotic pressure of sucrose
as the pressure force. (11) M. Demarty and E. SBlBgny, C. R. Hebd. Seances Acad. Sci., Ser. C, 276, 1549 (1973).
Kumamoto and Klmizuka
Donnan salt concentration, water content, and ion-exchange capacity. According to the frictional model of membrane transport proposed by Spiegler,12 the transport processes in the membrane are described by balances between the thermodynamic forces on the system and frictional interactions among the components. The advantage of this model is that the frictional coefficients thereby defined are provided with a physical frictional interpretation. From the magnitude of the frictional coefficients, some inferences were drawn as to the interactions between species in the membrane. The degree of coupling (qJ introduced by Kedem and Caplan13J4may also be used to examine the interactions between species in the membrane. By evaluating the magnitude of the qij coefficients, McCallum and Paterson16 have suggested the possibility of ion association between iodide and fixed charge in a strong base quaternary ammonium anion exchange membrane. We also evaluated the q v coefficients and examined the interactions between Ba2+and the fixed charge in the poly(su1fonic acid) cation-exchange membrane.
Theory The system considered is a homogeneous ion-exchange membrane of uniform thickness immersed in an aqueous binary electrolyte solution. Different electric potentials, pressures, and concentrations are allowed on both sides of the membrane. The membrane contains four chemical species: (1)counterion, (2) co-ion, (3) water, and (4) fixed ionic group on the membrane matrix. According to nonequilibrium therrnodynamics,l6 the dissipation function 9 for an isothermal system of the four components can be written as eq 1, where j , is the flux of 4
9=
C j,Xi r=l
(1)
species i relative to the center of mass and X,is the thermodynamic force on i which conjugates with the flux. The x axis is taken in the direction of the membrane thickness, and the flows are considered to occur only in this direction. Then X, is equal to the negative gradient of electrochemical potential of species i, -d$,/dx, in the membrane. Not all of the X,values are independent because of the Gibbs-Duhem equation:17 4
c cixi = 0
i=l
where ci is the concentration of component i in the membrane. If X4 is eliminated by use of eq 2, then eq 1 becomes 3
@ =
c JiXi
i=l
(3)
where Ji is the flux of i relative to component 4 and (4) Ji = ji - Cj4/C4 If the gradients of electrochemical potential within the membrane are small and the system is close to equilibrium, the fluxes and forces are connected by linear relations as (12) K. S. Spiegler, Trans. Faraday SOC., 54, 1408 (1958). (13) 0. Kedem and S. R. Caplan, Trans. Faraday SOC., 61,1897 (1965). (14) S. R. Caplan, J. Phys. Chem., 69,3801 (1965); J. Theor. Biol. 10, 209 (1966); 11, 346 (1966). (15) C. McCallum and R. Paterson, J . Chem. SOC.,Faraday T r a m . 1 , 72, 323 (1976). (16) D. D. Fitta, “Non-Equilibrium Thermodynamics”, McGraw-Hill, New York, 1962. (17) D. G. Miller, J. Phys. Chen., 70, 2639 (1966).
Transport Properties of Cation-Exchange Membrane
The Journal of Physical Chemistry, Vol. 85, No. 6, 198 1 837
(5) where 1, values are phenomenological or mobility coefficients. The number of independent lij coefficients may be reduced from nine to six by making use of the Onsager reciprocal relation, lij = Zji (i # j ) . 1 8 For the determination of the six 1, coefficients, six independent measurements have to be done. For simplicity, the membrane system is treated as discontinuous here.* Then, X imay be written as Xi = -Ajii/d = -(Vi* + Api z;FA$)/d (6)
+
where Vi, zi, d, and F are the partial molar volume, the valency of ion i , the membrane thickness, and the Faraday constant, respectively. AP,A$, and Api are the differences in the pressure, potential, and chemical part of the chemical potential of ion i , respectively, between the external solutions. Determination of Phenomenological Coefficients (lij). We shall first consider the case when under electrical force alone there are no gradients of chemical potential and pressure. The electric current I through the membrane is related to the ion flows and can be written by Ohm's law as I = ( z l J 1+ z2Jz)F = - K ~ A # / ~ (7) where K, is the specific conductivity of the membrane. Upon substitution of eq 5 into eq 7,K, may be represented in terms of lij by eq 8. The transport number ( t i )of ion K, = (Z12111 22122112 222122)F (8)
+
+
i is the fraction of the current carried by the ith ion, and the transport number ( t 3 )of water is the number of moles of water transferred by 1 faraday of electricity in the direction of the current: ti = z i F J i / I (i = 1, 2 ) (9) t 3 = FJ3/I (10) Substituting eq 5 and 7 into eq 9 and 10 yields t i = (z,21;i + zlz2112)F/~, ( i = 1, 2 ) (11) =
+ Z2123)F/Km
(12) We shall deal next with the case of the hydrostatic pressure difference under no electrical force, while retaining the restriction of no gradient of chemical potential. Then, upon substituting eq 6 into eq 5, we obtain (Ji), - L L 1 ( V I A P + z l F A $ ) / d li2(V'2AP + zzFA$)/d - Zr3V3AP/d ( i = 1, 2, 3) (13) where the subscript p refers to the flux under the gradient of hydrostatic pressure. Introducing eq 13 into I = (zl(J1), + z2(Jz),)F= 0, solving for A$, and substituting eq 8, 11, and 12 into the result gives an equation for the streaming potential, (A$)p, as ( A $ ) , = - [ t 1 V s / ( z 1 d + V 2 / ~ 2+ t3V3laP/F (14) t3
(2,113
where p, is the partial molar volume of a salt defined as V , = vIV1 + v2V2, and vl and v2 are the numbers of cations and anions, respectively, per molecule of salt. Furthermore, substituting eq 14 into eq 13 and rearranging yields the equations for the fluxes of salt and water due to the (18)Recently Foley and MearesIghave confirmed the validity of the Onsager reciprocal relation for the transport phenomena across ion-exchange membranes. (19) T. Foley and P. Meares, J . Chem. SOC.,Faraday Trans. I , 72, 1105 (1976).
gradient of hydrostatic pressure as
In obtaining these results, we have used eq 11and 12. Now the total volume flux, (JV),, without electric current is obtained from the salt and water fluxes by using the relation: J , = VsJs+ V3J3 (17) Since a hydrodynamic permeability coefficient (L,) is defined as L, = -(Jv),/(U) (18) substitution of eq 15 and 16 into eq 18 gives
We shall consider finally the case when the membrane separates two salt solutions of different concentration under no gradients of applied electric potential and pressure. Thus, upon substituting eq 6 into eq 5, we may obtain ( J i ) c = -lil(AP1 + zlFA$)/d - l i ~ ( A ~+ ~ lz2FA$)/d p (i = 1, 2, 3) (20) li3Ap3/d where the subscript c refers to the flux under the gradient of chemical potential. In the same way as in obtaining eq 14 from eq 13, we have an equation for the membrane potential (A#), in a concentration cell: (A$), = -(tiAps/ZiVi + A P Z / Z + Z t 3 A ~ 3 ) / J ' (21) where ps is the chemical potential of a salt defined as p, = vlpl + V Z / ~ ~Substituting . eq 21 into eq 20,we obtain as before the equations for the fluxes of salt and water with no electric current:
(23) Furthermore, introducing eq 22 and 23 into eq 17 yields as an equation for the total volume osmotic flux, (Jv)c,
638
The Journal of Physical Chemistry, Vol. 85,No. 6, 1981
Equations 11, 12, 19, 22, and 24 provide the six independent equations from which the six independent lij coefficients can be determined. We can solve these equations to obtain expressions for the lij in terms of the six independent quantities: K,, tl(tz),t3,L,, (Jdc,and (JJC. The procedure is as follows. Solve eq 19,22, and 24 for t12Krn/(z12J? - 111, tltBKrn/(zlF) - 113, and t3'KrnlP - 133, which gives the expressions for 111, 113, and 133. Substitute the expressions for 111 and E,, obtained in this way into eq 11 (i = 1) and 12, respectively, which yields the expressions for 112 and Z23. Moreover, introduce the expression for 112 into eq 11 ( i = 2), which gives an expression for lZ2. These results are expressed as follows:
Kumamoto and Kirniruka
where Illji is the minor of the determinant corresponding to the term lp, and 111 is the determinant of the matrix of the Zij coefficients. Moreover, the rijcoefficients involving interactions between the mobile species and the fixed ionic group are obtained from eq 32.l' 4
c
i=l
0'
=0
cirij
= 1, 2, 3, 4)
(32)
According to the frictional model of membrane transport as proposed by Spiegler,lJ2 the macroscopic thermodynamic force, X , , acting on 1 mol of i species must be balanced by the frictional forces, Fi,, acting between 1 mol of the same species and the other species present; namely
Xi = -E Fij j#i
(i = 1, 2, 3, 4)
(33)
Furthermore, according to the simple law of friction, the interactive force, Pi,, is assumed to be directly proportional to the difference between the velocities, ui and uj, of the two species as
Fij = -fij(ui - ~
j )
(34)
in which f i j is the frictional coefficient between 1mol of species i and an infinite amount of species j . When the balance of forces is _taken over unit volume of the system, a relationship, Fij/Cj= -Fji/Ci,holds by Newton's law of action and reaction. Thus
c.Jj= C.f..
(35)
J JC
where ps is a volume fraction of the solute in the solution of a mean concentration defined as cs= -Ap3/( v3Ap,) on both sides of the membrane, namelyz0 = -VSAP~/(~~AP~)
which is equivalent to Onsager's reciprocal relationship. If the membrane is taken as the velocity reference, Ji = Ciui, so that upon substitution of this relation and eq 34 into eq 33, we obtain
(28)
It can be noted from eq 25-27 that the six equations for l i . coefficients are represented by the sum of four terms: the first term associated with the product of the specific conductivity and the transport number, the second with the salt flux, the third with the osmotic volume flux, and the fourth with the hydrodynamic permeability coefficient. If the external solution concentrations are dilute enough, ps becomes very small as compared with unity. Then if ( J J cand L, are not so large, eq 25 becomes eq 29. These
loij coefficients relevant to the ionic species alone correspond to those derived under an assumption of no water flowz1or to solvent-fixed transport coefficients in binary electrolyte system^."^^^ Frictional Model of Membrane Transport. Equation 5 makes i t possible to give an alternative set of expressions in which the forces are represented as linear functions of the fluxes. Upon solving the set of eq 5 for the forces, we obtain
where the relations between the rij and lij are given by ri, = IZlji/lll (i, j = 1, 2, 3) (31) (20) A. Katchalsky and P. F. Curran, "Non-Equilibrium Thermodynamics in Biophysics", Harvard University Press, Cambridge, MA, 1965. (21) H. Kimizuka and K. Kaibara, J. Colloid Interface Sci., 52, 516 (1975).
Because the Ji (i = 1 , 2 , 3 ) values are independent of one another, a comparison of eq 30 and 36 yields
Cirii = Cfij -
( i , j = I, 2, 3, 4)
j#i
C.r.. = -f.,j I a1
(i
# j;
i, j
1, 2, 3, 4)
(37) (38)
where eq 2 and 32 have been used. It may be noted from eq 37 and 38 that the rii coefficients represent the interactions between i and all of the other species present, whereas the rij (i # j ) coefficient represents only the single interaction between i and j species. Experimental Section Membrane. The membrane used in this work is a cation exchanger of homogeneous sulfonated styrenedivinylbenzene copolymer, which was reported in a previous paper.s Membrane Resistance. The electric resistance (R,) of the membrane was measured with a two-compartment Lucite cell similar to that of Kaibara e t a1.22 After the membrane had been allowed to stand overnight with the same solutions to be measured, mercury was introduced from the reservoirs into the cell and established contact with the membrane surfaces. Platinum-wire electrodes dipping in mercury established electrical contact. The entire system was placed in an air thermostat a t 25.0 f 0.5 "C. The resistance was measured with a universal (22) K. Kaibara, K. Saito, and H. Kimizuka, Bull. Chem. SOC.Jpn., 46, 3712 (1973).
Transport Properties of Cation-Exchange Membrane
bridge No. 4265B and a decade capacitor No. 4440B (Yokogawa-Hewlett-Packard,Ltd.). The measurement was made on external solutions ranging in concentration from to 1 M. The effect of frequency on the membrane resistance was also investigated at several frequencies between 100 and 2000 Hz. R , showed an exponential decrease with an increase in frequency and did not change significantly above 1000 Hz. Therefore, a frequency of 1000 Hz was used in the measurement. The results were reproducible to f1%. All subsequent experiments were carried out on an another membrane sample which was cut from adjacent areas of the original sheet.23 Ionic Transport Number. The cell used was the same as that used in the measurement of salt flux8and was fitted with a pair of large Ag-AgC1 plate electrodes. Through the electrodes, the electric current was delivered in the cell from a regulated dc supply (Yokogawa Electric Works Co., Ltd. Type 2853). The concentration change was estimated from the analysis of aniodic half-cell only by the same method as described in the experiment of salt flu^.^,^ The change in concentrations before and after the passage of electricity in each half-cell was 5 % Preliminary investigations showed that transport numbers of Ba2+(fBa) at and 10-1 M were affected little by current density in the ranges of 0.1-0.5,0.5-2, and 2-5 mA/cm2, respectively. Therefore, the current densities of 0.3,1, and 4 mA/cm2 were used in the concentration ranges of 10-3-5 X 10-2-5 X and 10-l-1 M, respectively. The transport numbers were also affected little by stirring the bulk solutions or reversing the direction of current flow. The reproducibility of fBa was *2%, and that of ECl was *12% above loW2 M. Electroosmotic Flux. Electroosmotic flux was measured with the same cell as that for the ionic transport number. An L-shaped calibrated capillary, 1.5 mm in diameter, was connected vertically to each compartment. The movement of the liquid meniscus in the capillary was followed by means of a traveling microscope. The actual increase in volume, AV, on the cathodic side of the membrane when 1 faraday is passed through the cell was evaluated by eq 39, where AVOis the observed volume change, vAgcI and
The Journal of Physical Chemistry, Vol. 85, No. 6, 198 1 639
v 0
I
1
2
4
6
AP(atrn1
Figure 1. The volume flux ((J&) against the osmotic pressure of raffinose (AP).
The reproducibility oft, was &3%. Hydrodynamic Permeability Coefficient. This measurement was made by a method similar to that for the electroosmotic flux by use of the salt diffusion cell with an L-shaped calibrated capillary, 0.9 mm in diameter, connected vertically to each compartment. Since the water content of the membrane used in this work is low, as shown in the previous paper,8 the volume flux with an applied pressure may also be considered very small. The experiment using high hydrostatic pressure differences is accompanied by many difficulties, as pointed out by Krlimer and me are^.^ Therefore, the osmotic pressure of raffinose was used as a pressure force.= The osmotic pressure waa calculated from the data for the osmotic coefficient of raffmose obtained by Ellerton et al.,29where the coefficient was assumed not to be affected by the presence of BaC12. A series of measurements at pressures from 1 to 6 atm were carried out with the external solutions having the concentrations from to 1 M. The concentrations of raffinose used were 0.05-0.2 M. All of the volume fluxes observed were linear functions of the applied osmotic pressure. Figure 1shows a typical plot of the volume flux against the osmotic pressure. A hydrodynamic permeability coefficient (L,) was calculated from a slope of such a straight line of gradient accurate to f l l % . The numerical values of L, can be in error by 10% because of the presence of BaClz affecting the activity of water in the A V = AVO+ Vka - V A ~Z- J ~ ~ V B ~ C(39) ~ ~ / ~ raffinose solution.30 There are possibilities for the raffinose to permeate through the membrane or affect the mean activity of VAg are the molar volumes of crystalline AgCl and Ag, respectively, VBaC12 is the partial molar volume of BaC12 BaC12. However, even in the case of the most concentrated solution of the added raffinose on one side of the memin the solution, and EBa is the transport number of Ba2+ in the membrane.26 Then a water transport number (Ew) brane, no raffinose was detected on the other side by a defined as the number of moles of water transferred per polarimeter at the end of the experiment. Furthermore, faraday was calculated by Zw = AV/l8.0. Preliminary the membrane potential in a concentration cell of BaClz was unaltered by addiog raffinose on one side of the investigations showed that Zw at and lo-’ M were membrane. This may indicate that the raffinose does not affected little by current density in the ranges of 1-3 and 2-4 mA/cm2, respectively. Therefore, all electroosmotic flux experiments in the concentration range of 10-!-1 M (27) LakshminarayanaidP has observed a discrepancy of 7% between hydrodynamic permeability coefficients obtained by using the pressure were conducted at the current density of 3 mA/cm2. The force N2 gas pressure and the osmotic pressure of sucrose. water transport numbers were also affected little by stirring (28) N. Lakshminarayanaiah, J. Phys. Chem., 74, 2385 (1970). the bulk solutions or reversing the direction of current flow. (29) H. D. Ellerton, G. Reinfelds, D. E. Mulcahy, and P. J. Dunlop,
-
.
J.Phys. Chern., 68, 398 (1964).
(23) The permeability for a salt calculated from the observed salt fluxes described later gave almost a linear plot against concentrations with the same slope as obtained by use of the different membrane sample* cut from the same sheet,%showing that the change in transport properties owing to the difference of membrane samples cut from the same sheet might be small because of the homogeneity of the membrane used here. (24) E. Kumamoto and H. Kimizuka, unpublished result. (25) An estimated value of the volume change due to the electroosmotic flow described later was very small compared with the volume of the bulk solution during the experiment. Therefore, an increase in concentration in the cathodic half-cell must be equal to a decrease in the anodic one. (26) N. Lakshminarayanaiah, “Transport Phenomena in Membranes”, Academic Press, New York, 1969.
(30) So far as we know, there are no data for the activity coefficients of solutes or the osmotic coefficient for the BaC12-raffinose system. Therefore, a NaC1-sucrosea1 or CaC12-sucrose32system was used as a check. Even at 1 mol kg-I of sucrose, the activity coefficient of NaCl is affected within only 5% in the concentration range of NaCl less than 3 mol kg-’ (the ionic strength in this concentration range corresponds to that of BaCl, solution used here). On the other hand, in the concentration range of 0.1-0.3 mol kg-’ of sucrose, the osmotic coefficient in the solution of sucrose alone is affected within 10% in the presence of CaC12ranging in concentration from 0.2 to 1 mol kg-l. Such a large effect of salt on osmotic coefficients would give rise to large errors of L,. (31) R. A. Robinson, R. H. Stokes, and K. N. Marsh, J. Chem. Thermodyn., 2 , 745 (1970). (32) H.-J. Jansen and H. Schonert, Ber. Bunsenges. Phys. Chem. 79, 632 (1975).
640
Kumamoto and Kimiruka
The Journal of Physical Chemistry, Vol. 85, No. 6, 1981
t 0.4 10-2
10-3
10-1
, 10-3
1
C,(M)
I
I
16'
10-2
1
C&M)
Flgure 2. The concentrations of Ba2+, CI-, and water (Cm,Ca, and in the membrane as a function of-the external solution concenCBA; (A) Ccl: (0) C,. tration (Cs):(0)
c,)
I
Flgure 4. The transport number of Ba2+ (Tib) in the membrane as a function of the external solution concentration (CJ.
I
8t
0
10-2 10-1 I C&M) 10-3
162
10-1
C&M)
Flgure 3. The specific conductivity of the membrane (K,) as a function of the external solution concentration ( C J .
cause a change in the mean activity of BaClP3O Salt and Volume Osmotic Fluxes. The apparatus and the procedure for measuring the salt flux were the same as those used in our previous study.* Volume osmotic flux was measured with the same cell as that for the hydrodynamic permeability coefficient. The changes in concentration due to salt diffusion were negligibly small as compared with the bulk solution concentrations during the time required to determine the volume flux. The measurements were made with such a pair of concentrations, C,I and C?, that (C,' C:)/2 values are the concentrations at which R,, fBa, & ,, and L were measured and that C t / C p values are in the range oP2-3, Le., 0.015-0.005,0.03-0.01, 0.07-0.03, 0.15-0.05, 0.3-0.1, 0.7-0.3, and 1-0.5 M. The reproducibilities of salt and volume osmotic fluxes were f 2 % and &5%, respectively.
+
Results and Discussion The concentrations of Ba2+,Cl-, and water (GB~,~ C I and , in the membrane may be calculated from the Donnan salt concentration (C,) and the physical characteristics of the membrane which were reported in a previous paper.' These results are plotted against the-external solution concentration (C,) in Figure 2. The C C values ~ increase i_n proportion to C,. It can also be seen that the CBa and C , values are practically constant a t C, less than 0.1 M, but above this concentration the former increases and the latter decreases with increasbg concentrations, The reason for such a dependence of CBa on C, is that C, cannot be negligible as compared with the fixed charge concentration at C, larger than 0.1 M. Also, the decrease of C , at high external salt concentrations stems from the decreased water content.
c,)
1
1
Figure 5. The water transport number (7,) in the membrane as a function of the external solution concentration (Cs).
Specific Conductiuity. Figure 3 illustrates values of the specific conductivity ( K ~ of ) the membrane calculated from the observed membrane resistance (R,) and the crosssectional area and thickness of the membrane as a function of C,. The calculated K, value increases gradually almost linearly with log C, up t o C, nearly equal to 0.1 M, but beyond this concentration i t increases remarkably. This concentration is approximately equal to one at which CBa begins to increase abruptly, as indicated in Figure 2. Therefore, the abrupt increase of K , may be due to a greater contribution of the increased Donnan salt. Ionic Transport Number. The transport number of Ba2+( f B a ) in the membrane is plotted against C, in Figure 4. The values of tBaare nearly equal to unity at C, less than 0.02 M, indicating that the membrane is highly permselective for the counterion at dilute concentrations. As C, increases beyond 0.02 M, the fBa value decreases markedly and becomes 0.58 at 1 M. The reason for such a concentration dependence of fBBe may be the same as that for the abrupt increase in K,. Water Transport Number. Values of the water transport number (Ew) in the membrane are plotted against C, in Figure 5. The value of t, stays a constant of ca. 7.3 in the concentration range 0.014.1 M and then decreases to 4.4 at 1 M with increasing concentration. Such an abrupt decrease of fw may be attributed to the steep concentration change of Ba2+and water in the membrane as indicated in Figure 2, similar to the case of K, and f B a . The magnitude of f, is comparable to that observed by Lakshminarayanaiah and Siddiqi,33who used the membranes with low water contents. (33) N. Lakshminarayanaiah and F. A. Siddiqi, 2. Phys. Chem. (Frankfurt am Main), 78, 150 (1972).
Transport Properties of Cation-Exchange Membrane
The Journal of Physical Chemistry, Vol. 85, No. 6, 1981 641
r
I
91
10-1
10-2
t
I
1
C&M)
Flgure 6. The hydrodynamic permeability coefficient (L,)as a function of the external solution concentration (CJ.
/'
151
Figure 9, The 4, ( i , j = 1, 2, 3) coefficients as a function of the external solution concentration (CJ. 1 = Ba2+, 2 = CI-,3 = H20. Solid and broken llnes represent positive and negative values, respectively.
CZ(M) Flgure 7. The salt flux ((J& as a function of the mean salt concentration (C,")on both sides of the membrane. 15
Figure 10. The p,//I,/(i, j = 1, 2) values as a function of the external /022//22 1 solution concentration (CJ: (0)/ o l , / / l l ; (A) /o12//,2; (0) = Ba2+, 2 = Ci-.
I
-
10
c
ln
E
Y
b
0
X -
5:2
0
I
0.1
0.3
0.5
AC&M)
Figure 8. The volume osmotic flux ((&) as a function of the concentration difference (AC,)between the two external solutions.
Hydrodynamic Permeability Coefficient. Figure 6 illustrates the behavior of the hydrodynamic permeability coefficient (L,) against C,. It can be seen that the L, value slightly decreases and then increases with the increase in concentration. Thus, in contrast to K,, fBa, and f w , there is no tendency for L,. to change abruptly with the concentrations in the vicinity of 0.1 M. Salt and Volume Osmotic Fluxes. Figure 7 gives the salt flux data against the mean salt concentration, C," (=
