52 MEMBRANB POTENTIALS MEMBRANE POTENTIALS OF AN ION EXCHANGE MEMBRANE BY M. NAGASAWA AND I. KAGAWA Department of Applied Chemistry, Nagoya University Chikusa-ku, Nagoya, Japan Received 16th Junuary, 1956 By using the ion exchange membrane, Nepton CR-51, measurements were taken of the membrane potentials with different concentrations of simple electrolytes (NaCl, Na,SO,) and polyelectrolyte (sodium polyvinylsulphate), and a comparison made with theories already reported. It was observed that the membrane potential has a linear relationship with the logarithm of Na+ activity at lower concentrations with a slope slightly lower than RT/F. It is concluded that such results are due to the abnormal behaviour of ionic activities within membrane, and it is pointed out that they are important for determining an unknown activity of Na+ by the membrane potential method.The membrane potential of the rigid membrane, such as collodion, Cellophane ion exchange membranes and others, has been widely discussed by several authors on the basis of the theory of Meyer 1 and Teorell.2 Some years ago the more funda- mental equation of membrane potential, eqn. (l), was obtained by Schlogl and Helfferich 3 and, independently, by Nagasawa 4 : where the terms denote the average activities of ions within the membrane and I* their mobilities, and al, a2 the activities of the electrolyte on both sides of the membrane.M. NAGASAWA AND I . KAGAWA 53 If it is assumed that the ionic distribution between the outer and inner phases of the membrane is determined by the Donnan membrane equilibrium,l3 a: and a? in eqn.(1) may be approximately calculated from the following : - - in which A is the average concentration of counter ions of the membrane. Then, if eqn. (2) is introduced into eqn. (1) under the condition that A is constant, Meyer and Teorell’s equation (as already reported by Helfferich 5 ) can be derived from eqn. (1) as follows : I , - I- I+ + 1, u= - and x = (4a2 + &)*. Strictly speaking, it must be remembered, however, that the available range of eqn. (3) is considerably limited because the ionic distribution between two phases is given by the Donnan equilibrium only for membranes having homogeneous structure (see appendix), and also because the integration of eqn. (1) may be im- possible under the condition that A is constant if the activity coefficients of moveable ions within the membrane are not allowed to remain either constant during the variation of outer concentration of electrolyte, or equal to their activity coefficients in the outer solution.If the activity of ions within the membrane must be calculated from a relation other than the Donnan equilibrium, some other equation should be obtained. For the membrane having rigid porous structure, the following equation has already been given by Nagasawa and Kobatake 4 : E = [. log? - cc log (--) a1 + P + (“‘) log pQ)] F a2 a2 + 18 I+ i- 1- a;! + p 9 (4) where the relations were used for the calculation of membrane potential, and a, p and k’s are constant, depending upon the membrane and the electroIyte.Constants a and fi can be easily determined from the following linear relationship between transport number of cation and activity of electrolyte derived from equating eqn. (4) and Nernst’s equation of diffusion potential, Eqn. (4) was reported to be in good agreement with experiments using collodion, Cellophane, glass membrane, etc. In the present paper we show that the activities of ions in ion exchange membrane are better expressed by eqn. (5) than by eqn. (2) and that the observed values of membrane potential are in satisfactory agreement with calculated values from eqn. (4), despite the fact that ion exchange resins are usually supposed to have a comparatively homogenous structure. Moreover we stress the fact that the result thus obtained is mainly because of the abnormal behaviour of ionic activities within the membrane, and therefore we point out the importance of taking into account in full detail their abnormal behaviour for the purpose of discussing the theory of membrane potential and of determining an unknown ionic activity by the membrane potential method.54 MEMBRANE POTENTIALS In recently published papers it was shown that the transport of water with ions through membranes has considerable influence upon the membrane potentials. In the present paper the authors have neglected the discussion of water transport through membranes.Nevertheless, we find satisfactory agreement between experiments and theory. Therefore we may conclude that the deviation of mem- brane potential of ion exchange membranes from the potential of an ideal electrode is caused mainly by the presence of negative ions within the membrane and the abnormal behaviour of ionic activities.EXPERIMENTAL The essential part of the apparatus is shown in fig. 1. Electrolyte solutions wash the surfaces of the ion exchange membrane. The membrane potential appearing on both sides of the membrane was conducted by saturated KCl bridges and calomel electrodes, and measured by an ordinary potentiometer. If to electrode the flow of electrolyte solution is stopped, the value of the potential difference gradually decreases with time, probably because of the variation of electrolyte concentration in the thin layer near the membrane surface. The membrane used in this experiment is Nepton CR-51 which was kindly contributed by Dr.Wayne A. McRae. The electrolytes are NaCI, Na~S04, and Na-polyvinylsulphate (Na-PVS) prepared by the usual method. The Na+ activity of Na-PVS solution was determined by a Na-amalgam electrode, using the technique already reported by Lewis.7 In our experiments, however, the lower limit of activity which could be determined by this method was 0.001 N because the surface of the amalgam was continuously kept clear by dropping amalgam into Na-PVS solution drop-by-drop and the potential of the electrode was quickly determined using the vacuum tube voltmeter constructed of vacuum tube UX-54 and an ordinary potentiometer. A more precise description of the measurement of Na+ activity by the Na-amalgam electrode will be reported in another paper. Na+ activities in low molecular salt solutions were calcu- lated from the average activity coefficients of the salt in Landolt’s table, making use of Lewis’s hypothesis, that ions have the same activity coefficients if the ionic strength in the solutions are the same.rat, KC, bridqc membrane ‘---€ell for measurement Of membrane potential. RESULTS A series of experiments is shown in fig. 2 in which the membrane potential of different concentrations of NaCl were measured, maintaining the ratio of concentrations on both sides of the membrane at constant. Another series is shown in fig. 3 in which they were measured by maintaining the concentration on one side of the membrane constant. Inspection of fig. 3 shows that the membrane potential has an approximately linear relationship with log a1 at low activity of NaCl, but that it deviates from linearity at higher concentrations.The slope of the linear relationship between E and log a1 is nearly equal to but slightly lower than RVF. At activities of NaCl lower than 0.01 N, the relationship between membrane potential and log a1 can be given as (6) E = Eo - a(RT/F) log a1, where Eo is a constant depending upon the nature of the membrane and concentration of electrolyte used for the standard, and u is a constant nearly equal to unity, depending to a small degree upon the nature of the negative ion. TABLE RELATION BETWEEN a AND DEGREE OF ESTERIFICATXON OF Na-PVS no I I1 I11 w NaCl deg. of esterif. 0.740 0.71 1 043 1 0.301 a 0.816 0816 0883 0.927 0947M. NAGASAWA AND I. KAGAWA 55 Eqn.(6) in addition agrees fairly well with experimental results using electrolytes other than NaCI, i.e., Na2S04 and Na-PVS, if ON^+ is introduced into eqn. (6) instead of al. Here it is observed that the value of C( varies with the nature of the negative ion, and that as the degree of esterification of polyvinylsulphate decreases, the values of cc approach the value for NaCI. 0 0 - 5 I FIG. 2.-Membrane potential with NaCI, here C,/C, = const., temp. 25" C. (1) ExPt. values ; (2) values calc. from eqn. (4) ; (3) values of RT/F log (aib2)- E (mV) P FIG. 3.-Membrane potential with NaCl, here C2 = 0.1 N, temp. 25" C. (1) Values calc. from eqn. (4); (2) line of E = EO - (RT/F) log a1. There is a similar relation for H+, the activity of which can be easily determined by measuring the pH.The relation between membrane potential and H+ activity is shown in fig. 6. This behaviour of the membrane for H+ activity is exactly the same as that of a56 MEMBRANE POTENTIALS glass electrode.- fIt is usually found with the glass electrode that the membrane potential can be represented by an equation analogous to eqn. (6) and that it begins to deviate from linear relation at pH = 2, FIG. 4.-Linear relation between membrane potential and Naf activity of Na2SO4: here C2 is 0.1 N of NaC1, temp. 25" C. (1) Expt. values; (2) line of E = EO - (RT/F) log U N ~ + . FIG. 5.-Linear relation between membrane potential and Na+ activity of Na-PVS ; here C2 is approximately 0.01 N of NaCl, ~ temp. 25" C. (1) Expt. values; (2) lines of E = Eo - (RT/F) log UN*+.DISCUSSION If we consider that ion exchange equilibrium between resin phase and solution is based on the Donnan theory of membrane equilibrium as has been generally accepted, we should regard the structure of ion exchange resins as nearly homogene- ous. In this light, the present authors originally thought that experimental mem- brane potentials of ion exchange membrane should be compared with eqn. (3) of Meyer and Teorell. However, a satisfactory agreement between experiments and theory was not found. First, although the theory predicts that the membrane potential will approach lim (RTIF) log (al/a2) as the concentration of electrolyte decreases, the experimental membrane potential tends toward a value lower than lim (RT/F) log(al/a2), as indicated in fig. 2 and as already reported by others ; secondly, eqn.(3) can not explain the linear relationship between E and log al. The solid line in fig. 7 indicates the values calculated from eqn. (3) usingA = 0.1 and U = 1 which are most suitable for experi- ments. Here U has little effect on the calculated values except at higher NaCl concentrations. However, these calculated values are not in satisfactory agree- ment with experiments. Moreover as shown in fig. 8, the experimental values of fig. 2 do not agree with the values calculated using the same values of A and Uas in fig. 7. It is also difficult to find the suitable values of A and U for experiments represented in fig. 2 by the usual method. The linear relationship between E There are three principal points of disagreement. C+O c+oM .NAGASAWA A N D I. KAGAWA 57 and log a N+ in experiments using electrolytes other than NaCl could not be even qualitatively explained from eqn. (3) although, strictly speaking, such comparison between experimental results and theory is not valid. Thirdly, the value A = 0.1 may be too low for the activity of the counter ion in an ion exchange resin when compared with its analytical value even if the fact is taken into account,9~10 that the activity coefficient of sodium ion is usually lower within ion exchange resins than in the simple salt solution. FIG. 6.-Linear relation between membrane potential and H+ activity of polystyrene sulphonic acid ; here C2 is 0.00969 N HCI, temp. 35" C, (1) Expt. values; (2) line of E = EO + (RT'F)pH.FIG. 7.-Comparison between calculated values of eqn. (3) and experimental values (NaCl); here C2 = 0.1 N. (1) Expt. values; (2) calc. values. In contrast, comparing the experimental values with eqn. (4) and (6) derived for the porous, rigid membrane we can obtain a satisfactory agreement. We can see in fig. 9 a linear relation between E and a1 from which a = 0.924 and = 1.268 can be obtained by employing eqn (6). The values calculated from eqn. (4) by introducing a = 0-924-and fi = 1.268 are shown by solid lines in fig. 2 and 3.58 MEMBRANE POTENTIALS Here, we used the same value of l + / L within the membrane as that in the simple salt solution. The value of I + / L exerts little influence on the calculated values. That the experimental values are not consistent with eqn.(3) but satisfactorily so with eqn. (4), is due seemingly to the abnormal behaviour of the ionic activity in the resin, although it may be considered that this appears due to the porous, FIG. S.-Comparison between calculated values of eqn. (3) and experimental values (NaCl) ; here Cl/C:! = 2. (1) Expt. values; (2) values calc. using A = 0.1, U = 3 ; (3) values calc. using A = 0.1, u = 2. FIG. 9.-LLinear relation between E and al. rigid structure of the resin. That is to say, by measuring c$ and a& in the experi- ment of the ionic distribution between membrane phase and outer solution and, further by assuming that the activity coefficient of the anion is little affected by the resin, charge, as may be presumed from the published work of Kagawa and Katsuura,ll we can obtain the value of the activity coefficient of cation within the resin j-? and consequently of 3 using -- the following equation : f' +f c+ * c- +:=&.(8)M. NAGASAWA AND I. KAGAWA 59 The values of 3 so obtained along with the calculated values from eqn. (2) are shown in fig. 10. Thus it can be concluded that the Donnan equilibrium is not adequate in explaining the behaviour of 3, which is much more important than the behaviour of 2 for the purpose of discussing membrane potential. A further conclusion is that the linear approximation in eqn. (3, although not yet entirely satisfactory, may be better than eqn. (2). Similar abnormal behaviour of r;” or within ordinary ion-exchange resins has already been observed by Gregor,9 and also by Kanamaru, Nagasawa and Nakamura.10 FIG.10.-Abnormal behaviour of a? within the ion exchange membrane ; here (1) expt. values of e, (2) expt. values of a?; (3), (3) calc. values of a: and a? assuming A = 1 ; (4), (4)’ assuming A = 0 5 ; (9, (5)’ assuming A = 01. - - From the theoretical point of view it is impossible to prove quantitatively the linear relationship between membrane potential and Na+ activity of polyvinyl sulphate because the polymer negative ion cannot enter the membrane and therefore we cannot predict the effect of the polyelectrolyte upon the Na+ activity within membrane. Nevertheless, we can expect that the experiments with polyvinyl- sulphate at the lower concentration of polyelectrolyte are fairly well expressed by eqn. (6) for the three following reasons : (i) as > af within the ion exchange membrane, it can be seen from eqn.(1) that has greater importance than af for the membrane potential. Therefore, it may be expected that a1 and a2 in eqn. (3) and (4) are nearly equal to the activity of Na+ in the solution. (ii) A linear ap- proximation such as eqn. (5) has widest applicability for all kinds of behaviour of Na+ activity within the membrane. On the contrary, the following equation which is usually used for discussion of the membrane potential of an ion exchange mem- brane at low Na+ concentration can be derived only from the Donnan approxima- tion, the applicability of which is very limited : (9) It is natural to suppose that eqn. (6) is more useful than eqn. (9) for discussing the membrane potential of a Na+ solution of unknown activity. (iii) With the glass electrode, eqn.(7) is also consistent with the experiments with all electrolytes. Eqn. (6) may be important for the determination of Na+ activity by the mem- brane potential method. E = Eo - (RT/F) log aNa+.60 MEMBRANE POTENTIALS APPENDIX When we apply Donnan’s theory to the distribution of electrolyte between the membrane phase and the outer solution, we must assume two equations at the same time, i.e. a: at= a2, (10) where a$ denote the positive and negative ion activities at a point having an electrical poten- tial $ within the membrane phase, and a the average activity of electrolyte in the outer solution, so that If for the purpose of simplifying the following discussion, we assume the membrane phase to be composed of pores of radius r and also if we introduce a$/a = f ( r ) and aT/a = $(r) into eqn.(10) and (ll), we obtain a: = a exp (- qb/kT), af = a exp (E$/kT). (12) f ( r > * *(r) = 1, (13) ’(‘) * into eqn. (14), 1 *(r> Introducing f ( r ) = - obtained from eqn. (13), and K(r) = 1; $(r) . I’ dr we have finally From eqn. (15) we can obtain r2k2(r) - 4rk(r) + 4 = 0. k(r) = 2/r, that is, Moreover, it follows from eqn. (16) that $(r) = a exp (- et,h/kT) = constant. Therefore we should apply the Donnan equilibrium only to an electrolyte distribution between an outer solution and a membrane having a constant value of potential $ i.e. a homogeneous membrane. The authors thank Dr. Wayne A. McRae of Ionics, Incorporated, for his gift of the sample used in this study. Thanks are also due Mr. S. Shimoyama for experimental assistance. 1 Meyer and Sievers, Helv. chim. Acta, 1936, 19, 649, 665, 987. 2 Teorell, Proc. Soc. Expt. Biol., 1935,33, 282. 3 Schlogl and Helfferich, 2. Elektrochem., 1952, 56, 644. 4 Nagasawa, J. Chem. SOC. Japan (Pure Chem. Sect.), 1949,70, 160 5 Helfferich, 2. Elektrochem., 1952, 56, 644. 6 Spiegler, J. Electrochem. SOC., 1953, 100, 303 C. 7 Lewis and Kraus, J. Amer. Chem. 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