FIXEDCHARGE DENSITY GOVERNING MEMBRANE PHENOMENA
2447
Effective Fixed Charge Density Governing Membrane Phenomena. IV. Further Study of Activity Coefficients and Mobilities of Small Ions in Charged Membranes by Tetuo Ueda,* Naoki Kamo, Naobumi Ishida, and Yonosuke Kobatake Faculty of Pharmaceutical Sciences, Hokkaido University, Sapporo, 060, Japan (Received January 17, 1Q72) Publication costs borne completely by The Journal of Physical Chemigtry
The mobilities of counter- and coions in charged membranes were determined experimentally by use of the flux measurements of radioisotope in a wide range of salt concentrations. The mobility of coions was identical with that in the bulk solution in the whole range of concentration studied, while that of counterions decreased very much with decrease of the external salt concentration. The concentration dependencies of both mobilities and activity coefficients of mobile ions were found to be the same, and they agreed with the “additivity rule” found empirically in the field of polyelectrolyte studies. The amount of counterions bounded in the vicinity of membrane skeleton, however, depended on the salt concentration, which was different from that observed in the polyelectrolyte solutions. The identity between the concentration dependencies of the mobility and the activity coefficient of counterions leads to a great simplification of the theoretical analysis of various membrane phenomena and enables us to evaluate the fixed charge density effective to the membrane phenomena from a simple measurement of the membrane potential a t an arbitrary salt concentration. Theoretical values of various membrane phenomena other than the membrane potential were evaluated by use of the effective fixed charge density determined from the membrane potential data and were compared with the corresponding experimental data. The agreement between theory and experiment was satisfactory.
Introduction In a different series of papers’ concerned with the physicochemical studies of membrane phenomena, we stressed repeatedly that the nonideal behaviors of small ions in a charged membrane should be properly taken into consideration for quantitative description of the transport processes in the charged membrane. The nonideality of small ions in the membrane phase stems from the strong ionic interaction between movable ions and the charges fixed on the membrane skeletons. This situation is essentially the same as that in a polyelectrolyte solution. In part I of this series12 the activity coefficients of movable ions in a charged membrane with known fixed charge density were determined experimentally, and they were found to be expressed by the equation
where yk0, C-, X , and 4 stand for the mean activity coefficient of an electrolyte component in the bulk solution, the concentration of the anion in the membrane, the stoichiometric fixed charge density, and a numerical constant less than unity, respectively. Equation 1 has the same functional form as the “additivity rule” found empirically in the field of polyelectrolyte solution ~ t u d i e s . ~+X is generally referred to as the thermodynamically effective charge density
of the membrane or of the polyelectrolyte component, and 4 represents the fraction of free counterions which is not bounded in the vicinity of the polymer chains. Although the fraction of free counterions was considered to be constant irrespective of the external salt concentration in a polyelectrolyte solution, t#I in the charged membrane was shown to be dependent on the salt concentration. Furthermore, in part I, the mobilities of movable ions in the membrane were determined experimentally by combining the data of the membrane potential and the salt flux together with the analytical amount of cation and anion species adsorbed into the membrane phase. The results were represented as
where utostands for the mobility of ion species i in the bulk solution, and t#I’X is the charge density of the membrane effective to the mobilities. Of special interest is the identity of the concentration dependencies of mobilities and activity coefficients of small ions in the membrane phase, i.e., t#I and 4’ in eq 1 and 2 are (1) Y. Toyoshima, AM.Yuasa, Y. Kobatake, and H. Fnjita, Trans. Faraday Soc., 63, 2803, 2814, 2828 (1967). (2) N.Kamo, Y. Toyoshima, Y. Kobatake, and H. Nozaki, Kolloid2. 2. Polym., 284, 914 (1971). (3) A. Katchalsky, 2. Alexandrowics, and 0. Kedem, “Chemical Physics of Ionic Solutions,” Wiley, New York, N. Y., 1966, p 296. The Journal of Physical Chemistry, Vol. 76, No. 17, lQ78
T. UEDA,N. KAMO,N. ISHIDA, AND Y. KOBATAKE
2448 identical in the whole range of the salt concentration studied. In part I, however, we determined y t and ut (i = +, -) only for the case of KCl as the external electrolyte component. To generalize the conclusion described above, it is necessary to survey various combinations of membrane and electrolyte pair other than that employed in part I. This paper purports to confirm eq 1 and 2 for different electrolyte species with the same membranes studied in part I. When the concentration dependencies of the mobilities and activity coefficients of small ions in the charged membrane always follow eq 1 and 2 with 4 = C$’, great simplification is obtained in the derivation of the theoretical equations for various membrane phenomena, and also the parameter 4 X characteristic to a given pair of membrane and electrolyte can be determined only from a measurement of the membrane potential. The other transport phenomena for the same pair of membrane and electrolyte can be described theoretically by using the value of C$X. The comparison between theory and experiment provides further confirmation of the applicability of eq 1and 2.
Experimental Section Materials. The membranes used were the same as those used in the previous papers, which were collodionbased polystyrenesulfonic acid membranes. It was noted that both stoichiometric charge density, X , and the water content of each of these sample membranes stayed constant irrespective of the species and concentration of the electrolyte component in the external aqueous solution.6 The relevant characteristics are listed in Table I. Table I: Some Characteristics of Membrane Used X,
Water content,
LO,
Membrane
equiv/l.
wt %
cm
PS- 1 PS-2 PS-3
0.224 0.116
0.78 0.86 0.85
0.053 0.101 0.090
0.0438
The salts used were LiC1, KC1, NaC1, and RbC1. Potassium chloride of analytical grade was purified by recrystallization. Sodium chloride of standard reagent and lithium chloride and rubidium chloride of analytical grade were used as delivered. Radioactive isotopes used as tracers were sodium22 and chlorine-36 in NaCl aqueous solution, and they were purchased from the Radiochemical Center. The water used as solvent was obtained by passing distilled water through both cation and anion exchangers. Activity Coe$cients of Xmall Ions in the Membrane. The system considered here is a negatively charged The Journal of Physical Chemktry, VoE. 76,N o . 17, 1972
membrane immersed in an aqueous solution of a 1:1 type electrolyte. The spatial distribution of the fixed charges in the membrane is assumed to be uniform. The condition of the equilibrium between the membrane and the solution phases leads to
a+a- = a2 = (yko)2C2
(3)
Here, a t denotes the activity of ion species i in the membrane, and a and C are the activity and the concentration of the electrolyte component in the bulk solution, respectively. As shown in part I, the contribution of the difference in the osmotic pressures in two phases t o eq 3 was less than 0.6% in the whole range of concentration studied. The activity coefficients of small ions in the membrane, y+ and y-, are defined by a+ = r+C+, a- = r-C-. Thus, eq 3 is rearranged to give CZ/(C+C-)
r+r-/(r*o>2 =
(4)
Equation 4 reads that the activity coefficient of the electrolyte component in the membrane, y+y-, can be determined from the analysis of C- and C+. The concentration of anion in the membrane, C((71- ion for the present case), was determined by titration against AgN03 by the usual way or by the potentiometric titration as described in part I. The value of C+ was calculated from the electroneutrality conX. dition, i.e., C+ = CMobilities of Small Ions in the Membrane. The mobilities of movable ions can be determined by measuring the isotope fluxes. The tracer method for the determination of mobility in the membrane has the following two advantages over the method proposed in part I. (a) The experiments can be performed with no gradient in the concentration of the electrolyte component, and (b) the movement of the local center of mass does not occur in the system in problem, and hence the ambiguity which stems from the mass movement is not included in the observed mobility. The slight inaccuracy due to isotope effect in electrolyte solutions may be neglected in the membrane systems encountered here. In the measurement of the isotope flux, the effect of the unstirred liquid film adjacent to the membrane surface on the observed flux must be taken into account. However, Lak~himinarayanaiah~ showed that the contribution of this layer to the isotope flux is negligibly small in an ordinary membraneelectrolyte system. Then the following equation is obtained for the flux of the isotope, J*
+
(5)
where C, stands for the concentration of the cold ion species i in the membrane, L the effective thickness (4) N. Lakshminarayanaiah, “Transport Phenomena in Membranes,” Academic Press, New York, N. Y., 1969, p 131.
FIXED
CHARGE DENSITY GOVERNING MEMBRANE
of the membrane, C the concentration of the external solution, and AC* the difference of tracer concentration in two compartments. Equation 5 tells that the measurements of the tracer flux enable us to evaluate the mobility of species i in the membrane provided that the concentration C, of the cold ion species i in the membrane is known. Two solutions of identical concentration were separated by the membrane. The geometrical area of the membrane was 1.04 cm2. Each compartment contained 40.0 ml in volume of the salt solution. The solutions were stirred vigorously and were changed several times every 2 or 3 hr before introducing the radioisotope. This procedure permits the membrane and bulk solution to be in equilibrium. Then, the solution of radioactive isotope (0.1 ml) was added to one compartment. With a determined time interval the solutions were picked up by a micropipet (0.20 ml) from each compartment. The radioactivity of the solutions containing 22Na or a6C1was counted by a scintillation spectrometer (Aloka model LSC 501). The radioactivity in the hot compartment, (CPA!I)2, stayed constant within the experimental error, and that in the cold compartment, (CPiVI)~, increased linearly with time after a short period of time from the onset. When the slope of this linear portion is denoted by 8, the mobility u, can be evaluated by the equation
(6,) c,
C L
ui
=
=
J*
f.
.(")(")(">"RT Ao C, (CPA/I)z
(6)
where V is the volume of electrolyte solution in each compartment, A. and Lo are the geometrical area and thickness of the membrane, respectively, and f is the tortuosity factor. In general, the effective area, A , is smaller than Ao, and the effective thickness of the membrane, L, is larger than Lo. Then f is defined by the equation5
f
(Ao/Lo)/K (7) Here K stands for AIL. The value off may be constant irrespective of the species and concentration of ions when the degree of swelling of the membrane is not changed by the salt concentration as encountered in the present study. Since the effects of the fixed charges in the membrane on the activities and mobilities are diminished when the concentration of the external solution is high enough in comparison with the fixed charge density, X , the value off is able to be determined from the limiting value of the concentration of the external solution. Then we have =
2449
PHENOMENA
Here, uiostands for the mobility of ion species i in the bulk solution at a concentration C. Once the value o f f is determined, the absolute value of the mobility of i ion in the membrane can be evaluated from eq 6 with use of data on C, in the membrane a t a given concentration of the external salt solution. Measuremenis of Membrane Potential and Dc Resislance. The cells and procedures adopted for the measurements of the membrane potential and dc resistance were the same as those used in the previous stUdy.2*6 All experiments were performed in an air oven regulated at 25 0.2".
*
Results and Discussion Activity Coeficienls in the Membrane. Introducing the results of t'he analysis of C+ and C- in the membrane into eq 4, we obtain the value of y+y-/(y*")2. Figure 1 shows the data for y+~-/(y*")~as a function of C - / X in various pairs of NaCl and LiCl and three membranes with different X . In Figure 2, the data of y+y-/(~*")~for KC1 in the same membrane as in Figure 1 are plotted against C / X , which are taken from part I. It is seen that the data points for different pairs of X and C follow a single sigmoid-shaped curve when plotted against C-/X for each electrolyte species studied. I n part I, we showed that the value of 9 in eq 1 for a pair of KC1 and the membrane is varied from 0.4 0.05 in a concentrated solution to 0.1 in a dilute RCl solution. In Figures 1 and 2, two dashed lines represent the values calculated from eq 1 with (b being 0.4 and 0.1, respectively. From these two figures, it is considered that the value of (b for NaCl and LiCl varies with the concentration of external solution as in the
*
r
I
1.0 -
0.8
I-
% '*v- 0 . 6 u \
J .f 0 . 4 -
0.20' I 0-2
I 0-1
c-/ x
I
I O
Figure 1. Plots of y + y - / ( y * ' ) Z against log ( C - / X ) for systems of three kinds of membrane and NaCl and LiCl solution: a, PS-1; 0, PS-2; 0, PS-3 for NaCl, and 0, PS-2 for LiCl. (5) J. 5. Maokie and P. Mears, Proc. Roy. Soc., Ser. A , 232, 498 (1955). (6) N.Kamo, Y . Toyoshima, and Y . Kobatake, Kolloid-2.2. Polym., 249, 1061 (1971).
The Journal of Physical Chemistry,Vol. 76,No.17, 1972
T. UEDA,N. KAMO,N. ISHIDA, AND Y. KOBATAKE
2450 I
I
1.0-
0.86.. *+I
?-
0.6-
\
r:
_____------
0.4-
0.2-
I
I
20
30
I
0-
I
1
I
1
10-2
I 0-1
I
10
10
0
I
40
IIC
C./X
Figure 2. Plots of 71-7/ ( Y + ' ) ~ against log. (c- /x) for three kinds of membranes: 6, PS-1; 0-, PS-2; 0, PS-3 for KC1.
KC1 solution. This fact is slightly different from the "additivity rule" found in polyelectrolyte solutions, where 4 is believed to be constant irrespective of the concentration of added salt.* Precise comparison of the data shown in Figures 1 and 2 reveals that the value of 4 for NaCl and LiCl is slightly differeht from that for KC1 when the concentration is low enough, i.e., C / X
