J. Phys. Chem. B 1999, 103, 173-177
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Membrane Potential across a High Water Content Anion-Exchange Membrane Separating Two Solutions with a Common Counterion but Two Different Co-ions Masayasu Tasaka,* Ryotaro Kiyono, and Dong-Suk Yoo† Department of Material Science and Engineering, Graduate School of Science and Technology, Shinshu UniVersity, Wakasato, Nagano 380-8553, Japan ReceiVed: August 6, 1998; In Final Form: NoVember 2, 1998
A theory for bi-ionic potential, especially for bi-co-ionic potential, which is a potential difference across a membrane due to the difference in co-ions, is proposed. A high water content interpolymer membrane of poly(ethyleneimine) and poly(vinyl alcohol) was prepared as a low-charged anion-exchange membrane. The potential difference ∆ψ for the system of KCl/membrane/1/2MgCl2 or 1/2CaCl2 was measured by changing the external electrolyte concentrations cs and analyzed according to the theory. The bi-co-ionic potential ∆ψ increased with increasing cs at low concentrations, and after the potential ∆ψ reached a maximum value, the potential ∆ψ decreased with increasing cs at high concentrations; that is, a bell-shaped dependence of ∆ψ was observed against cs. This is because the low mobility of co-ions with charge number +2 in the charged membranes increases with increasing cs and finally reaches the mobility in the free electrolyte solutions.
Introduction A potential difference is observed across a membrane if the membrane separates two electrolyte solutions with a common co-ion but two different counterions at the same ionic strength. The membrane potential is called the bi-ionic potential. This potential is of interest in connection with the analysis of the selectivity of counterions in the membrane, and many works on the bi-ionic potential due to counterions have been carried out.1-5 It may be suitable to call it bi-counterionic potential. On the other hand, a potential difference due to the difference in co-ions may also be observed across a low-charged membrane having high water content if the membrane has a low transport number of counterions and separates two electrolyte solutions with a common counterion but two different co-ions at the same ionic strength. It may be called bi-co-ionic potential. The selectivity of co-ions in the low-charged membranes is of interest in connection with the separation between co-ions having different charge numbers, especially for a low-charged surface layer on cation-exchange membranes in electrodialysis of sea water. The selectivity between the cations having different charge numbers has been improved by sorption of polycations on the surface of cation-exchange membranes.6,7 In this study, a theory for bi-co-ionic potential is developed. Low-charged anion-exchange membranes having high water content were prepared, and the potential difference in the system of KCl/ membrane/1/2MgCl2 or 1/2CaCl2 was measured and analyzed according to the theory. Theory We shall now consider a membrane system constructed with an anion-exchange membrane separating two electrolyte solutions with a common counterion but two different co-ions [KCl * To whom correspondence should be addressed. Phone: +81-26-2264101. Fax: +81-26-223-9249. E-mail:
[email protected]. † Visiting Researcher from Department of Chemical Engineering, College of Engineering, Kangwon National University, Chunchon 200-701, South Korea.
Figure 1. Schematic diagram of the concentration profile of two coions in the anion-exchange membrane for the system KCl/membr/ 1/ MgCl . 2 2
and 1/2MgCl2 or 1/2CaCl2]. The subscripts 1, 2, 3, and 0 represent two co-ions K+ and 1/2Mg2+ or 1/2Ca2+, counterion Cl-, and water species, respectively. In the steady state, we may assume that
j˜ 3(0) µ˜ 3(0) ) µ
(1)
µ˜ 3(0) ) µ03 + RT ln a3(0) + z3Fψ(0)
(2)
j˜ 3(0) ) µ µ j 03 + RT ln aj3(0) + z3Fψ h (0)
(3)
where µ˜ i is the electrochemical potential of species i and the overbar indicates the membrane phase. R is the gas constant, F the Faraday constant, T the absolute temperature, ai the activity, zi the charge number, µ0i the chemical potential at the standard state, and ψ the electric potential; (0) and (δ) refer to the two surfaces of the membrane stretching from x ) 0 to x ) δ (Figure 1). The potential difference at the two side surfaces of the membrane is given by
∆ψ(0) ≡ ψ h (0) - ψ(0) ) -
RT aj3(0) 1 0 (µ j 3 - µ03) ln (4) z3F z3F a3(0)
10.1021/jp9833092 CCC: $18.00 © 1999 American Chemical Society Published on Web 12/10/1998
174 J. Phys. Chem. B, Vol. 103, No. 1, 1999
∆ψ(δ) ≡ ψ(δ) - ψ(δ) )
Tasaka et al.
1 0 RT aj3(δ) (5) (µ j 3 - µ03) + ln z3F z3F a3(δ)
RT aj3(δ)a3(0) ∆ψ(0,δ) ≡ ∆ψ(0) + ∆ψ(δ) ) ln z3F a3(δ)aj3(0)
(6)
Substituting eqs 13 to 17 into eq 18, we obtain
[
()
grad cj1(x) ) -
On the other hand, the mass flux of species i is written as
-Ji )
∑jLij grad µj˜ j
(7)
∆ψ h )-
j˜ 1 + L12 grad µ j˜ 2 + L13 grad µ j˜ 3 -J1 ) L11 grad µ
][
]
l2cj2(δ) + l3cj3(δ) RT l1 - Rl2 - (1 - R)l3 ln F l1 - Rl2 + (1 - R)l3 l1cj1(0) + l3cj3(0)
Therefore, from eqs 6 and 20 the total membrane potential across the membrane becomes
≈ (l11 grad aj1 + l12 grad aj2 + l13 grad aj3)RT + h (8) (l11cj1z1 + l12cj2z2 + l13cj3z3)F grad ψ -J2 ≈ (l21 grad aj1 + l22 grad aj2 + l23 grad aj3)RT + (l21cj1z1 + l22cj2z2 + l23cj3z3)F grad ψ h (9) -J3 ≈ (l31 grad aj1 + l32 grad aj2 + l33 grad aj3)RT + h (10) (l31cj1z1 + l32cj2z2 + l33cj3z3)F grad ψ Applying the condition of no electric current, I ) ∑iziFJi ) 0, for 1-1 and 2-1 electrolytes across anion-exchange membranes (z1 ) z2/2 ) 1 ) -z3) we obtain
l1 grad aj1 + l2 grad aj2 - l3 grad aj3 ) h (11) -(F/RT)(l1c1 + l2c2 + l3c3) grad ψ where
l1 ) l11 + 2l21 - l31 l2 ) l12 + 2l22 - l32 (12)
Figure 1 shows the schematic diagram of ionic concentration profile in the membrane, where we assumed that eqs 13 and 14 hold. In general, a parameter R will be in a range between 0 and 1, and R becomes unity at the limit of very porous membranes that have no selectivity for two co-ions on the both sides of the membranes.
cj2(δ) ) Rcj1(0)
(13)
cj2(x) ) R{cj1(0) - cj1(x)}
(14)
∆ψ ) ∆ψ(0,δ) + ∆ψ h )
) cj1(x) + R{cj1(0) - cj1(x)} + φX (15)
dcj2(x) ) -Rdcj1(x)
(16)
dcj3(x) ) (1 - R)dcj1(x)
(17)
If we assume grad aji ) grad cji, from eq 11 we obtain
l1 grad cj1 + l2 grad cj2 - l3 grad cj3 ) h (18) -(F/RT)(l1cj1 + l2cj2 + l3cj3) grad ψ
( )[
] ]
RT l1 - Rl2 - (1 - R)l3 RT aj3(δ)a3(0) ln z3F a3(δ)aj3(0) F l1 - Rl2 + (1 - R)l3 l2cj2(δ) + l3cj3(δ) ln (21) l1cj1(0) + l3cj3(0)
[
If the activities of co-ions in the solutions on the both sides of the membrane are the same, a3(δ) ) a3(0), the first term of eq 21 becomes zero. In addition, if the two concentrations of co-ions on both sides of the membrane are the same at least, c3(δ) ) c3(0) even if a3(δ) * a3(0), the relationship aj3(0)/a3(0) ≈ aj3(δ)/a3(δ) holds and the first term of eq 21 becomes negligible. The value of [{l1 - Rl2 - (1 - R)l3}/{l1 - Rl2 + (1 - R)l3}] is larger than zero and smaller than unity if 0 < R < 1. Therefore, if there is no specific rejection for the ions with zi ) +2 on the surface of the membrane having high water content and R ) 1, we obtain
∆ψ ) ∆ψ h
()[
)-
]
l2cj2(δ) + l3cj3(δ) RT ln F l1cj1(0) + l3cj3(0)
(22)
Equation 22 corresponds to Henderson’s equation of the liquid junction potential.8 If cations with a large charge number are completely rejected by the fixed charges due to the electric repulsion and/or due to the steric hindrance, and the value of R tends to zero, we obtain
( )[ ] [
∆ψ h )-
cj3(x) ) cj1(x) + cj2(x) + φX ) (1 - R)cj1(x) + Rcj1(0) + φX
F grad ψ h (19) RT
(20)
j˜ 1 + l12cj2 grad µ j˜ 2 + l13cj3 grad µ j˜ 3 ) l11cj1 grad µ
l3 ) -l13 - 2l23 + l33
×
Integrating eq 19 from one side of the membrane (0) to the other (δ), we obtain
( )[
If we assume grad µ j 0 ) 0 in the membrane, we obtain
]
l1 - Rl2 - (1 - R)l3
(l2 + l3)Rcj1(0) + l3φX + {l1 - Rl2 + (1 - R)l3}cj1(x)
]
l3cj3(δ) RT l1 - l3 ln F l1 + l3 l1cj1(0) + l3cj3(0)
(23)
The value of ∆ψ h is nearly equal to zero because of l1 ≈ l3 for KCl/membrane/chloride compound solution systems. This conclusion at the limiting case seems to be reasonable because coion with zi ) +2 cannot directly affect on the membrane phenomena. The ratio of mobilities of co-ions in charged membranes may be different with that in the free electrolyte solutions. Then, we may assume that l1/l3 ) l01/l03 and l2/l3 ) θ(l02/l03) where l0i is the mobility in the free electrolyte solutions and θ is a parameter expressing the effect of the difference in circumstances on the mobility in the membrane phase. Equation 22 reduces to the following:
Membrane Potential across an Anion-Exchange Membrane
Figure 2. Theoretical curves of the bi-co-ionic potential calculated by eq 24 against the external reduced concentration cs/(φX) for the system KCl/anion-exchange membrane/1/2MgCl2. Curve a is calculated with θ ) 1.0, curve b with θ ) 0.5-1.0, and curve c with θ ) 0.5. The mobility ratio is l01:l02:l03 ) 0.96:0.69:1.00.
()[
]
θ(l02/l03)cj2(δ) + cj3(δ) RT ∆ψ ) ln 0 0 F (l 1/l 3)cj1(0) + cj3(0)
(24)
The value of θ will be small when the transport number of counterions is unity at low external electrolyte solutions and will tend to unity at high concentrations. Figure 2 shows the dependence of the bi-co-ionic potential calculated by eq 24 on the reduced concentrations cs/(φX) for the system KCl/membr/1/2MgCl2 across an anion-exchange membrane. Then, the θ values changed from 0.5 to 1 with the reduced concentration. The values of cj1, cj2, and cj3 were estimated from the following assumption
J. Phys. Chem. B, Vol. 103, No. 1, 1999 175
Figure 3. Dependence of transport number of anions on the lower electrolyte molality of KCl, 1/2CaCl2, and 1/2MgCl2. Electrolytes: O, KCl (measured at first); 4, KCl (measured after the measurements for 1/ CaCl and 1/ MgCl ); 0, 1/ CaCl ; ], 1/ MgCl . 2 2 2 2 2 2 2 2
Electrolytes. The solutions were prepared from the specialgrade reagent of Wako Pure Chem. Ind. Ltd. without any purification, and the concentration was determined by Mohr’s method. Measurements of Membrane Potential. The membrane potential cell was constructed from two sections made of poly(methyl methacrylate) resin, and a calomel electrode was connected to each section. The membrane, whose effective area was 0.78 cm2, was mounted between the half-cells, and KCl and CaCl2 or MgCl2 were placed at the two sides of the membrane. The solution inlet and outlet were inserted in each half of the cell to disturb a diffusion layer on the membrane surface. The membrane potential was measured with a PD 325-1 potentiometer (Shimadzu Seisakusho Ltd.) at 25.0 ( 0.5 °C.
cj1(cj1 + φX) ) cs2
(25)
Results and Discussion
cj1 + φX ) cj2 + φX ) cj3
(26)
l01:l02:l03 ) 0.96:0.69:1.00
(27)
The transport numbers of counterions were estimated from the concentration membrane potential with molality ratio 2. The concentration membrane potentials for KCl were measured twice before and after the measurements for MgCl2 and CaCl2. There is no essential difference between two measurements of membrane potentials in the experimental errors. Therefore, the exchange of counterions in the membrane may be taken as complete. Figure 3 shows the dependence of the transport number of counterions in the membrane AM-9/1 on the lower electrolyte concentration for KCl, 1/2MgCl2, and 1/2CaCl2, where the molality ratio was kept at 2. The transport number of counterions decreased largely between 0.01 and 0.1 mol kg-1 of electrolytes. For KCl solutions, the effective concentration of fixed charges can be approximately estimated by the following equations:5
where the mobility ratio in the free electrolyte solutions was estimated from the limiting ionic mobility in aqueous solution at 298.2 K.9 Figure 2 shows the typical bell-shaped dependence of bi-co-ionic potential against the external electrolyte concentration. Experimental Section Membrane. A high water content interpolymer membrane of poly(ethyleneimine) (PEI) and poly(vinyl alcohol) (PVA) was used as a low-charged anion-exchange membrane. The nine parts of PVA with a degree of polymerization of 2000 was dissolved in 70 vol % dimethyl sulfoxide at 130 °C and 1.27 × 105 Pa, and then the one part of PEI was added to the solution. After the mixture was well mixed for 1 min at 150 °C, it was cast on the glass plate and maintained at 130 °C and 1.27 × 105 Pa for 15 min and at room temperature and atmospheric pressure for 1 day. Moreover, the formed membrane on the plate was dried at 50 °C for 1 h and 100 °C for 8 h and then dipped into the distilled water for 3 days in order to detach the membrane from the glass plate. The thickness of the membrane was 0.27 mm, and the water content was 1.02 g H2O/g dry membrane.
φX )
2 av cs (tCl ) 3/4) x3
(28)
cjK(cjK + φX) ) c2s
(29)
cjCl ) cjK + φX
(30)
3 Here, cav s (tCl ) /4) is the average value of the external salt concentration where the value of the transport number of
176 J. Phys. Chem. B, Vol. 103, No. 1, 1999
Tasaka et al.
Figure 4. Dependence of bi-co-ionic potential across the PEI and PVA membrane for the system KCl//1/2MgCl2. The solid line (a) is calculated with φX ) 0.073 mol dm-3 and θ ) 0.1-1.0 and the broken line (b) with φX ) 0.022 mol dm-3 and θ ) 0.1. The solid dot represents experimental data.
Figure 5. Dependence of bi-co-ionic potential across the PEI and PVA membrane for the system KCl//1/2CaCl2. The solid line (a) is calculated with φX ) 0.073 mol dm-3 and θ ) 0.1-1.0 and the broken line (b) with φX ) 0.022 mol dm-3 and θ ) 0.1. The solid dot represents experimental data.
counterions is 3/4. The estimated value of φX was 0.073 mol dm-3. Strictly speaking, the value of φX is dependent on the counterions and the concentration of the external electrolyte solution. The value of φX decreases with decreasing the concentration of the external electrolyte solutions. Therefore, in the following analysis of the experimental data the concentration of ions in the membrane was approximately estimated using the values of φX ) 0.073 mol dm-3. In addition, the analysis was carried out using the lower value of φX ) 0.022 mol dm-3. Figure 4 shows the dependence of the membrane potential on the concentration of the external solutions for KCl/membr/ 1/ MgCl system. Here, the theoretical solid lines (a) are 2 2 calculated by eq 24 with l01:l02:l03 ) 0.96:0.69:1.00,9 φX ) 0.073 mol dm-3, and varying the θ value from 0.1 to 1.0 as shown in Figure 4. Considering the variation of θ as shown in the upper part of Figure 4, the decrease in bi-co-ionic potential at high salt concentrations can be explained by eq 24 with fairly good agreement. When the external salt concentration cs is lower than 0.1 mol dm-3, the value of θ is 0.2 or less. This means that the mobility of 1/2Mg2+ in the membrane is 0.2 times as low as in the free electrolyte solutions. Therefore, the high mobility ratio of co-ions K+ and 1/2Mg2+ in the membrane would play an important role on the selectivity between co-ions K+ and 1/ Mg2+ at the lower concentrations of MgCl than 0.1 mol 2 2 dm-3. The θ values increased clearly with increasing the external electrolyte concentrations after the external electrolyte concentration became larger than the φX value. This will be because the electrostatic interaction between fixed charges and co-ions decreases with increasing the electrolyte concentrations. The state of co-ions in the membrane phase becomes similar to that in the external solution at high electrolyte concentrations and the value of θ tends to unity. The bell-shaped dependence of ∆ψ having maximum at the middle electrolyte concentration cannot be expected using the change in R. Moreover, the bell-shaped dependence of ∆ψ can be numerically expected if the change in φX is fairly high. However, the large change in φX is not real in the small range of the external electrolyte concentration. However, the discrepancy between the experimental data and the theoretical line (a) at lower concentrations cannot be
explained even if the value of θ varied. The broken line (b) is the line calculated by eq 24 with θ ) 0.1 and φX ) 0.022 mol dm-3. The discrepancy from the line (a) at low concentrations might be caused by the decrease of the effective fixed charges. Similar characteristic feature was observed for KCl/membr/ 1/ CaCl system as shown in Figure 5. In a strict sense, the value 2 2 of φX is smaller at dilute electrolyte concentrations and becomes larger at high electrolyte concentrations.10 Therefore, the estimated theoretical curves with φX ) 0.073 mol dm-3 must slide toward the left side at the dilute electrolyte concentrations. The concentration of co-ions in the membrane becomes higher than that estimated in this work when the lower value of φX is adopted. In these cases, the potential difference due to the difference between the two co-ions will become larger. The curve (b) was also calculated by eq 24 with θ ) 0.1, and φX ) 0.022 mol dm-3. If we can consider the concentration dependence of φX, the deviations of the experimental data from the theoretical line (a) become smaller in Figures 4 and 5. Conclusions 1. A bell-shaped dependence of bi-co-ionic potential on the external salt concentration was experimentally observed. 2. A theory of the salt concentration dependence of bi-coionic potential ∆ψ is presented. The typical bell-shaped dependence of ∆ψ can be explained by the change in the mobility ratio of co-ions by the theory. 3. It becomes clear that the extreme decrease of the mobility of co-ions with zi ) +2 in the membrane serves the high selectivity between the co-ions. List of Symbols The overbar refers to the membrane phase, and the subscripts 1, 2, 3, and 4 refer to co-ions K+ and 1/2Mg2+ or 1/2Ca2+, counterion Cl-, and water species, respectively. ai ci F
activity of component i concentration of component i (mol cm-3) Faraday constant
Membrane Potential across an Anion-Exchange Membrane Ji li l0i lij R T X zi R ∆ θ µi µ˜ i
mass flux of component i in the membrane phase (mol cm-2 s-1) coefficient defined by eq 12 mobility in the free electrolyte solution coefficient defined by eq 8 gas constant absolute temperature (K) concentration of fixed charges (mol cm-3) charge number of component i parameter defined by eq 13 difference across the membrane parameter expressing the nonideality for mobility in the membrane phase chemical potential of component i (J mol-1) total chemical potential of component i (J mol-1)
J. Phys. Chem. B, Vol. 103, No. 1, 1999 177 φX ψ
effective concentration of fixed charges (mol cm-3) electric potential (V)
References and Notes (1) Gregor, H. P.; Sollner, K. J. Phys. Chem. 1946, 50, 53. (2) Dray, S.; Sollner, K. Biochim. Biophys. Acta 1955, 18, 341. (3) Marshall, C. E. J. Phys. Chem. 1948, 52, 1284. (4) Toyoshima, Y.; Nozaki, H. J. Phys. Chem. 1970, 74, 2704. (5) Tasaka, M.; Iwaoka, S.; Yamagishi, K.; Ikeda, Y. J. Membrane Sci. 1985, 24, 29. (6) Sata, T.; Izuo, R. J. Membrane Sci. 1989, 45, 209. (7) Sata, T.; Yamaguchi, Y.; Matsusaki, K. J. Membrane Sci. 1995, 100, 197. (8) Henderson, P. Z. Phys. Chem. 1907, 59, 118. (9) Parsons, R. Handbook of Electrochemical Constants; Butterworth Sci. Publ.: London, 1959. (10) Tasaka, M.; Aoki, N.; Kondo, Y.; Nagasawa, M. J. Phys. Chem. 1975, 79, 1307.
