- Email: [email protected]
Conductivity of ion-exchange membranes—I. Convection conductivity and other components
CONDUCTIVITY OF ION-EXCHANGE CONVECTION CONDUCTIVITY COMPONENTS
MEMBRANESI. AND OTHER
ANNA NARFBSKA and STANISLAWKOTER Institute of Chemistry, Nicolaus Copernicus University, ul. Gagarina 7, 87-100 Torub, Poland (Receiued 14 February
1986; in
revisedform 19 August 1986)
Abstract-The conductivity of ion-exchange membranes has been discussed on the grounds of irreversible thermodynamics of transport. Taking for calculations the conductance coefficients, the membrane conductivity ( K,) has been separated into the conductivity of counterions against water (K;). the convection conductivity (K;) and the conductivity of sorbed electrolyte (K,). The calculations have been presented for the perfluorinated cation-exchange membrane N&on 120 in contact with NaCl and NaOH solutions. It has been found that the convection conductivity is considerable, covering about 50 % of K,.
INTRODUCTION The conductivity and selective transport of ions of a charge opposite to the charge of a polymer matrix are the main properties which decide about the successful use of ion-exchange membranes in any laboratory or industrial devices. The precise theory of a membrane conductivity has not been elaborated yet and it would be difficult if possible at all. The membrane morphology and complexity of transport phenomena are responsible for the state of this problem. The main features of the system may be described as follows. (i) Artificial polymer membranes are physically inhomogeneous. Consequently the local concentration of conducting ions may vary from that close to zero up to 12-13 moldme3. The tortuous route for ions is another consequence of the inhomogeneity of a polymer matrix. (ii) Due to Donnan sorption of an electrolyte from an adjacent solution, a membrane presents binary mixture. The concentration distribution of fixed charges results in the distribution of the concentration of an absorbed electrolyte in a swollen membrane. (iii) Since only counterions are mobile, during their flow they impart momentum to water molecules, thus generating a convective flow of the solution within a membrane. The convective contribution to membrane conductivity is difficult to assess quantitatively. Due to the complex membrane structure and the specifity of transport behaviour, some simplifications in the theory of membrane conductivity are obvious. There are two ways to approach the problem. Some authors model the topology of membranes and get the conductivity equations ascribing the corresponding electrical circuits to the model. Such models have been published by Spiegler[l], Arnold and Koch[2] and others[3, 41. Another possible approach to the membrane conductivity comes from the irreversible thermodynamics of transport. The necessary set of experimental data and phenomenological transport equations make it possible to calculate the transport coefficients describ-
ing quantitatively the flow of each separate component within the membrane. The conductance (I;,), resistance (rik) and friction (fik) coefficients supply the information on the energy needed by any species to overcome the resistance imposed on transport by the membrane interior. In this paper we present the discussion on the membrane conductivity based on the irreversible thermodynamics, mainly on conductance coefficients Zik. Having these coefficients we were able to separate the total membrane conductivity K, to the conductivity of the charged polymer ?$ and its constituents K;+ up” (for explanation see text) and to the conductivity of an absorbed electrolyte K,, . The calculations were done applying the results on ions and water transport across the perfluorinated cation-exchange membrane Nafion 120 (DuPont, U.S.A.) in contact with sodium chloride and sodium hydroxide solutions of concentrations 0.14 M and temperatures 298, 313 and 333 K[5, 61.
THEORY The system considered here is a macroscopically homogeneous ion-exchange membrane immersed in an aqueous solution of a binary electrolyte. The swollen membrane consists of four chemical species: counterions (l), co-ions (2), fixed ionic groups attached to the polymer network (m) and water (w). According to the linear nonequilibrium thermodynamics, in mechanical equilibrium, the fluxes of species i relative to the membrane Ji, and thermodynamic forces Fk, which are negative gradients of electrochemical potentials of k (- VP,), are linearly related[7]:
449
450
ANNA
Here I;,_ rik are the molar conductance coefficients and: Fi=
-Vbi=
NARFBSKAANDSTANI%LAW
and resistance
J1 = [zilil
(2)
valency and partial molar electric potential, pressure F, R are Faraday and gas electrical force is reduced to:
+z21i2]F(-V+)
acts alone
i = 1,2,w.
(3)
The current density I can be written as: I = F (zlJ,
+z2Jz)
where K, is the specific brane [8, 93: K, = F’(Z:&,
Introducing
+
2Z,Z$,Z
+Z;&).
(5)
Ji
Zl
= ciziF(-vti)
lil + diZ
=
zi ci
1,2
(6)
we can express
K,
in the well-known
%I = F’(z:C,
form:
U, +zfc;&).
(7)
Here C, , E2are the concentrations of counterions and co-ions, respectively. They are related to the concentration of fixed charges C, by the following equation: zici
+zsEs
+z,E,
Substituting equation (8) into divide K, into two parts: K, =
= 0.
(7) we can formally (9)
KP+KK,,
where repis the conductivity ing the charged matrix conductivity):
of counterions neutraliz(or the ionic polymer
~~ = F2(-z,z,)c,u~ and K,, is the conductivity
(8)
(10)
of sorbed electrolyte:
KS, = F’(-zzz)E~(z,ti,
-zz2&)
= FZv~z~~,,(z~~ 1 -z2t&). (11) In the above equation C., denotes the concentration of a sorbed electrolyte and vi is the number of moles of ion 1 formed from 1 mole of an electrolyte. The well-known phenomenon is that in an electric field the flowing ions produce the flow of water usually toward the direction of counterions[lO]. As counterions experience less resistance due to the solvent movement in the same direction, they move faster and co-ions move slower than they would if the flow of water were zero. The mobility of water molecules in an electric field, that is the electro-osmotic mobility, can be described with the help of equation (3) as: V,
U”=F(_vG)= Introducing
JW c,F(--VIII)
the mobilities
21 Ll + z2Lz C
1
_
~iw=ziF~_v~~=u~-~uw
,‘T=K'+KW P P
where water:
K;
is the conductivity
of ions against water UT
1,2
P
of counterions
ic; = FZ(-z,z,)c,u;
(13)
(14) against (15)
and K~W is the convection conductivity-the excess of membrane conductivity due to water transport [lo]: K;
=
Fz( -z,)E,,,U,p
(16)
The second component in equation (9), K,,, does not depend on U,. This can be easily verified introducing equation (13) into (11). The convection conductivity up” is equal to the difference between K,,, and the conductivity calculated under the assumption that the water mobility against the membrane is zero. In practice this condition (Jw = 0) can be obtained by imposing an adequate pressure difference. It must be stated, however, that the difference between isobaric conductivity and the conductivity of a membrane under the pressure, for which J, = 0, is not equal to K&Y. The detailed analysis of this fact in terms of the phenomenological conductance coefficients is given in the Appendix. It is the comment to the statement in Wiedner’s and Woermann’s paper
Clll.
The solution presented here is valid for cationexchange membranes. For the anion exchange membranes a factor w = - 1 should be introduced into the equation defining the mobility of water [equation (12)] and, consequently, to all the equation with U,. The factor w accounts for the direction of electroosmotic flow of water.
EXPERIMENTAL The calculations of components of the membrane conductivity were performed applying the phenomenolo&al conductance coefficients I+ The methods and equipments used for the determination of the membrane properties necessary to compute liL, that is membrane conductivity, concentration membrane ootential. electroosmotic. diffusion, osmotic and hydrodynamic flows, were’described in Ref. [12]. The corresnondine results were nublished in Refs f5, 61. In computation of lik we followed the procedure described by Meares et a/.[ 13, 141. For computing In the experimental data were taken first to calculate the differential discontinuous coefficients L,,. These coefficients represent the transport across a membrane in equilibrium with a single electrolyte solution of molality inext. The matrix of molar conductance coefficients In was calculated from the equation: [lik] = d,r-‘Ca,p]T-‘T.
(12)
i=
we can write for K~:
(4)
of the mem-
of species i defined as:
Vi
ui=Zi(F(_V$) i=
= K,( - Vti)
conductivity
the mobility
into equation (10) vi-v,
-((RTVlnai+ziFV$++-iVp)
where ni, rir i& are activity, volume of ion i; J/,p, Tare and absolute temperature; constants, respectively. In the case when the (VP, Vc = 0) equation (la)
KOTER
(17)
Here I is the matrix transforming molar fluxes into practical fluxes and d, is the membrane thickness (for more details concerning the computation of L,, and Iii, see Refs [12-141).
Conductivity of ion-exchange membranes-I
451
The concentration dependence of In at temperatures 298, 313 and 333 K is presented in Figs 1 and 2 for Nafion 120 membrane equilibrated in NaCl and NaOH solutions.
RESULTS
AND
DISCUSSION
All the conductivities constituting IC,, ie of the membrane ionic polymer K~, its constituents the convection conductivity K;, and the conductivity of counterions against water K;, and of sorbed electrolyte rcc,,are seen in Fig. 3. The following conclusions can be drawn. At low sorption the membrane conductivity K, and the conductivity of an ionic polymer K,, (Fig. 3, dotted line) denote the same and have the same numerical values. According to the expectation, since in contact with both electrolytes the chemical composition of the polymer is the same, that is the fixed charges are neutralized by sodium counterions, rcP is the same in NaCl and NaOH at each temperature. The membrane conductivity increases with increased sorption of an electrolyte. The decrease of K~ and the increase of K,, results in the irregular shape of the membrane conductivity K,, not easily understandable before resolution. The conductivities of both electrolytes present within the swollen membrane K~, are below the conductivities of free aqueous solutions of these electrolytes of corresponding concentrations. Both the tortuous routes and the lower mobilities of ions within the electrostatic field of an ionic polymer are responsible for this difference. The mobilities of ions in
Fig. 1. Conductance coefficients Iii for the Nafion 120 membrane us the concentration of NaCI in external solutions at temperature: 298 K (o), 313 K (0) and 333 K (0).
m, Fig. 2. Conductance coefficients le for the Nation 120 membrane vs the concentration of NaOH in external solutions at temperatures: 298 K (o), 313 K (o) and 333 K (e).
Fig. 3. Nation 120 membrane conductivity and its components DSthe concentration of NaCl (left side of figure) and NaOH (right side of figure) in external solutions at temperatures 298, 313 and 333 K.
452
ANNA
NAR$BSKAAND.~ANI%AW
Nafion 120 membrane will be discussed in Part II of this paper. Finally, the most important result of resolution concerns the convective component K~W.The convection conductivity covers 5&55 y0 of the total membrane conductivity and even more at increased temperature. This means that the flowing water doubles the ability of the membrane to transport the ionic current. Once more, the conclusions confirm the substantial role that water plays in the transport behaviour of a membrane. In order to discuss the temperature dependence of component conductivities, the activation energy for the conductivities of ionic polymer (icp) and sorbed electrolyte (Key) have been calculated and presented along with the activation energy for aqueous solutions of both electrolytes:
NaCl NaOH * Calculated from Ref. [15].
E.(kJ mol-‘) (KS.) (%I)
(K.)*
15.1
14.2
16.2 16.5
13.8
for infinite dilution,
12.3
taking the data
The results of E, for K~ and K,, correspond to the molality of an electrolyte solution tnext = 0.5 for which the Donnan sorption is high enough for K,, to be precisely calculated. These results prove that not only the concentration but also the temperature dependence of ICYin both electrolytes are the same within the experimental error. As far as the conductivity of sorbed electrolytes is concerned E, is just above that for free solutions reflecting the resistance exerted by an ionic polymer on the conductivity of electrolyte within a gel phase.
KOTER
11. G. Wiedner and D. Woermann, Ber. Bunsenges. physik. Chem. 79, 868 (1975). 12. A. Narebska, S. Koter and W. Kujawski, Desalination 51,
3 (1984). 13. H. Krlmer and P. Meares, Biophys.
J. 9, 1006 (1969). 14. T. Foley, J. Klinowski and P. Meares, Proc. Royal Sot. Iand. A336, 327 (1974). 15. R. A. Robinson and R. H. Stokes, Electrolyte Solutions, 2nd edn, p. 465. Butterworths, London (1959). 16. F. Helfferich, Ion Exchange, Chap. 7-l. McGraw-Hill, New York (1962). 17. J. W. Lorimer. J. Chem. Sot. Far&w Trans. II 74, 75 (1978). 18. G. Schmid, Z. Elekrrochem., Ber. Bunsenges. physik. Chem. 56, 181 (1952).
APPENDIX Here we present some comments on the definition of the convection conductivity. In this work, the increase of the conductance of a membrane due to water transport, given by the product of fixed charges concentration and the mobility of water [equation (16)], is called the aonvection conductivity. In the monograph by HelfTerich [ 163 the definition of convection conductivity difFers from ours by assuming the mobility of the local center of gravity of the pore liquid in a membrane. For both definitions the formula for the membrane conductivity can be written in a general form:
i= 1.2 where iif = iii - ii./ri, U. is an arbitrary reference mobility and the product Fz( -z,)C,,,U. denotes the convection conductivity. Applying the definition of an arbitrary reference velocity [ 171 u. = &+D,, where wi denotes weight factors whose sum is unity, and the equation for flux Ji = C,vi = (z, lil + z21i2) F(-V@) [equation (3)], we can express the convection conductivity in terms of phenomenological conductance coefficients I, in the following way: Q,,
= F2( - z,,,)~_,ii,, =
REFERENCES 1. M. C. Sauer, P. F. Southwick, K. S. Spiegler and M. R. Wyllie, Ind. Engng Chem. 47, 2187 (1955). 2. R. Arnold and D. F. A. Koch, Aust. J. Chem. 19, 1299 ( 19661 3. k. N&ebska, R. W6diki and S. Koter, Angew. Makromol. Chem. 86, 157 (1980). Chem. 4. R. W6dzki and A. Narebska, Angew. Makromol. 88, 149 (1980). 5. A. Narebska, S. Koter and W. Kujawski. J. Membr. Sci. 25, 153 (1985). 6. A. Narebska, W. Kujawski and S. Koter, J. Membr. Sci., in press. Separation Processes (Edited by 7. P. Meares, in Membrane P. Meares), Chap. 1. Elsevier, Amsterdam (1976). and P. F. Curran, Nonequilibrium 8. A. Katchalsky Thermodynamics in Biophysics, p. 142. Harvard University Press, Cambridge, Massachusetts (1965). 9. R. Paterson and C. R. Gardner, J. Chem. Sot. A, 2254 (1971). 10. N. Lakshminarayanaiah, Transport Phenomena in Membranes, Chap. 5. Academic Press, New York (1969).
=
F2( -z,,,)~,,,(w,~~
ii, + wzz2ii2
+ w,fi,)
(A2)
Equation (AZ) presents a general formula for the convection conductivity. If U. is the mobility of water, then w, = 1, wI = w = 0 and equation (AZ) is reduced to equation (16). If ii. is the mobility of the center of mass, then wi = M&/&M&; M,, molecular weight of a substance i. The convection conductivity calculated by means of this frame of reference is less than 18 % below that presented in this paper [equation (16)] for NaCl solutions and 15% for NaOH solutions. The difference is even smaller, within the range of a few per cent only, if the center of volume is chosen as a frame of reference (Wi = v,z~/z,l&c,). Another definition of the convection conductivity was given by Wiedner and Woermann[ll] while quoting the paper by Schmid[l8]. They defined the convection conductivity as the difference between the conductivity of the membrane under isobaric conditions [K,,,(AP = 0)] and the conductivity under theconditions ofzero volume flow [K,/J, = O)]. With the symbols used in the present paper the conductivity defined by Wiedner and Woermann can be
Conductivity represented K&J
of ion-exchange
as:
= rc,,,(Ap = O)--K,(J,
= 0) (A3)
membranes-I
453
Equation (A3) is not comparable to equation (A2) and it presents another definition of the convection conductivity. In numerical values rccoay [equation (A2)] is more than twice that of K,,,, calculated for the systems discussed here. However, contrary to these authors[lI], we attribute the difference not to the experimental errors but to the more fundamental differences in definitions.
MEMBRANESI. AND OTHER
ANNA NARFBSKA and STANISLAWKOTER Institute of Chemistry, Nicolaus Copernicus University, ul. Gagarina 7, 87-100 Torub, Poland (Receiued 14 February
1986; in
revisedform 19 August 1986)
Abstract-The conductivity of ion-exchange membranes has been discussed on the grounds of irreversible thermodynamics of transport. Taking for calculations the conductance coefficients, the membrane conductivity ( K,) has been separated into the conductivity of counterions against water (K;). the convection conductivity (K;) and the conductivity of sorbed electrolyte (K,). The calculations have been presented for the perfluorinated cation-exchange membrane N&on 120 in contact with NaCl and NaOH solutions. It has been found that the convection conductivity is considerable, covering about 50 % of K,.
INTRODUCTION The conductivity and selective transport of ions of a charge opposite to the charge of a polymer matrix are the main properties which decide about the successful use of ion-exchange membranes in any laboratory or industrial devices. The precise theory of a membrane conductivity has not been elaborated yet and it would be difficult if possible at all. The membrane morphology and complexity of transport phenomena are responsible for the state of this problem. The main features of the system may be described as follows. (i) Artificial polymer membranes are physically inhomogeneous. Consequently the local concentration of conducting ions may vary from that close to zero up to 12-13 moldme3. The tortuous route for ions is another consequence of the inhomogeneity of a polymer matrix. (ii) Due to Donnan sorption of an electrolyte from an adjacent solution, a membrane presents binary mixture. The concentration distribution of fixed charges results in the distribution of the concentration of an absorbed electrolyte in a swollen membrane. (iii) Since only counterions are mobile, during their flow they impart momentum to water molecules, thus generating a convective flow of the solution within a membrane. The convective contribution to membrane conductivity is difficult to assess quantitatively. Due to the complex membrane structure and the specifity of transport behaviour, some simplifications in the theory of membrane conductivity are obvious. There are two ways to approach the problem. Some authors model the topology of membranes and get the conductivity equations ascribing the corresponding electrical circuits to the model. Such models have been published by Spiegler[l], Arnold and Koch[2] and others[3, 41. Another possible approach to the membrane conductivity comes from the irreversible thermodynamics of transport. The necessary set of experimental data and phenomenological transport equations make it possible to calculate the transport coefficients describ-
ing quantitatively the flow of each separate component within the membrane. The conductance (I;,), resistance (rik) and friction (fik) coefficients supply the information on the energy needed by any species to overcome the resistance imposed on transport by the membrane interior. In this paper we present the discussion on the membrane conductivity based on the irreversible thermodynamics, mainly on conductance coefficients Zik. Having these coefficients we were able to separate the total membrane conductivity K, to the conductivity of the charged polymer ?$ and its constituents K;+ up” (for explanation see text) and to the conductivity of an absorbed electrolyte K,, . The calculations were done applying the results on ions and water transport across the perfluorinated cation-exchange membrane Nafion 120 (DuPont, U.S.A.) in contact with sodium chloride and sodium hydroxide solutions of concentrations 0.14 M and temperatures 298, 313 and 333 K[5, 61.
THEORY The system considered here is a macroscopically homogeneous ion-exchange membrane immersed in an aqueous solution of a binary electrolyte. The swollen membrane consists of four chemical species: counterions (l), co-ions (2), fixed ionic groups attached to the polymer network (m) and water (w). According to the linear nonequilibrium thermodynamics, in mechanical equilibrium, the fluxes of species i relative to the membrane Ji, and thermodynamic forces Fk, which are negative gradients of electrochemical potentials of k (- VP,), are linearly related[7]:
449
450
ANNA
Here I;,_ rik are the molar conductance coefficients and: Fi=
-Vbi=
NARFBSKAANDSTANI%LAW
and resistance
J1 = [zilil
(2)
valency and partial molar electric potential, pressure F, R are Faraday and gas electrical force is reduced to:
+z21i2]F(-V+)
acts alone
i = 1,2,w.
(3)
The current density I can be written as: I = F (zlJ,
+z2Jz)
where K, is the specific brane [8, 93: K, = F’(Z:&,
Introducing
+
2Z,Z$,Z
+Z;&).
(5)
Ji
Zl
= ciziF(-vti)
lil + diZ
=
zi ci
1,2
(6)
we can express
K,
in the well-known
%I = F’(z:C,
form:
U, +zfc;&).
(7)
Here C, , E2are the concentrations of counterions and co-ions, respectively. They are related to the concentration of fixed charges C, by the following equation: zici
+zsEs
+z,E,
Substituting equation (8) into divide K, into two parts: K, =
= 0.
(7) we can formally (9)
KP+KK,,
where repis the conductivity ing the charged matrix conductivity):
of counterions neutraliz(or the ionic polymer
~~ = F2(-z,z,)c,u~ and K,, is the conductivity
(8)
(10)
of sorbed electrolyte:
KS, = F’(-zzz)E~(z,ti,
-zz2&)
= FZv~z~~,,(z~~ 1 -z2t&). (11) In the above equation C., denotes the concentration of a sorbed electrolyte and vi is the number of moles of ion 1 formed from 1 mole of an electrolyte. The well-known phenomenon is that in an electric field the flowing ions produce the flow of water usually toward the direction of counterions[lO]. As counterions experience less resistance due to the solvent movement in the same direction, they move faster and co-ions move slower than they would if the flow of water were zero. The mobility of water molecules in an electric field, that is the electro-osmotic mobility, can be described with the help of equation (3) as: V,
U”=F(_vG)= Introducing
JW c,F(--VIII)
the mobilities
21 Ll + z2Lz C
1
_
~iw=ziF~_v~~=u~-~uw
,‘T=K'+KW P P
where water:
K;
is the conductivity
of ions against water UT
1,2
P
of counterions
ic; = FZ(-z,z,)c,u;
(13)
(14) against (15)
and K~W is the convection conductivity-the excess of membrane conductivity due to water transport [lo]: K;
=
Fz( -z,)E,,,U,p
(16)
The second component in equation (9), K,,, does not depend on U,. This can be easily verified introducing equation (13) into (11). The convection conductivity up” is equal to the difference between K,,, and the conductivity calculated under the assumption that the water mobility against the membrane is zero. In practice this condition (Jw = 0) can be obtained by imposing an adequate pressure difference. It must be stated, however, that the difference between isobaric conductivity and the conductivity of a membrane under the pressure, for which J, = 0, is not equal to K&Y. The detailed analysis of this fact in terms of the phenomenological conductance coefficients is given in the Appendix. It is the comment to the statement in Wiedner’s and Woermann’s paper
Clll.
The solution presented here is valid for cationexchange membranes. For the anion exchange membranes a factor w = - 1 should be introduced into the equation defining the mobility of water [equation (12)] and, consequently, to all the equation with U,. The factor w accounts for the direction of electroosmotic flow of water.
EXPERIMENTAL The calculations of components of the membrane conductivity were performed applying the phenomenolo&al conductance coefficients I+ The methods and equipments used for the determination of the membrane properties necessary to compute liL, that is membrane conductivity, concentration membrane ootential. electroosmotic. diffusion, osmotic and hydrodynamic flows, were’described in Ref. [12]. The corresnondine results were nublished in Refs f5, 61. In computation of lik we followed the procedure described by Meares et a/.[ 13, 141. For computing In the experimental data were taken first to calculate the differential discontinuous coefficients L,,. These coefficients represent the transport across a membrane in equilibrium with a single electrolyte solution of molality inext. The matrix of molar conductance coefficients In was calculated from the equation: [lik] = d,r-‘Ca,p]T-‘T.
(12)
i=
we can write for K~:
(4)
of the mem-
of species i defined as:
Vi
ui=Zi(F(_V$) i=
= K,( - Vti)
conductivity
the mobility
into equation (10) vi-v,
-((RTVlnai+ziFV$++-iVp)
where ni, rir i& are activity, volume of ion i; J/,p, Tare and absolute temperature; constants, respectively. In the case when the (VP, Vc = 0) equation (la)
KOTER
(17)
Here I is the matrix transforming molar fluxes into practical fluxes and d, is the membrane thickness (for more details concerning the computation of L,, and Iii, see Refs [12-141).
Conductivity of ion-exchange membranes-I
451
The concentration dependence of In at temperatures 298, 313 and 333 K is presented in Figs 1 and 2 for Nafion 120 membrane equilibrated in NaCl and NaOH solutions.
RESULTS
AND
DISCUSSION
All the conductivities constituting IC,, ie of the membrane ionic polymer K~, its constituents the convection conductivity K;, and the conductivity of counterions against water K;, and of sorbed electrolyte rcc,,are seen in Fig. 3. The following conclusions can be drawn. At low sorption the membrane conductivity K, and the conductivity of an ionic polymer K,, (Fig. 3, dotted line) denote the same and have the same numerical values. According to the expectation, since in contact with both electrolytes the chemical composition of the polymer is the same, that is the fixed charges are neutralized by sodium counterions, rcP is the same in NaCl and NaOH at each temperature. The membrane conductivity increases with increased sorption of an electrolyte. The decrease of K~ and the increase of K,, results in the irregular shape of the membrane conductivity K,, not easily understandable before resolution. The conductivities of both electrolytes present within the swollen membrane K~, are below the conductivities of free aqueous solutions of these electrolytes of corresponding concentrations. Both the tortuous routes and the lower mobilities of ions within the electrostatic field of an ionic polymer are responsible for this difference. The mobilities of ions in
Fig. 1. Conductance coefficients Iii for the Nafion 120 membrane us the concentration of NaCI in external solutions at temperature: 298 K (o), 313 K (0) and 333 K (0).
m, Fig. 2. Conductance coefficients le for the Nation 120 membrane vs the concentration of NaOH in external solutions at temperatures: 298 K (o), 313 K (o) and 333 K (e).
Fig. 3. Nation 120 membrane conductivity and its components DSthe concentration of NaCl (left side of figure) and NaOH (right side of figure) in external solutions at temperatures 298, 313 and 333 K.
452
ANNA
NAR$BSKAAND.~ANI%AW
Nafion 120 membrane will be discussed in Part II of this paper. Finally, the most important result of resolution concerns the convective component K~W.The convection conductivity covers 5&55 y0 of the total membrane conductivity and even more at increased temperature. This means that the flowing water doubles the ability of the membrane to transport the ionic current. Once more, the conclusions confirm the substantial role that water plays in the transport behaviour of a membrane. In order to discuss the temperature dependence of component conductivities, the activation energy for the conductivities of ionic polymer (icp) and sorbed electrolyte (Key) have been calculated and presented along with the activation energy for aqueous solutions of both electrolytes:
NaCl NaOH * Calculated from Ref. [15].
E.(kJ mol-‘) (KS.) (%I)
(K.)*
15.1
14.2
16.2 16.5
13.8
for infinite dilution,
12.3
taking the data
The results of E, for K~ and K,, correspond to the molality of an electrolyte solution tnext = 0.5 for which the Donnan sorption is high enough for K,, to be precisely calculated. These results prove that not only the concentration but also the temperature dependence of ICYin both electrolytes are the same within the experimental error. As far as the conductivity of sorbed electrolytes is concerned E, is just above that for free solutions reflecting the resistance exerted by an ionic polymer on the conductivity of electrolyte within a gel phase.
KOTER
11. G. Wiedner and D. Woermann, Ber. Bunsenges. physik. Chem. 79, 868 (1975). 12. A. Narebska, S. Koter and W. Kujawski, Desalination 51,
3 (1984). 13. H. Krlmer and P. Meares, Biophys.
J. 9, 1006 (1969). 14. T. Foley, J. Klinowski and P. Meares, Proc. Royal Sot. Iand. A336, 327 (1974). 15. R. A. Robinson and R. H. Stokes, Electrolyte Solutions, 2nd edn, p. 465. Butterworths, London (1959). 16. F. Helfferich, Ion Exchange, Chap. 7-l. McGraw-Hill, New York (1962). 17. J. W. Lorimer. J. Chem. Sot. Far&w Trans. II 74, 75 (1978). 18. G. Schmid, Z. Elekrrochem., Ber. Bunsenges. physik. Chem. 56, 181 (1952).
APPENDIX Here we present some comments on the definition of the convection conductivity. In this work, the increase of the conductance of a membrane due to water transport, given by the product of fixed charges concentration and the mobility of water [equation (16)], is called the aonvection conductivity. In the monograph by HelfTerich [ 163 the definition of convection conductivity difFers from ours by assuming the mobility of the local center of gravity of the pore liquid in a membrane. For both definitions the formula for the membrane conductivity can be written in a general form:
i= 1.2 where iif = iii - ii./ri, U. is an arbitrary reference mobility and the product Fz( -z,)C,,,U. denotes the convection conductivity. Applying the definition of an arbitrary reference velocity [ 171 u. = &+D,, where wi denotes weight factors whose sum is unity, and the equation for flux Ji = C,vi = (z, lil + z21i2) F(-V@) [equation (3)], we can express the convection conductivity in terms of phenomenological conductance coefficients I, in the following way: Q,,
= F2( - z,,,)~_,ii,, =
REFERENCES 1. M. C. Sauer, P. F. Southwick, K. S. Spiegler and M. R. Wyllie, Ind. Engng Chem. 47, 2187 (1955). 2. R. Arnold and D. F. A. Koch, Aust. J. Chem. 19, 1299 ( 19661 3. k. N&ebska, R. W6diki and S. Koter, Angew. Makromol. Chem. 86, 157 (1980). Chem. 4. R. W6dzki and A. Narebska, Angew. Makromol. 88, 149 (1980). 5. A. Narebska, S. Koter and W. Kujawski. J. Membr. Sci. 25, 153 (1985). 6. A. Narebska, W. Kujawski and S. Koter, J. Membr. Sci., in press. Separation Processes (Edited by 7. P. Meares, in Membrane P. Meares), Chap. 1. Elsevier, Amsterdam (1976). and P. F. Curran, Nonequilibrium 8. A. Katchalsky Thermodynamics in Biophysics, p. 142. Harvard University Press, Cambridge, Massachusetts (1965). 9. R. Paterson and C. R. Gardner, J. Chem. Sot. A, 2254 (1971). 10. N. Lakshminarayanaiah, Transport Phenomena in Membranes, Chap. 5. Academic Press, New York (1969).
=
F2( -z,,,)~,,,(w,~~
ii, + wzz2ii2
+ w,fi,)
(A2)
Equation (AZ) presents a general formula for the convection conductivity. If U. is the mobility of water, then w, = 1, wI = w = 0 and equation (AZ) is reduced to equation (16). If ii. is the mobility of the center of mass, then wi = M&/&M&; M,, molecular weight of a substance i. The convection conductivity calculated by means of this frame of reference is less than 18 % below that presented in this paper [equation (16)] for NaCl solutions and 15% for NaOH solutions. The difference is even smaller, within the range of a few per cent only, if the center of volume is chosen as a frame of reference (Wi = v,z~/z,l&c,). Another definition of the convection conductivity was given by Wiedner and Woermann[ll] while quoting the paper by Schmid[l8]. They defined the convection conductivity as the difference between the conductivity of the membrane under isobaric conditions [K,,,(AP = 0)] and the conductivity under theconditions ofzero volume flow [K,/J, = O)]. With the symbols used in the present paper the conductivity defined by Wiedner and Woermann can be
Conductivity represented K&J
of ion-exchange
as:
= rc,,,(Ap = O)--K,(J,
= 0) (A3)
membranes-I
453
Equation (A3) is not comparable to equation (A2) and it presents another definition of the convection conductivity. In numerical values rccoay [equation (A2)] is more than twice that of K,,,, calculated for the systems discussed here. However, contrary to these authors[lI], we attribute the difference not to the experimental errors but to the more fundamental differences in definitions.