J. Chem. SOC., Faraday Trans. 1, 1984, 80, 3379-3390 Transport of Acetate and Chloroacetate Weak Electrolytes through a Thin Porous Membrane in Counter-current Electrolysis BY KYOSTI KONTTURI,* TUOMO OJALA AND RRKKO FORSSELL Department of Chemistry, Helsinki University of Technology, SF-02150 ES 15, Finland Received 19th March, 1984 The transport of weak electrolytes through a porous membrane during counter-current electrolysis has been studied both theoretically and experimentally. The theoretical model is based on the Nernst-Planck equations and the obtained transport equations for the ion constituents are used to solve the fluxes of the ion constituents. Expressions for the diffusion coefficients and transport numbers are also presented. Experiments and calculations for two binary systems (acetic acid in water and monochloroacetic acid in water) and for three ternary systems (acetic acid and sodium acetate in water, monochloroacetic acid and sodium mono- chloroacetate in water and acetic acid and monochloroacetic acid in water) were performed at a total concentration of 0.01 mol dm-3 or less.The theoretically obtained fluxes of the ion constituents are compared with the experimental fluxes and it is shown that the theoretical model is able to predict the transport phenomena. It has been shown that counter-current electrolysis in a thin porous membrane can be used to separate ions of different In these studies only strong electrolytes in water were considered. It has been verified both theoretically and experimentally that the ionic equivalent conductance determines the order in which the ions can be separated and that the separation efficiency is exponentially dependent on solvent flow through the porous membrane (convection).Furthermore, the small contribution of electric migration to separation in the case of strong electrolytes has been deduced. For weak electrolytes the case is different. It is obvious that electric migration has a significant effect on the transport phenomena occurring in a mixture of several weak electrolytes of different dissociation constants. This becomes clear when it is realized that the transport number depends directly on the amount of ionic species present. In the present work three different kinds of weak electrolyte system were studied. The first system [a weak acid (acetic acid or monochloroacetic acid) in water] was studied to test the feasibility of the Nernst-Planck equations derived from a kinetic The second system [a weak acid and its sodium salt (acetic acid and sodium acetate or monochloroacetic acid and sodium monochloroacetate) in water] was studied because it forms an interesting buffer solution, i.e.a system in which there can be different pH values on either side of the porous membrane. This arrangement may have useful applications in the separation of zwitterions, which have different isoelectric points. The third system [two weak acids (acetic acid and monochloroacetic acid) in water] was studied in order to verify that by using counter-current electrolysis in a thin porous membrane weak acids can be separated efficiently when their dissociation constants differ from each other.33793380 COUNTER-CURRENT ELECTROLYSIS IN A POROUS MEMBRANE THEORY TRANSPORT EQUATIONS The transport processes of weak electrolytes can be described by equations obtained by a macroscopic approach such as the thermodynamics of irreversible processes.6-8 These transport equations are analogous to those used to describe the transport of strong electrolytes except that the concept of the ion must be replaced by the conept of the ion constituent, as defined by Noyes and Falk.g However, these transport equations, in spite of their general nature and applicability, are not as useful in practice since the dependences of diffusion coefficients and transport numbers on concentrations are rarely known.1° Because the Nernst-Planck equations for dilute solutions describe the transport phenomena through a thin porous membrane in counter-current electrolysis fairly accurately for the case of strong electrolytes,ll the same treatment is also used for systems of weak electrolytes.For every ionic species i we have a Nernst-Planck equation of the form where ji is the flux of ion i, Ai its molar conductivity, zi its charge number, ci its concentration and v the velocity of the solvent in a fixed coordinate system. There is one Nernst-Planck equation for every ion, with each equation containing the common quantity usually denoted by d$/dx and interpreted as the gradient of the electric potential. The concentrations of the ions are subject to the condition of electroneutrality: and the ionic fluxes must obey the n i-1 c zici = 0 equation for electric current ( I ) : n i-1 zi ji = I/F.(3) For every undissociated species of weak electrolyte we may write Fick’s law in the form where the subscript u denotes the undissociated species of the weak electrolyte j and Duj is its diffusion coefficient. Duj cannot be measured but it can be estimated by linear extrapolation of the data at high concentrations,12 because in a concentrated solution the degree of dissociation of weak electrolytes is vanishingly small. The dissociation constants form a link between the ionic species and undissociated species and, when only monovalent weak acids are considered, we have where Ki is the dissociation constant, cH+ is the concentration of hydrogen ion and ci and cuj are the concentrations of the ionic and undissociated species of the weak acid, respectively.For the ionic species and for the undissociated species the fluxes in eqn (1) and (4) are not stationary, i.e. constant, because of the changes in concentration in the porousK. KONTTURI, T. OJALA AND P. FORSSELL 3381 membrane. This is the case even though the transport process itself is in a stationary state. The fluxes are constant only for the ionic constituents. For instance, in the system of acetic acid (HOAc) + sodium acetate (NaOAc) +water the fluxes in eqn (1) and (4) are jH+, joAc-, jNa+ and juHOAc. and the stationary fluxes are j H =jH++juHOAc, Before deriving the transport equations with the aid of eqn (1)-(5) the following notation is presented.The subscripts T1 and T2 denote acetic acid and monochloro- acetic acid, - 1 and ul the dissociated anion and undissociated species of acetic acid, and - 2 and u2 the dissociated anion and undissociated species of monochloroacetic acid, respectively. With this notation the following expressions for the concentrations and fluxes of the ion constituents can be written. j o ~ c = jOAc- +juHOAc and jNa = jNaf- Binary systems : cT1 = CH = cH++c,,; jH = jH++jul for acetic acid in water. To obtain the corresponding expressions for the system of monochloroacetic acid in water subscript 1 is replaced by subscript 2. Ternary systems : cT1 = CH = cH+ + c,,; CNa = CNa+; jNa =ha+ j H = j H + +jul for acetic acid + sodium acetate + water. Again the corresponding expressions for monochloroacetic acid + sodium monochloroacetate + water are obtained by re- placing subscript 1 by subscript 2.For the system acetic acid + monochloroacetic acid+water we have c, = cT1 = c-, + c,, ; j, = j-, +jul c, = cTZ = c2+jU2; j, = j-,+j,,. The concentrations c, ahd c, are the concentrations of the acetate-ion constituent and monochloroacetate-ion constituent, respectively. The symbols for the dissociation constants are chosen to be K , for acetic acid and K, for monochloroacetic acid (Kl = 1.75 x Taking into account the condition of electroneutrality, eqn (2), the relationship between the ionic fluxes and electric current, eqn (3), the dissociation constants, eqn (9, the Nernst-Planck equations, eqn (l), Fick's law, eqn (4), and the Nernst-Einstein relation for ions and K2 = 1.36 x mol dmP3).we obtain, after the elimination of the electric-potential gradient, the transport equations for the different systems. For the binary system acetic acid + water we obtain where the transport number of the hydrogen-ion constituent has the form and the diffusion coefficient of acetic acid is3382 COUNTER-CURRENT ELECTROLYSIS IN A POROUS MEMBRANE The diffusion coefficient as expressed by eqn (9) is the weak electrolyte analogue of the Nernst-Hartley equation. It gives the same numerical values for the diffusion coefficient as the equation presented by Muller and Stokes,13 indicating that the same assumptions were made when deriving both of these equations. The transport equation and estimates for the transport quantities in the case of monochloroacetic acid + water are obtained from eqn (7)-(9) by substituting subscript 1 with subscript 2.The values for the ionic diffusion coefficients or ionic conductivities have been presented by Robinson and Stokes14 and the values for the diffusion coefficients of undissociated species have been presented by Vitagliano and Lyons12 and Holt and Lyons15 in the case of acetic acid and by Garland et aZ.lS in the case of monochloroacetic acid. The numerical values are presented in table 1 . Using the numerical data of table 1 and eqn (8 b) the estimates for diffusion coefficients were calculated and were found to be in a good agreement with measured data.l2? 14-16 For the ternary system consisting of acetic acid + sodium acetate +water the transport equations are where the transport numbers of the hydrogen-ion constituent and sodium ion are tNa = and the expressions for the diffusion coefficients are In eqn (1 2) and (1 3) the concentrations of the hydrogen and acetate ions are obtained using and the condition of electroneutrality. The corresponding equations for the system of monochloroacetic acid + sodium monochloroacetate + water are obtained by sub- stituting subscript 1 with subscript 2.The diffusion coefficients in the system acetic acid + sodium acetate + water have cH+ = +{ 'd K1 + (CNa+ + K1)21 - (K1 + cNa+)) (14)K. KONTTURI, T. OJALA AND P. FORSSELL 3383 Table 1. Diffusion coefficients of the species at infinite dilution at 25 "C species Oil cm2 s-l ref.H+ 9.3 1 14 Na+ 1.33 14 OAc- 1.09 14 CH2C1C00- 1.12 14 HOAc" 1.20 12 CH,ClCOOH" 1.08 16 a Undissociated species. been measured at different concentrations by Leaist and Lyons." They also developed a theoretical model to estimate the diffusion coefficients, and good agreement was found between calculated and measured values. Even though Leaist and Lyons used the methodology of the thermodynamics of irreversible processes their assumptions and results are almost the same as ours. The small differences are due to the fact that Leaist and Lyons took into account the influence of activity coefficients, but we do not. However, the results obtained are not essentially different. For the ternary system acetic acid + monochloroacetic acid + water the transport equations are where the transport numbers of the acetate-ion constituent and monochloroacetate- ion constituent are3384 COUNTER-CURRENT ELECTROLYSIS IN A POROUS MEMBRANE where A = (Kl + K2) cH+ + c$+ + K , c-, + Kl cT2 + cH+(c-, + c-,) + K, K2.The concen- trations of hydrogen ion, acetate ion and monochloroacetate ion in eqn (1 7) and (1 8) are obtained as follows. The concentration of acetate ion (eel) is calculated from (19) Eqn (19) has three roots, of which the one existing in the concentration range (0, cT1) is the value of the acetate-ion concentration required. After obtaining the value of c-, the hydrogen-ion concentration can be calculated from and the condition of electroneutrality is used to obtain the concentration of the monochloroacetate ion. The diffusion coefficients in eqn (13) and (18) are ternary analogues of the Nernst-Hartley equations for weak electrolyte systems.The transport numbers in eqn (8), (12) and (17) are estimates of the stoichiometric transport numbers, which are measured by Hittorf s method, i.e. the only measurable transport numbers in the above systems [cf. ref. (18)]. TRANSPORT PROCESSES The above equations are now used to model the transport processes through a thin porous membrane which is part of the counter-current electrolysis cell presented schematically in fig. 1 . In the stationary state the mass balance for ion constituent i, which cannot go through the ion-exchange membrane, in compartment a is simply -Aji = cf F/a (21) where A is the surface area of the porous membrane and cf is the concentration of ion constituent i in compartment a.Using eqn (21) we can, after analysing the concentration of ion constituent i and after measuring the volume flow determine the experimental value ofj,. Comparing this value with the theoretically obtained value of ji we are able to deduce the feasibility of the theory. Solving for the fluxes ji in the above equations leads us to a two-point boundary value problem of a set of ordinary differential equations. This problem does not have an analytical solution, except in the binary case, and therefore numerical methods have to be used. Two different procedures for solving the problem have been p r e ~ e n t e d l ~ ? ~ ~ and are used in the present calculations, that based on the shooting method described in the ref.(19) being applied most often. These numerical methods are also applied to the binary case, because the form of the analytical solution is dependent on the parameter to be solved and therefore its use is unwieldy. As mentioned above, the binary systems are studied only to find out how well the Nernst-Planck equations approximate the diffusion coefficient and transport number. In the ternary systems our main interest is in studying the separation of weak electrolytes. The separation efficiency is characterized by the selectivity ratio ( S ) where subscripts 1 and 2 denote ion constituents chosen so that S 3 1 and superscriptsK. KONTTURI, T. OJALA AND P. FORSSELL 3385 Fig. 1. Schematic drawing of the counter-current electrolysis cell. A thin porous membrane (M) divides the cell into two compartments a and p.The solutions in compartments a and pare well mixed. When cations are separated the polarity of electric current is chosen as presented in the figure and anionexchange membranes (AM) are used to separate the electrode compartments (E) from the a and j? compartments. If anions are separated the polarity must be changed and AM must be replaced by cation-exchange membranes. This means that the counterion in the ion-exchange membrane is the common ion of the system studied. In the binary case we have only one cation, i.e. H+, which moves towards the cathode and no separation exists. vf denotes the stream into compartment of the weak electrolyte solution being studied and Vb the flow out of the compartment.The concentrations in compartment /3 are kept constant by circulating the weak-electrolyte solution being studied so rapidly that no essential changes in concentration take place. Vo denotes the water flow jnto compartment a. Part of this water flows out of the compartment on the product stream ( Va) while. the rest flows as convection through the porous membrane ( Vc = Vo - Va). a and /3 denote the compartments a and /3. Unfortunately, unlike in strong-electrolyte systems where the selectivity ratio is exponentially dependent on convection,23 no simple relationship was found for the selectivity ratio in weak-electrolyte systems. EXPERIMENTAL The weak-electrolyte systems studied were HOAc + H,O, CH,ClCOOH + H,O, HOAc + NaOAc + H,O, CH,ClCOOH + CH,ClCOONa + H,O and HOAc + CH,ClCOOH + H,O.The apparatus used in the measurements is schematically described in fig. 1. A detailed representation of the experimental set-up has been presented previ~usly.~ For every measurement the total concentration in compartment p was kept constant (ca. 0.01 mol dmP3). The concentratjons in compartment a were determined by analysing the concentrations in the outflow stream Va. The system was deduced to have reached the stationary state when the concentrations in the outflow stream Va remained unchanged. For the binary systems the measurements were carried out by changing the electric current and convection as well as the outflow rate so that large variations in the magnitude of the flux through the membrane could be obtained. In the ternary systems the electric current and convection were changed while the outflow rate was kept constant.Sodium was analysed by flame photometry and the acids were mostly analysed by potentiometric titration. When the concentration of hydrogen was too small for potentiometric determination the acid concentration was calculated from pH measurements. In the case of two weak acids a computer program of Partanen21 was used to interpret the titration curve and to obtain the concentrations of the individual acids. The accuracy of these analyses was estimated to be better than 5% in every case. 110 F A R 13386 COUNTER-CURRENT ELECTROLYSIS IN A POROUS MEMBRANE RESULTS AND DISCUSSION BINARY SYSTEMS The results of the measurements in the binary systems HOAc+H,O and CH,ClCOOH + H,O with varying electric current, convection and outflow rate are reported in table 2, which also includes the quantity N calculated with the help of eqn (7)-(9).The quantity N is defined as N = N,+t,I, where NH is the dimensionless flux of the hydrogen-ion constituent and I, is the dimensionless electric current (for the definition of N , and I, see the footnote to table 2). The theoretical value of NH is calculated using the experimental value of I,, which is very accurate. The last column of table 2 is the ratio of the difference in experimental and theoretical hydrogen flux to electric current. This quantity is designated by qH (a type of current efficiency of ion-exchange membranes) and can be explained as follows. In the measurements we have to use ion-exchange membranes to separate the products of the electrode reactions from the rest of the system (see fig.1). For our binary systems these membranes were anion-exchange membranes through which hydrogen ions always leak introducing extra flux into the balance equation, eqn (21). This leakage is a complicated function of the electric current, the concentrations in compartment a and the electrode compartments (cg, cg) and the time of equilibration t,, i.e. VH =Ar, CF, c?, tE)- Usually tE differs from t g , which is the time required to reach the stationary state in the membrane. This difference is due to the fact that it was impossible to maintain constant values of the parameters ( vo, vc, 3 and A / l ) when the measuring time was lengthened (the usual measuring time in these experiments was a few days).The effect of electro-osmosis for the various v is ca. 0.1 % or less (assuming that 20 mol water are transported for 1F) and it can therefore be neglected as a source of error. Other souces of error are the theoretical model itself, the analysis of hydrogen in dilute solutions and the determination of the flow rates. The membrane constant ( A / l ) , needed in the calculation of fluxes, depends on convection and stirring in compartments a and a, and for acetic acid and monochloroacetic acid slow changes in the structure of the porous membrane (made of cellulose acetate or nylon) caused variations in membrane constant. The effect of the changes in A / l was studied with the aid of a theoretical model and it was found to be small for binary systems but large for ternary systems.In table 2 the membrane constant A/Z is taken to be 13.1 cm, which corresponds to the geometric value.29 22 Taking into account these errors we conclude that the theoretical approach based on the Nernst-Planck equations describes the behaviour of the system sufficiently well. Furthermore, note that theoretical calculations can be arranged to fit the experimental data exactly by changing the boundary concentrations on both sides of the porous membrane within the range of experimental error. However, this procedure always decreases the boundary concentration in compartment a and therefore it cannot be interpreted merely as an experimental error. On the other hand, the leakage of hydrogen through the anion-exchange membrane always occurs and so exact corres- pondence between measured and theoretical data is not possible.system -4 UC Ye YP measured theoretical measured theoretical H20 0.95 0.48 0.78 0.93 0.48 0.31 0.37 0.54 0.08 1 .oo 0.50 0.35 0.35 0.50 0.95 4.45 2.85 0.60 1.45 0.98 1.66 1.04 0.89 1.51 2.85 4.35 0.26 1 .oo 1 S O 1.13 1.05 1.42 7.60 4.66 0.71 1 .oo 3.93 3.31 2.87 3.49 9.49 1.91 1.17 1 .oo 2.55 2.25 5.95 6.25 2.85 0.98 1.03 0.94 1.59 1.03 0.95 1.515 Definitions of dimensionless parameters when c, = 0.01 .mol dm3, Do = 2 x lop5 cm2 s-l, A / l = 13.1 cm, Fis Faraday's constant and 1 membrane: uc = VCl/(ADo) (convection); ua = Val/(AD,) (outflow); I,.= Il/(c,D,F) (electric-current density); & = CJC, (//(Do co); N = NH + t , I,.; N,(measured) = uaca/co; AN, = N,(measured) - N,(theoretical).CH2C1COOH + H,O 0.95 1 .oo 0.43 1 .oo 0.45 0.27 0.40 0.57 2.85 4.35 0.30 1 .oo 1.29 1.29 1.26 1.263388 COUNTER-CURRENT ELECTROLYSIS IN A POROUS MEMBRANE TERNARY SYSTEMS The results of our measurements for the three different ternary systems HOAc + NaOAc + H20, CH2C1COOH + CH2C1COONa + H20 and HOAc + CH2C1COOH + H20 are reported in table 3, which also includes the theoretically calculated fluxes. In every measurement the outflow rate via had the same value and A/Z was taken as 13.1 cm. The sources of error are essentially the same as in the binary case. As can be seen from the results the theoretical model predicts the course of the transport phenomena. The model fails to explain the smaller fluxes quantitatively, but gives results of the same order of magnitude.HOAc + NaOAc + H,O Since acetic acid is very poorly dissociated at the studied concentrations (ca. 0.01 mol dm-3), its transport through a porous membrane in counter-current electrolysis consists mainly of diffusion and convection, the contribution from electric migration being small. However, some contribution from electric migration exists, as can be seen when inspecting the first three measurements in table 3. When the electric current is doubled and the convection is kept constant, the concentrations of acetic acid and sodium acetate are also doubled. The amount of totally dissociated sodium acetate in compartment a increases. This increase is due to electric migration, i.e. to the much greater transport number of the sodium-ion compared with the hydrogen-ion constituent (note that the concentration of hydrogen is ten times greater than that of sodium in compartment p).Diffusion and convection alone cannot explain this increase. In the last three measurements for the system HOAc + NaOAc + H20 the selectivity ratio S [see eqn (22)] is very high (in fact infinite): no hydrogen is left in compartment a because it has been partially neutralized by hydroxide ions, the products of the hydrolysis of sodium acetate. This means that the acetic acid concentrations in these three measurements are calculated not real values. Nevertheless, they demonstrate the flow of the hydrogen-ion constituent through the porous membrane. However, we must remember that the tabulated, rather inaccurate theoretical values for the flux of the hydrogen ion are obtained using a poor theoretical model, since the model does not take hydrolysis into account.Hydrolysis can be included in our model but it complicates the problem too much compared with the advantage gained. The measurements for the system HOAc + NaOAc + H20 show that using counter- current electrolysis with a thin porous membrane different buffer solutions of different pH values can be maintained on both sides of the porous membrane. This may be of use in the separation of zwitterions and pH-dependent complexes. CH2C1COOH + CH,ClCOONa + H 2 0 The behaviour of this system is very similar to that of the previous one, but because the dissociation constant of monochloroacetic acid is approximately a hundred times greater than that of acetic acid the contribution of electric migration to the transport is much stronger. To verify this, compartment p was filled with a solution containing a hundred times more acid than its sodium salt.Thus in this solution we have more hydrogen ions than sodium ions. Furthermore, the transport number of the hydrogen ion ought to be greater since A,+ 7A,,+. According to the calculations and the measurements the amount of acid in compartment a is increased. This is due to electric migration and both diffusion and convection have only a small effect on this increase. The theoretical model predicts the behaviour of this weak electrolyte system fairly well, but the increasing influence of activity coefficients can clearly be seen.Our modelTable 3. Experimental and calculated results for the ternary systems measured theoretical system - I , vc Yp Yg yz" y! Nl N2 Nl N2 S (1) HOAC+ 1.98 1.22 3.96 1.16 (2) NaOAc + H 2 0 3.96 2.12 7.91 4.18 11.87 4.18 15.43 4.37 (1) CH2C1COOH + 3.96 1.87 3.96 4.12 (2) NaOAc + H,O 7.91 1.94 7.91 4.22 (2) CH2C1COOH + H 2 0 11.87 4.24 (1) CH2C1COOH + 5.93 4.45 0.12 0.2 1 0.10 0.004 0.005 0.003 0.76 0.49 1.43 0.9 1 0.025 0.014 1 .oo 1 .oo 0.98 0.96 0.95 0.98 0.97 0.90 0.95 0.95 1.02 1 .oo 0.25 0.52 0.41 0.62 0.89 1.07 0.0016 0.000 8 1 0.0045 0.00 16 0.063 0.17 0.096 0.095 0.097 0.088 0.088 0.089 0.01 1 0.012 0.012 0.012 0.94 0.93 0.25 0.50 0.36 0.41 1 .oo 0.45 0.20 0.82 0.13 9.8 x 1.75 6.6 x 7.1 x 10-4 1.21 8.2x 10-3 5.9 x 10-4 2.17 5 . 4 ~ 10-3 1.68 3.5 x 10-3 2.1 1 3.03 9.5 x 10-3 4.20 1.91 3.5 x 10-3 3.22 1.10 1.8 x lob3 1.49 0.048 0.12 - 0.098 0.028 0.34 - 0.052 0.53 1.17 1.13 1.78 2.8 1 3.81 5.5 x 10-4 -2.2 x 10-4 16.6 x 10-4 2.4x 10-4 0.36 0.56 ?? 21.3" * 26.1" 41.0" 5 18 700" 19 400" Y 5 40 600" 5.4b 8.3b $ 4.0b > 2 6.6b $ 2.7" 13.3" 6 B ? Definitions of dimensionless parameters, when c, = 0.01 mol dmP3, Do = 2 x lop5 cm2 sP3, A / l = 13.1 cm, Fis Faraday's constant and I is the thickness of the membrane: uc = VCI/(AD,) (convection); ua = Val/(AD,) (outflow, ca. 2.0); I, = Il/(c,D,F) (electric-current density); 6 = ci/co (concentration); Ni (theoretical) = JiI/(Doco); N,(measured) = uac$/co; the meaning of subscripts 1 or 2 can be seen in the first column.a S = Y; Yf/Y,f'Y;. s = Yp Y!/ Yg Yz". w w 00 \o3390 COUNTER-CURRENT ELECTROLYSIS IN A POROUS MEMBRANE does not take into account the dependence of the activity coefficients on concentration and therefore greater deviations between theoretical and experimental results compared with the system HOAc + NaOAc + H20 can be expected.Monochloracetic acid is a stronger electrolyte than acetic acid, which is why the effect of activity coefficients is more pronounced in the system CH,ClCOOH + CH,ClCOONa + H20. HOAc + CH2C1COOH + H,O This system was studied in order to show that counter-current electrolysis can be used efficiently when separating weak acids from each other. In spite of the great difficulties in the analysis of these acids we are able to achieve our goal, i.e. to show that separation by this method is possible.The selectivity ratios ( S ) are much lower than predicted by calculations (note that N , > 0), which can be partly explained by the leakage of acetic acid through the cation-exchange membrane from the electrode compartment to compartment a. To obtain more quantitative results the experimental set-up must be improved. CONCLUSIONS Our theoretical model predicts the transport of weak electrolytes through a thin porous membrane in counter-current electrolysis when the total concentration is ca. 0.01 mol dm-3. In the case of weak electrolytes transport across a porous membrane is characterized mainly by electric migration, i.e. the separation can be estimated by considering transport numbers, and diffusion and convection determine the separation only in the case where the transport numbers of the individual ion constituents are of the same order of magnitude as in the case of strong electrolytes. A. Ekman, P. Forssell, K. Kontturi and B. Sundholm, J. Membr. Sci., 1982, 11, 65. K. Kontturi, P. Forssell and A. Ekman, Sep. Sci., 1982, 17, 1195. P. Forssell and K. Kontturi, Sep. Sci., 1983, 18, 205. W. Nernst, Z. Phys. Chem., 1888, 2, 613; 1889,4, 129. M. Planck, Ann. Phys., 1890, 40, 561. D. G. Miller, J. Phys. Chem., 1966, 70, 2639; 1967, 71, 616; 1967, 71, 3588. A. Ekman, S. Liukkonen and K. Kontturi, Electrochim. Acta, 1978, 23, 243. K. Kontturi, Acta Polytech. Scand., 1983, 152. A. Noyes and K. Falk, J. Am. Chem. Soc., 191 1, 33, 1437. lo E. Cussler, Multicomponent Dryusion (Elsevier, Amsterdam, 1976). l1 K. Kontturi, P. Forssell and A. H. Sipila, J. Chem. Soc., Faraday Trans. 1, 1982, 78, 3613. l2 V. Vitagliano and P. A. Lyons, J. Am. Chem. Sac., 1956, 78, 4538. l3 G. T. Miiller and R. H. Stokes, Trans. Faraday Soc., 1957, 53, 642. l4 R. A. Robinson and R. H. Stokes, Electrolyte Solutions (Butterworths, London, 2nd edn, 1959). l5 E. L. Holt and P. A. Lyons, J. Phys. Chem., 1965, 69, 2341. l6 C. W. Garland, S. Tong and W. A. Stockmayer, J. Phys. Chem., 1965, 69, 2469. l7 D. G. Leaist and P. A. Lyons, J. Phys. Chem., 1981, 85, 1756. l9 A. H. Sipila, A. Ekman and K. Kontturi, Finn. Chem. Lett., 1979, 97. 2o K. Kontturi and A. H. Sipila, Finn. Chem. Lett., 1983, 1. 21 J. Partanen, to be published. 22 T. Hashitani and R. Tamamushi, Trans. Faraday SOC., 1967, 63, 369. R. Haase, Thermodynamics of Irreversible Processes (Addison-Wesley, London, 1969). (PAPER 4/443)