TRANSPORT PHENOMENA IN ION-EXCHANGE MEMBRANES G. B. W I L L S Virginia Polytechnic Institute, Blacksburg, Va. E. N. L I G H T F O O T
Chemical Engineering Department, University of Wisconsin, Madison, W i s . Experimental equilibrium and transport data were obtained using a commercial cation membrane with several salts. Membrane swelling, electrolyte absorption, and differential conductivity were determined at concentrations ranging to 1N for the salts NaCI, NaN03, KN03, KCI, and AgNO3. Free diffusion and osmotic flux data were obtained in this concentration range for the same salts except KCI. Transport number and electro-osmotic flux data were obtained for AgNO3. Theoretically, a new set of approximate flux expressions are derived, based on a linearization of a frictional model, and consistent with the more usual irreversible thermodynamic treatments. The concept of a generalized mobility is introduced, and the close relationship of the frictional model to the Stefan-Maxwell equations is pointed out. The approximate flux expressions give a very satisfactory correlation of the experimental transport data.
recent years there has become available a highly selective membrane which is a sulfonated copolymer based on a polyethylene matrix. A commercially available cation membrane of this type has been studied (American Machine and Foundry Co., C103). A description of the method of preparation, the water content, and other data for the membrane studied can be found elsewhere (2). Reported here are measurements of equilibrium and transport processes for several 1-1 electrolytes a t moderate solution concentrations. I n particular, the fixed ion concentration, the concentration dependence of swelling, electrolyte absorption, conductivity, free diffusion, and selectivity have been determined in the concentration range of 0.1 to 1.01V for the electrolytes AgN03, KN03, NaN08, NaC1, and KCl. ITHIN
Experimental Procedures
Membrane Thickness. The thickness of the membrane was determined a t several concentrations for the five electrolytes studied: AgN03, KXO3, NaSO3, XaC1, and KCl. Following a 24-hour equilibration period with a n appropriate solution, the thickness was measured using a dial gage which had been calibrated using gage blocks. The gage sensitivity was approximately 0.0001 inch per division. Donnan Equilibria. For study of the absorption of electrolyte, membranes were equilibrated for 24 hourswith an electrolyte of a given concentration. The equilibrated membranes were then removed from the electrolyte, carefully wiped with blotting paper, and placed in distilled water for elution of the absorbed electrolyte. After standing for at least 24 hours, the membranes were removed and the eluate was analyzed. The AgN03 analysis was by the Volhard method. Except for very dilute solutions, KC1 and S a C l were determined by titration with AgN03 using an absorption indicator. Very dilute chloride liquors were made up to a standard volume and the potential of a silver-silver chloride electrode was determined with a potentiometer. The electrode was calibrated with solutions of known chloride content just before and just after the analysis. KN03 and N a K 0 3 were determined gravimetrically by evaporation of an aliquot in a platinum dish. The rate of approach to equilibrium during elution and the efficiency of the blotting procedure were checked in an experiment in which the amount of AgN03 eluted from the membrane was measured periodically. From the results, it was concluded that the 24 hours allowed for elution was more than adequate. That the resulting elution data appeared to extrapolate to a point of no elution at zero time was taken as confirmation that the blotting procedure removed all adhering liquor from the membrane surface. 114
I&EC FUNDAMENTALS
Membrane Conductivity. The conductivity cells were of the form shown in Figure 1. Platinized platinum electrodes were used for all determinations except for AgN03, when silver electrodes were used. EXPERIMEKTAL TECHNIQUE. The electrodes and the cell containing the membrane were separated and placed in a beaker of the solution to be studied. The beaker was covered and placed in a 25’ C. bath to equilibrate for 12 to 24 hours. The cell was then closed and replaced in the 25’ C. bath. The cell resistance was measured periodically until a constant value was attained. After the cell resistance had been obtained for solutions of various concentrations, the membrane was removed and the procedure repeated for the solutions alone, In this way the membrane resistance was obtained as the difference between the readings with the membrane and without the membrane. The solutions were analyzed by the methods described for the equilibrium data. The cell resistance was measured by a Leeds & Northrup Model 4960 60-cycle A.C. bridge. Originally all of the cells were designed for immersion in the 25’ C. water bath. However, some difficulty was experienced in sealing the cells without lubricating the O-rings and it had been found that stopcock grease in the system was a source of erratic and erroneous results. This sealing problem was solved by putting the cells in a beaker of the solution with which the membrane was to be equilibrated. Contamination of the membrane surface can be a source of considerable error. A simple test for contamination was the wetting properties of the membrane surface. If a membrane taken from the solution had a continuous film of liquid on its surface, the membrane was assumed to be clean. If the solution film ruptured and formed drops, the membrane was assumed to be contaminated. The “weep hole” in the cell was provided for pressure release, since the large O-ring on the electrode forms a seal before the electrode makes firm contact with the cell body. Free Diffusion. The electrodiffusion cell shown in Figure 2 was used to determine the free diffusion of electrolyte across the membra ne. Initially the lower cell was filled with a solution of known electrolyte concentration and the upper cell was filled with distilled water. Each half of the cell contained a pair of platinized electrodes, which were used for periodic measurement of the conductivity of the solution in each half of the cell. From these data it was possible to calculate the amount of electrolyte that had diffused across the membrane at any time during the experiment.
MATERIAL : PLEXIGLAS UNLESS NOTED OTHERWISE
t
TOP VIEW
WITH SEAL
$1 I-
W V -1 W
VIEW INE
u
t
I
rJ/
/
1
-
FULL SCALE
Figure 1 . Cross-!iectional view of membrane conductivity cell
n
0.5 I.o SOLUTION CONCENTRATION. MOLES/L.
"0
Figure 3. Amount of absorbed electrolyte in membrane a t various solution concentrations for five electrolytes studied
3.5-
3.07,
-~ 3
,
I
I
,
,
,
, , NaCI, , ,
U
.
=LE
= 112 FULL
2"
Figure 2.
Cross-sectional view of electrodiffusion cell
. 0,
0
-
NoNO, ,
I
l
l
1
0.5 I.o SOLUTION CONCENTRATION, EQ./L.
Figure 4. Conductivity of membrane in five electrolytes studied at various solution concentrations VOL. 5
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FEBRUARY 1966
115
The osmotic flow of solvent was measured by collecting the overflow from the bottom chamber in a 5-ml. buret which was connected to the bottom chamber by a I/d-inch diameter polyethylene tube as shown in Figure 2. A constant, positive head of about 4 inches of liquid was maintained in the bottom cell by adjusting the height of the buret. This pushed the membrane upward into what was observed to be a stable, convex shape. During the diffusion process the cell was immersed in a 25' C. bath. and the agitators were run continuously at 120 r.73.m.. giving predomizantly turbulent flow in the cells. With circulation through the interior cooling coils shown in Figure 2 the temperature inside the cells was less than 0.1' C. above that of the bath. In general, 24 to 72 hours were allowed per determination. Upon completion of each run, the contents of each half of the cell were sampled and analyzed to check the calibration of the platinum electrodes mounted in the cells.
Electrodiffusion Measurements. Ionic and solvent transport numbers were determined only for the electrolyte AgN03. The equipment was the same as that used to determine the free diffusion of electrolyte except for the addition of the silver electrodes shown in Figure 2. Initially, each half of the cell contained the same electrolyte concentration. Upon passage of current the electrolyte concentration fell in one half of the cell, while a concomitant increase was observed in the other half. From the initial and final concentrations in each half of the cell, the silver ion transport number across the membrane was calculated. The electro-osmotic solvent transport was determined from the overflow of solution from one of the cell compartments into a buret. Solution Phase Mass Transfer Effects. The stirrer in each compartment of the cell in Figure 2 was presumed to give agitation such that there was a negligible resistance to mass transfer in the solution phase. This presumption was checked by placing a silver electrode in the position normally occupied by the membrane and measuring the limiting current for the silver ions at this position. A solution 5.22 X lO-4N in AgSOs and approximately 1.OA' in NaN03 was used, and a limiting current of about 0.27 ma. per sq. cm. was observed. Assuming a mobility of 6.4 X 10-4 sq. cm./sec.-volt for silver ions, this corresponds to a solution phase mass transfer resistance equivalent to a stagnant film 4 X lop3 cm. thick. This gave negligible effects under the conditions used in this study. Details of how this calculation can be carried out are given elsewhere (8).
The uniformity of the membrane thickness was determined by measuring ten random samples cut from a large sheet. The maximum variation in thickness was =k2.5% of the average thickness of the ten samples. Donnan Equilibria. The membrane equilibrium data are given in Figure 3. The curves show the concentration of absorbed electrolyte as a function of the external solution concentration. The silver nitrate is the most strongly absorbed, the other nitrates are less strongly absorbed, and the chlorides
// I
d
NacP0 A
a/ W
- 1
I/
I
7 ,
0oW'
'
' '
' ' 0.05 I ' " ' 0.10 CONCENTRAT ION DIFFERENCE, EQA.
Figure 5. Free diffusional flux across membrane as a function of difference in concentration of invading electrolyte a t interfaces, AC2
31
Experimental Results
Membrane Thickness. The membrane thickness, 6, was found to be a linear function of the equilibrating solution concentration, (? equivalents per liter, for each salt studied. These data were correlated by a least square analysis assuming such a linear dependence. The best straight lines from the least square analysis were as follows: Salt
NaCl NaN03 KXOa KCl
AgNOs
Besf Straight Line, 6 =
60(7
- mc)
6 , cm. = 0.02155 (1 - 0.00645c) 6, cm. = 0.02155 (1 0.00707c) 6, cm. = 0.02147 (1 - 0.01133C) 6, cm. = 0.02147 (1 - O.O0827C_) 6, cm. = 0.02088 (1 - 0.00659C)
-
The thickness is but a slight function of solution concentration. However, the membrane thickness is significantly less for silver nitrate than for the remaining four salts. This effect has been noted before and has been attributed to association of the silver ions with the fixed charges on the lattice with a n attendant loss of water of hydration. 116
l&EC FUNDAMENTALS
0 0
0.06 0.1 0 CONCENTRATION DIFFERENCE, E0.R.
Figure 6. Osmotic solvent flux as a function of concentration difference of absorbed electrolyte
u z 6. P cn
2.2
/---
KNO,
I
1 .O5
.IO
0
ABSORBED ELECTROLI'TE, E W L . Figure 7. Differential and integral diffusion coefficients as a function of concentration of absorbed electrolyte
-- Integral diffusion coefficients - - - - Differential coefficients
exhibit the least absorption. However, in all cases the amount of absorption is surprisingly small. T h e fixed charge concentration was found to be 1.33 X 10-3 equivalent per cc. for the membrane in the silver form in equilibrium with distilled water Membrane Conductivity. Shown in Figure 4 are the experimental membrane conductivities as a function of the concentration of the equilibrating electrolyte. A modest dependence of conductivity on concentration is noted. Except for KCI, the curves through the points are calculated from an analysis discussed below. In the case of KC1 the curve is based on a least square analysis of the conductivity data. Free Diffusion. Figure 5 shows the free diffusional flux of electrolyte across the membrane as a function of the invading electrolyte concentration difference across the membrane. T h e curves through the experimental points are based on calculations discussed below. The corresponding osmotic solvent flux is shown in Figure 6. No data were obtained for the salt KC1. Figure 7 shows integral and differential diffusion coefficients of the various salts in the membrane, calculated from data. Electrodiffusion Measurements. Figure 8 shows the experimental cation transport number and the electro-osmotic solvent flux as a function of the external solution concentration. This later concentration can, of course, be related to the corresponding membrane phase concentration by use of the equilibrium data in Figure 3. These data are for the salt AgN03. The curve for the cation transport number is based on calculations discussed below. Theory
Ionic tranport in membranes can be treated in several ways depending on the models envisioned for the membrane structure and the transport processes. The simplest treatments assume that ionic fluxes are independent of one another and of solvent transport. These fluxes are calculated as the product of a mobility, a concentration, and a force, as in the follo\ving form.
61 = U G F i
(1)
The force Ft is set equal to the negative of the gradient of the total potential of species i, with this total potential taken as the sum of the chemical, electrical, and mechanical contributions. This treatment gives the Nernst-Planck flux equations. to which some add a convection term. An extensive review of this type of treatment is available ( 2 ) . A more complete approach to ionic transport is given by the irreversible thermodynamic treatments which allow for coupling between the fluxes as shown by Equation 2.
- 1
0
I
(
'
I
I
0.5 I .o SOLUTlON CONCENTRATION, EW L .
Figure 8. Cation and solvent transport numbers as a function of solution concentration for electrolyte
AgNO3
T h e L,, can be concentration-dependent but are independent of the forces. A well-known treatment consistent with Equation 2 is a frictional model put forth by Spiegler (7). I n this model certain frictional parameters are introduced, and there results a prediction for the form of the concentration dependence of the L z j . I n the model used here, the membrane is treated as a homogeneous phase, and the treatment is similar to Spiegler's in that frictional forces are envisioned. However, simplifications are introduced and there are considerable departures from the Spiegler treatment. Furthermore, the results obtained here cannot be derived from the usual irreversible thermodynamic treatments. Although the frictional model gives results conVOL. 5
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FEBRUARY 1966
117
sistent with irreversible thermodynamic treatments, a great deal more information is forthcoming-a prediction of the composition dependence of phenomenological coefficients. The frictional model used here, with slight modification, can be used to derive the Stefan-Maxwell equations. These latter flux equations are likewise consistent with the usual irreversible thermodynamic treatments, but, of course, they predate Onsager’s (6) contributions by over a half century ( 5 ) . Finally, treating the membrane as a completely homogeneous phase has important ramifications that must not be overlooked. The final results obtained here for a homogeneous membrane will not be consistent with those obtained for the more frequently encountered pore model. Close attention must be paid to this point, since this difference is not readily evident in the preliminary formulations. I n the well-known Spiegler model for transport processes in membranes the external force F, on species i is balanced by the sum of the frictional drag forces. I n turn, these frictional forces are taken proportional to the relative velocity of species i with respect to species j . Thus Spiegler writes the frictional force exerted on 1 mole of species i by species j as follows: Ji, =
--‘&,(Vi
- V,)
(3)
Here the frictional force on 1 mole of species i by species j is taken in a somewhat different form
The ,Vz3 are those of Kressman (4) and are not to be confused nith coefficients appearing in the usual irreversible thermodynamic treatments. The Ni,are treated as constants here and there are no constant coefficients in irreversible thermodynamics. However, the iV,, are closely related to the Dij found in the Stefan-Maxwell equations for diffusion in ideal gases (7). If one replaces the volumetric concentration, C, in Equation 4 with the mole fraction of j , x j , and continues with the derivation, it will be found that Equation 7 is identical to the Stefan-Maxwell equations with the N i j related to the D,, as follows: Dij
RT/N,j
(5)
For ideal gases Fishould be taken as follows : Fi
=
-v(RT In x i ) =
RT -VXi xi
(6)
Since the fluxes and velocities are taken with respect to fixed coordinates, Equation 7 is easily rewritten in terms of I&EC FUNDAMENTALS
Now a defined mobility, Ui, and a defined ideal or NernstPlanck flux, &*, are introduced.
(9)
+** = UiCtFi
(10)
Substitution of Equation 9 and Equation 10 into Equation 8 gives the following expression.
4t
=
4i*
+ CJ~
l ~ ~ i ~ $ jj
z
i
(11)
I
No approximations have been made; only the form of Equa-
The reciprocal relation for the Ni, follows from a simple force balance; recourse to Onsager’s relation is unnecessary. The rationalization of the form in Equation 4 is that the frictional force should be proportional not only to the relative velocity, but also to the probability of encounters of this 1 mole of species i with species j . This collision or encounter probability is taken proportional to the concentration of species j. The velocities in Equation 4 are of course drift velocities, and these are but small average perturbations of the thermal motion of the species. It is through exchange of these small drift momenta that the frictional forces arise. The net external forces, F,, are equal to the sum of Equation 4 over the species present with a change of sign, since the external force and the net frictional force are in opposite directions. As in the Spiegler model, the matrix is treated as a component species, except that its velocity is zero.
118
the fluxes. Thus there are two equations relating the forces to the fluxes: Equation 2 involving the L t j and Equation 7 involving the N i j . From these two equations the L,, can be expressed in terms of the N t j and the concentrations. I t follows that the frictional model predicts the concentration dependence of the L,,, since the X i j are considered constants. I t is at this point that the homogeneous membrane model differs from the pore model. For the pore model, and for solutions in general, Equation 2 would be summed only from 1 to n - 1 where n is the number of components, so that the fluxes are given in terms of an independent set of n - 1 forces. All n forces are not independent, because the Gibbs-Duhem equation must be satisfied. Here the sum is carried out from l to n, where n is the number of mobile species. However, there is a n additional species, the matrix rn, which allows these n forces to be independent. Corresponding to the GibbsDuhem equation is an over-all force balance which includes a force F, on the matrix. However, this latter equation is not needed here. Equation 2 is summed over the n-mobile species, and with the homogeneous membrane model, the n forces are independent. The Lij given here are different from those one would find in treatments of membranes by a pore model, or in considerations of solutions where all the components are mobile. A considerable departure is taken from the Spiegler treatment at this point. Instead of solving the set of liner equations given by Equation 7 for the individual velocities, Equation 7 is rearranged so that each velocity appears only once.
tion 7 has been changed to give Equation l l . The approximation is now made that the fluxes, $ j , in the second term on the right side of Equation 11 can be replaced by the corresponding ideal fluxes, @r*. One can then rewrite Equation 11 in terms of the forces, F,, and the defined mobilities, U,, by use of Equation IO.
This last result is, of course, an approximation to the result one would get by a rigorous solution of Equation 5 . By comparing Equation 1 2 with Equation 2 the following forms are gotten for the Li5.
The advantage gained is that Equation 12 is much simpler than the corresponding rigorous solution. I n light of the modest flux coupling that has been observed experimentally, this type of simplification appears particularly suitable. For correlation of the data here several further approximations are made: The treatment is restricted to a single 1-1 electrolyte, only ionic fluxes are considered, and finally mem-
brane swelling and certain ion-solvent interactions are neglected. The defined mobilities in Equation 9 are rewritten as follows. I
J
The last term in Equation 15 is taken as the reciprocal of a limiting mobility, U,o, I n the first term to the right of the equality sign, membran: swelling and solvent-ion interactions are neglected. This amounts to assuming U, to be constant except for a single concentration-dependent ion-ion interaction term. Upon recognizing C2o as zero, Equation 15 can be rewritten for the ionic species as follows :
l/Ul
+
== C2A-12
1/u10
+ l/U,O
1/U2 == CzhT1p
(1 6) (17)
Equations 16 and 17 are inverted and approximated as follows:
u1 =
u10
[l
up =
L;20
[I
-
u 1 0
C2N121
(1 8)
r20
CzN12]
(19)
Using Equations 12, 18, and 19 the folio\\ ing forms are derived for the differential conductivity and cation transport number. Terms involving the squares of C2 or .VI? have been neglected, and constant activity coefficients have been assumed. k = F[C11110
t-
+ C2UZO - 2iZT1?C1C2(C~lo)(Uzo+ U10/2)] =
FC2[1/2'
- .V~~C~C~OCI~]/~
(20) (21)
With the further neglecl. of L23F3 terms in Equation 2, an approximation which amounts to assuming a zero flux for the \vater in calculating the ionic fluxes, the free diffusional flux is given by Equation 22.
+/F =
- I ( R T/ k ) (Cp + C J U
OL12'[[1
- Xi2C2(Lri0+ L1:,o)])dCs/dx (22)
It is now seen that there are only three parameters to be determined from experiment: Ulo, U20, and N 1 2 . These are presumed to be constantc, and they appear in fairly simple form. These expressions for ionic transport are not much more involved than those obtained from the simplest Nernst-Planck treatment. A Nernst-Planck type mobility is retained, but first-order correction terms have been added to account for interaction between the ionic fluxes. This treatment could be made more general by including ionic-solvent flux interaction, but the trend of these data is such that the neglect of solventionic flux coupling appears justified. However, there is interaction between the ions and the solvent; the mobilities contain such interactions Integration of Equation 22 is straightforward, since one has a n expression consisting of the ratio of two second degree polynomials. But the results of such a n integration are complex, and the term contained within the braces in Equation 22 is almost concentration-independent for the membrane studied here. This suggests that this grouping be defined as a membrane diffusion coefficient, D . This gives the following very simple expression for the free diffusional behavior.
$/F
-D(dC?/dx) (23) The slight dependence of D upon concentration can be corrected by use of a suitable average value. ==
transfer. Here an attempt has been made to preserve the customary units for familiar quantities such as diffusion coefficients, mobilities, concentrations, and conductivities. This results in a set of mixed units in some of the equations and close attention must be paid to the definitions in the nomenclature. However, if the units in the nomenclature are used, the equations as given should prove dimensionally correct. The concentrations appearing in the equations have the units of moles of species per cubic centimeter of membrane volume. Special note is made when solution concentrations or units of moles per liter are used. Calculation Procedures. Free diffusion, conductivity, and transport number data are available for the salt AgNOI. Only conductivity and free diffusion data were obtained for the other salts. According to the theory developed, three parameters need to be determined for each salt: CTio, L'20, and N1p. This can be done with only the conductivity and free diffusion data. The parameters so calculated can be used to predict the cation transport number, but only for AgN03 can these predictions be compared to experiment. Actually the required parameters can be extracted from any two of the three types of data available for AgN03, but since the transport number data are the least precise, the other tIvo types of data are the preferable choices for calculating U10, U 2 0 , and N12.
The parameter U1o is obtained directly from the conductivity data by extrapolation to infinite dilution. Letting this value of k be ko, the following expression is used to calculate C10. 0 ' 10
- ko/FC,n
(2.1)
T o extract Ut0 the free diffusion flux is plotted as a function of the co-ion concentration difference across the membrane. Such plots of flux us. concentration difference are nearly linear. The slope of these curves at the origin is set equal to DO. Returning to Equation 22 and taking the limit of ($/F)/(dC2;'dx) as Cz approaches zero gives the following expression for C ? O .
Cz0 = FD'/RT
(25)
The parameter ~ 1 ' 1is~ calculated by inserting G 1 0 and L'20 into Equation 20 and fixing N l z to give the best least squares fit of the conductivity data as a function of concentration. T h e calculated values of UP, U2O, and -VIZ for the four salts are summarized in Table I. Table 1.
Calculated Values
40,
So/!
NaCl
NaN03 KNOI AgN03
UP, sq. Crn./Sec.-Vo/t
sq. Crn./Sec.-Vo/t
3.182 3.358 4.864 1.166
X X X X
2.780 1.502 2.580 0.927
10-5 10-5
10-5 10-5
X X X X
10-5 10-5 10-5
loe5
Niz, Vo/t-Crn.-Sec./Eq.
2.55 2.01 1.66 3.12
X 107 X lo7 X 107 X lo'
The integral and differential diffusion coefficients shown in Figure 7 are calculated from the C-10, C20, and VI^ values listed in Table I . Comparing Equations 22 and 23 one finds the following expression for D,the differential diffusion coefficient.
D = (RT/k)(C2
+ C ~ ) U I ~ U-~ .V12C2(Ui0 ~[I + Crzo)]
(26)
T h e integral diffusion coefficient is then defined as follows.
D* =
"L
DdC?/ " L d C 2
(27)
Correlation of Results
A note of caution should be given concerning the nomenclature. Units used in electrical measurements such as conductivity are different from those ordinarily used in mass
Discussion
The curves in Figures 4, 5, and 7 , and the cation transport number curve in Figure 8, are based on the parameters in Table VOL. 5
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FEBRUARY 1966
119
I. Equations 20, 21, and 22 have been used in these calculations. As can be seen, the curves are consistent with the data, Furthermore, the parameters in Table I appear reasonable and of the expected magnitude, However, corrections introduced by the h’12 terms cannot be easily estimated by inspection of Table I ; one must actually calculate the effect for the property of interest. But as a rough estimate, the 1V12 terms contribute corrections as large as 30y0 to the transport number data, 10% or less to the conductivity data, and several per cent to the free diffusion data. The magnitude to the correction depends upon the concentration of invading electrolyte, and as this concentration is reduced, the effect of the Nlz terms is diminished. As an alternative model, the data were analyzed assuming simple Nernst-Planck behavior with mobilities that were linearly dependent on concentration. This model has, of course, been widely used in the past. The free diffusion and conductivity data could be correlated by cation mobilities that decreased, and anion mobilities that increased with concentration. For example, the free diffusion and conductivity data for A g N 0 3 are correlated very well by the following concentration dependent mobilities. U1
1.166 X 10-5
U2 = 0.740 X
-
1.47 X 10%’2
(Cation)
+ 5.55 X 10-2C~
(Anion)
Similar expressions were found for the other salts. However, there are three serious objections to this four-parameter model. I t seems improbable that the cation mobilities decrease while the anion mobilities increase, and even less probable that the anion mobility increases with increasing concentration, and finally, the predicted cation transport number becomes much less than the measured values at the highest concentrations. For example, a cation transport number of about 0.86 is predicted for the membrane in equilibrium with a 1.ON solution of AgNOs, while the measured value was about 0.95. By way of contrast, the model actually used gave a good correlation of the conductivity, free diffusion, and transport number data throughout the concentration ranges using only three parameters. Water transport has not been analyzed, since the free water content of the membrane was not measured and it is not obvious how one could make such a measurement, a point more fully discussed elsewhere ( 3 ) . The total water content of the membrane is about 15% by weight, but most of this is probably present as water of hydration. This is borne out by the AgN03 electro-osmotic solvent flux, which amounts to only about 2 moles of water per faraday, and most if not all of this would be attributable to water of hydration carried by the cation. This suggests that either the ion-solvent coupling is slight or else the concentration of free water in the membrane is very small. The abnormally low osmotic flux found experimentally would further substantiate the latter. The correlations given here may prove inaccurate if extended to very dilute solutions. No measurements were made in solutions more dilute than about 0.1N. However, there is evidence that considerable changes may be taking place in the concentration range below 0.1S. For instance, the mem-
120
I&EC FUNDAMENTALS
brane activity coefficient6 appear to be rapidly decreasing in this range of concentration. The approximate flux expressions given by Equation 12 should be of general applicability and are not restricted to membranes or to any particular membrane model. However, for the membrane pore model, or for solutions in general, the forces appearing in Equation 12 are not independent. I n order to cast Equation 12 in the form of the equation normally used in these cases, the nth force should be eliminated by use of the Gibbs-Duhem equation, and this will give an additional contribution to the expressions given in Equations 13 and 14 for the Li,’s. Or, if one prefers to think in terms of a frictional model, an over-all force balance can be used to eliminate the nth force. The two approaches, of course, give exactly the same results. Nomenclature
C1 = cation concentration, eq./cc. Cz = anion concentration, eq./cc. C , = fixed charge, eq./cc. R = gas constant ’oules/mole-’K. T = temperature: J.K. F = faraday, coulombs/eq. F1 = force on cation, joule/eq.-cm. Fz = force on anion, joule/eq.-cm. k = conductivity, ohm-’ ern.-' @I = flux, coulomb/sq. cm.-sec. @I/F= flux, moles/sq. cm.-sec. t+ = anion transport number t - = cation transport number t, = solvent transport number, moles/faraday U1 = cation mobility, sq. cm./sec.-volt Uz = anion mobility, sq. cm./sec.-volt 6 = membrane thickness, cm. D = differential diffusion coefficient, sq. cm./sec. D* = integral diffusion coefficient, sq. cm./sec. N p j = friction coefficients, volt-cm.-sec./eq. L,, = coefficients in flux equations, eq./volt-cm.-sec. SUBSCRIPTS m = membrane matrix 1 = counter-ion 2 = co-ion 3 = solvent SUPERSCRIPT = species concentrations with membrane at equilibrium 0 with pure solvent Literature Cited
(1) Bird, R. B., Stewart, W. E., Lightfoot, E. N., “Transport Phenomena,” p. 570, Wiley, New York, 1960. (2) Helfferich, F., “Ion Exchange,” pp. 64, 582, McGraw-Hill, New York, 1962. A , , Tye, R. L., Trans. Vol. 2, p. 629, Dover, New York. 1952. (6) Onsager; L., Ann. N . Y.Acad. Sci. 46, 241 (1945). (7) Spiegler, K. S., Trans. Faraday Sot. 54, 1408 (1958). (8) Wills, G. B., Lightfoot, E. N., A. I. CI1.E. J. 7, 273 (1961).
RECEIVED for review January 4, 1965 ACCEPTED July 19, 1965 Based in part on Ph.D. thesis of G. B. Wills, University of Wisconsin, 1962.
