Jan., 1962
THEORY OF PARTICLE-DIFFUSION COKTROLI ED 103EXCHANGE
39
ION-EXCHANGE KINETICS. 111. EXPERIMENTAL TEST OF THE THEORY OF PARTICLE-DIFFUSION CONTROLLED ION EXCHANGE BY F, HELFFERICH Shell Development Company, Emeryville, California Received June 1.8, 1962
Experiments were designed to test the validity of the theory of particle-diffusion controlled ion exchange presented in previous publications. Ion-exchange rates and intra-resin concentration profiles were measured with phenolsulfonic acid cation exchangers and H + and N a + as the exchanging counterions. The experimental results are in very satisfactory agreement with liheoretical predictions based on individual intra-resin diffusion coefficients that were determined by independent conductivity measurements. I n particular, the predicted characteristic differences in rates and concentration profiles between forward and reverse exchange of two given ions having different mobilities were confirmed experimentally. The results thus establish the importance of electric potential gradients that were neglected in earlier theories. It is shown by qualitative uguments that the electric potential gradient also should cause transient changes in the co-ion and solvent contents of the resin in the course of ion exchange. These additional predictions also were confirmed by measurements.
Introduction
of the individual diffusion coefficients of the exchanging ions. To establish the superiority of the exchange of counterions between ion-exchange new theory unambiguously, it was necessary to resins and electrolyte solutions is controlled by choose a system for which the new and the older diffusion of the counterions either in the resin theories predict drastically different rates and conparticle (“particle-diffusion control”) or in an centration profiles, and to measure these accurately. adherent liquid diffusion layer (“film-diffusion Drastic differences between the theories are obcontrol”) . 2 The present study deals exclusively tained with counterions having markedly different mobilities. H+ and Na+ (mobility ratio with particle-diffusion controlled ion exchange. I n earlier theories of particle-diffusion controlled about 6 :1) were chosen as two such ions. Acion exchange, it was assumed that interdiffusion curate rate measurements with the usual spherical can be described in terms of one constant inter- ion-exchanger beads of 1 mm. diameter or less are diffusion coefficient, and the well-known mathe- difficult because of the short conversion time (half matical solutions for diffusion with a constant time, as a rule, less than a minute), and the task diffusion coefficient into, or out of, a sphere were of measuring non-stationary concentration proin a more recent theoretical files in such beads is almost hopeless. Therefore a ~ p l i e d . However, ~ treatment4 based on the Nernst-Planck flux the measurements were made with stacks of ionequations it was shown that interdiffusion gives exchanger discs which gave much longer converrise to an electric potential gradient which acts on sion times and could be taken apart to determine the diffusing ions and results in a variable inter- the concentration profiles. Beyond the calculation of counterion exchange diffusion coefficient and quite different rate laws. I n particular, tho new and the older theories differ rates and concentration profiles, as discussed in in the following points, In the exchange of two Part I, the Nernst-Planck equations can be used counterions A and B, of which A has the higher to predict transient changes in the eo-ion and solmobility, the new theory predicts: (1) Exchange of vent contents of the resin in the rourse of ion exB (initially in the solution) for A (initially in the change. Separate experiments were made to resin) is faster than the reverse exchange of A for verify these additional predictions. B (see Fig. 3 in Part I); (2) I n the exchange of B Experiments for A, a comparatively sharp concentration “front” Disc Preparation.-Cation-exchanger discs were prepared moves in toward the center of the bead, whereas, as described by Manecke,6 with modifications as previously in the reverse exchange, the beads are much more reported.6 This procedure is essentially as follows. Phenol sulfonated with concentrated HgSQ a t 110’. A mixture uniformly converted (see Fig. 4 in Part I). In is of the obtained phenolsulfonic acid solution with aqueous marked contrast, the earlier theories predict equal formaldehyde is left standing a t room temperature until rates and concentration profiles for forward and viscous and then is poured into inert plastic moulds (Mylar plastic sheets with circular holes) which are sandwiched bereverse exchange of two such ions. glass plates. Condensation is carried out in the Experimental results published so far are in- tween moulds ai, 70” (about 3 hours). The properties of the discs adequate for a quantitative test of the new theory. are listed in Table I. The limits given are deviations beTherefore the present study was undertaken to tween the individual discs. These deviations are greater provide such a test by a minimum of crucial experi- than the experimental errors. Ion-exchange Capacity of the Discs.-The discs were conments. For this purpose, ion-exchange rates and verted to the H + form by repeated batch equilibration with concentration profiles were determined experi- 1 N HCI and were washed with deionized water to remove mentally and were compared with theoretical sorbed HC1. H + then was displaced by successive batch predictions based on independent measurements equilibrations with 1 A’ S a C l and was determined in the
It long has k)een established that the rate of
(1) Part I , J . Chem. P h y s . , 28, 418 (1958); Part 11, ibid., 29, 1064 (1968). (2) G. E. Boyd, A. W.ddamson and L. S. Myers, Jr., J . Am. Chem. Soc., 69, 2836 (1947). (3) S?e, for euample, T. R. E. Kressman axid J. A. Kitchener, Discussions Faraday Soc., 7 , 90 (1949); D. Reichenberg, J . A m . Chem. Soc.. 7 5 , 589 (1953). (4) F. Helfferich and 111. S. Plesset. J Chem. Phys., 28, 418 (1958).
combined decanted solutions by titration with 0.1 AT NaOH. Co-ion Content of the Discs.-Go-ion contents of the discs were measured in equilibrium with 0.1 and 1 A T solutions of HCl, NaCl and HCl-NaCI mixtures a t 2 5 ” . After equilibration with the respective solution in a thermostat, ad-
(5) G. Manecke, 2. physik. Chem. (Leipzig), 201, 193 (1952). (6) F. Helfferich and H. D. Ocker, Z . phusik. Chem. (Frankfurt), 10, 213 (1957).
40
F. HELFFERICH
'
Vol. 66
platinized platinum electrodes, as described by Manecke7 and Lorimer,* was used. The cell consists of two half-cells I between which the disc is clamped. During the measurements, the cell is immersed in the equilibrating solution ke t in a thermostat. The resistance of the disc is determined t y +w subtracting the blank resistance of the cell (without disc) 0 5 from the resistance measured with the disc in place. The 6 resistances were measured with a Jones Bridge (Leads and Northrup, Research Model) in combination with a Tuned Audio Frequency Amplifier, a telephone receiver, and a decade condensor (10-8to 1 rF.) with 1000 cycles alternating 0 l ! ! l ! current. The average of three readings for every disc was 2 taken; the cell was taken apart and reassembled for every reading. The specific conductivity was calculated from the resistance and the known thickness and cross-section of the disc. Ion-exchange Rate and Concentration Profile Measurements.-Rates and intra-resin concentration profiles were measured at 25" for complete conversion of the discs from the Na+ form to the H + form with 0.1 N HC1, and from the H + form to the N a + form with 0.1 N XaCl. Prior to the experiments, the discs were converted to the monoionic Na+ form (or H + form) and extensively washed with deionized water. A stack of five discs and one inert Mylar disc of the same size were placed in a finger cot of thin rubber, with the inert disc a t the far end of the cot. The open end of the cot then was attached to a lucite cell in sukh a way that the surface of the first disc was exposed to the solution in the cell. Fig. 1. 25'. The cell was immersed in a thermostat, and the solution was ra idly circulated (flow rate about 10 to 20 ml./sec.) throu h a %astelloy-B heat exchanger in the thermostat, the cefl, Average Equivalent Fraction of Na+ i n Disc and a 1-liter reservoir bottle. A nozzle in the cell directed a 0,5 Yjolet jet of solution against the exposed surface of the first CLISC. The solution was renewed repeatedly, as re uired to keep the Na+ (or H+) concentration below 0.001%. Experiments were made with contact times between the resin and the solution of 1:20, 4 and 10 hr. Thereafter, the disc ' stack was disassembled, and the individual discs were analyzed for both H + and Na+. For this purpose, both these ions were displaced by successive batch equilibrations with 0.25 M MgC12. I n the combined decanted solutions, H + was titrated with 0.1 N NaOH, and N a + was determined in the flame photometer. All measurements were made with the same five discs in the same sequence. Difficulties were encountered in the conversion from the Na+ to the H + form; here, voids between the discs tend to develop because of solvent depletion of the discs, and it was necessary to comless the stack repeatedly by pressing it gently against the ucite cell. Measurements of Transient Changes in the Co-ion and Water Contents.-The transient behavior of the co-ion and water contents of the resh was studied at 25" for complete conversion from the Na+ to the H + form with 1N HCl, and from the H + to the Na+ form m t h 1 N NaCl. Single discs 0 5 1.0 rather than stacks were used. The disc was equilibrated with 1 M NaCl (or NCl), was wiped with filter paper, and Average Equivalent F m c t i o n of HI ~n D i s c . then was attached to a glass stirring rod in the following way. Fig. 2.-Non-equilibrium co-ion and water contents of A Tygon strip (obtained by splitting a Tygon tube lengthdisc in forward and reverse ion exchange. Equilibrium co- wise in half) was wrapped around the perimeter of the disc, ion and water contents (solid circles, broken lines) are given and both ends of the strip were secured to the glass rod by for comparison ( 1 M solutions, 25"). elastics. This "lollipop" was attached to a stirring motor and was immersed in 1 M HCl (or NaCl) in a thermostat, the herent liquid was wiped off n-ith filter paper and sorbed disc serving as the stirrer. Measurements were made with electrolyte was leached out by successive batch equilibra- 1, 2, 5 and 15 min. contact time between the disc and the tions with deionized water. C1- was determined in the com- solution. After removal from the solution, the disc was bined decanted solution by Volhard titration with 0.1. N analyzed for its counterion, co-ion, and water contents a8 described before. AgKOs and 0.05 N SR4SCN. Reagents.-Standard analytical-grade laboratory reaWater Content of the Discs.-Water contents of the discs were determined at 25' in equilibrium with water and with gents were used throughout. 0.1 and 1 nT solutions of HC1, NaCl and HCI-NaC1 mixtures. Restilts and Discussion After equilibration and removal of adherent liquid as described above, the discs were weighed in a stoppered weighThe experimental results are given in Tables ing bottle. The water content of the water-washed H +form I through 111and Figs. 1and 2. was determined directly by drying the disc to constant The foremost aim of this study was a quantitative weight in vacuo over PZOsa t 45" (9 days). The water contents of the other forms were calculated from their wet test of the theory by comparison of observed and weight, the water content and met weight of the water- predicted ion-exchange rates and intra-resin conwashed H + form, the ion-exchange capacity, and the H+, centration profiles. The theoretical predictions N a + and C1-contents. (7) G. Manecke and K. F. Bonhoeffer, 2. Elsktrochem., 66, 476 Electric Conductivity of the Discs.-The conductivities (1951). of the discs in the H + and N a + forms at 25" in equilibrium (8)lJ. W.Lorimer, E.1. Boterenbroodand J. J. Hermans, Discudaiose with 0.1 AT HC1 and NaCl solutions, respectively, were measured. A conventional lucite conductivity cell with Faraday SOC.,21, 141 (1956). 1.o
conversioll
Ht t o Ka' Form
lv'"
P
L-----i
THEORY OF PARTICLE-DIFFUSION CONTROLLED ION EXCHANGE
Jan., 1962
41
TABLE I DISC PROPERTIES Dimensions (Swollen in 0.1 M soln.) Ion-exchange capacity
H + form
Water content
Na+ form
H + form
C1- content
Na + form Specific conductivity Intraresin diffusion constants
H + form Na + form H+ Na +
a
Diameter 2.7 f 0.05 cm. Thickness 0.18 f 0.005cm. 1.10 f 0.035 meq./disc 2.08 f 0.18 meq./g. dry H + form 1.10 f 0.06meq./cm.s when swollen in 0.1 M soln. Water washed 49 f 2 mmoles/meq. disc (71% wt.) In 0.1 M HC1 43.6 f 2 In 1 M HCl 43.0a Water washed 48.4 & 2 In 0.1 M NaCl 44.5 f 2 In 1 M NaCl 43.3” 0.011 i 0.004 mmole/meq. disc In 0.1 M HCl I n 1 M HC1 .450° .010 f 0.004 I n 0.1 M NaCl .50 I n 1 M NaCl .132 i 0.014 Q-1 cm.-1 I n 0.1 M HC1 In 0.1 M NaCl .0195 0.0015i2-I cm.-’ Uncor. 3.0 X cm.2/sec. &lo% Cor. 2.4 X Uncor. 4.4 x 10- &lo% Cor. 3.5 x 10-8
Measured with one disc only.
this correction requires difficult additional measureTABLE I1 OBSERVED AND PREDICTED RATESOF IONEXCHANGE ments which were not made in the present study. Instead, two sets of diffusion coefficients were Experimmts with disc stacks and 0.1 M solutions at 25’ --Degree Time
Obsd.
of oonversion-PredictedUncor. C0r.a
Conversion from H + form to Na + form 1 hr. 20 min. 0.29 0.30 0.27 4 hr. .43 .53 .47 10 hr. .64 .83 .75 Conversion from N a + form to H + form 1 hr. 20 min. 0.21 0.24 0.21 .36 .41 .36 4 hr. 10 hr. .56 .62 .56 a Correction for convection conductivity is applied in calciilation of diffusion coustants. TABLE 111 OBSERVED ION-EXCHANGE RATES Time, min.
1 2
5 15
----Degree of conversionH + form Na form to N a + form t o H +form +
0.36 .51 .86 .97
0.23 .37 .66 .86
require the knowledge of the individual intra-resin diffusion coefficients of the two exchanging ions. These coefficients were calculated from the conductivity measurements with the discs in the monoionic H + and Naf forms. Calculation of counterion diffusion coeficients from conductivity measur ements is a standard procedure.9-11 Unfortunately, a correction for ‘‘convection conductivity”1°-12 must be applied. Calculation of (9) G. Manecke, et al., Z. EZektrochem., 66, 475 and 672 (1951);
Z.physik. Chem. (Leipzig), 201, 193 (1952); Z. physik. Chem. (Frankfurt), 2,336 (19S4). (10) G.J. Hills, et al., Trans. Faraday SOC.,61,719 a n d 1260 (1956). (11) K. 9. Spieglw a n d C. D. Coryell, J . Phyu. Chem., 66, 196 (1952);67,687 (1953).
calculated; the first without correction for con-. vection conductivity, and the second assuming 20% convection conductivity as estimated by Schlogl and SchOdell3 for an almost identical ion exchanger. Both these sets of coefficients were used for calculating theoretical rates and concentration profiles. Fortunately, errors in estimating the correction for convection conductivity have little effect on the predicted rates and hardly any of the predicted concentration profiles. The calculated intra-resin diffusion coefficients of Na+ and H + are included in Table I. The theoretical rates and concentration profiles presented in Parts I and 11 are for spherical beads and thus do not apply to the experiments in the present study. Numerical solutions were calculated for one-dimensional geometry, as in the present experiments, on an IBM-704 electronic computer using the procedure described in Part 11. The results obtained for the parameter values as in the experiments are included in Fig. 1 and Table 11. Kumerical results for other conditions are given in dimensionless form in Fig. 3 and Table IV. The comparison of the observed ion-exchange rates and concentration profiles with the predicted ones in Table I1 and Fig. 1 shows that the theory is remarkably successful. Qualitatively, the predicted differences in rates and concentration profiles between forward and reverse exchange are confirmed experimentally. The most striking features in which the new theory differs from the previous ones thus are verified. experimentally. Exact quantitative agreement cannot be expected (12) G. Schrnid. Z.Elektrochem.. 66, 181 (1952). (13) R. Sch16g1 and U. Schbdel, Z. physik. Ckem. (Frankfurt), 5, 372 (1955).
F. HELFFERICH
42
Vol. 66
TABLE IV CALCULATED DEQREESOF CONVERSION F( T ) , DIMENSIONLESS TIMEST = D A t / X o 2 , AND DIMENSIONLESS RATESdF( T)/d7 FOR VARIOUS VALUES O F THE MOBILITY RATIODA/DBa -DA/DB
10-
F(T)
7
dF/d+
0.05
0.006248 .02447 ,05512 .09758 .1519 .2185 .2971 .3876 .4900 .6038 .7267 .8596 1,003 1.157 1,326 1.514 1,729 1.994 2.374
4.103 2.645 1.373 1.032 0.8267 .6892 .5912 .5181 .4624 .4188 ,3908 .3626 .3365 .3104 .2825 .2509 .2130 .1650 ,09989
.1 .15 .2
.25 .3 .35 .4 .45 .5 .55 .6 .65 .7 .75
.8 .85 .9 .95
-DA/DB = 4r dF/ds 0.003640 ,01449 .03261 .05793 ,09047 .1302 ,1772 .2311 .2019 .3595 .4321 .5115 .5981 .6937
.go08 .9244 1.074 1.269 1.576
-DA/DB T
6.938 3.463 2.308 1.730 1.384 1.154 0.9910 .8709 ,7789 .7052 .6589 .6043 ,5512 .4964 .4376 .3728 .2997 .2159 .1177
0.002594 .01038 ,02334 .04150 .06485 ,09335 .1269 ,1655
zm
2-
dF/dr 9.715 4.827 3.222 2.414 1.930 1.616 1.386 1.220 1.190 0.9830 ,9115 .8241 .7373 .6481 ,5550 ,4570 ,3533 ,2431 ,1256
,2089 .2573 ,3095 .3673 ,4314 .5038 .5871 .6862 .8103 ,9798 1.260
-Da/Dn = I/*-. r dF/dr 0.001563 16.10 .006264 7.991 5.336 .01409 .02505 4.000 3.213 .03910 ,05619 2.691 ,07629 2.314 2.022 .09Y45 1.781 ,1258 1.568 ,1558 1.404 ,1892 .2274 1,220 1,044 ,2718 ,3241 0.8763 ,3872 ,7153 .5607 .4662 ,4125 .5700 ,7191 .2698 .1324 .9790
-DA/Dn r 0.001320 .005258 ,01183 .02100 ,03273 ,04701 .06388 ,08348 ,1061 .1322 ,1618 ,1963 ,2368 .2859 ,3450 .4203 ,5206 ,6662 ,9227
= 1/47 dF/dr
-DA/Dn
19.07 .9.517 6.358 4.786 3.846 3.215 2.74Y 2.377 2.063 1.786 1.571 1.343 1,132 0.9347 ,7542 ,8842 .4249 ,2750 ,1336
0,001121 22.55 .004456 11.26 .01001 7.532 ,01774 5.677 ,02764 4.550 ,03976 3.769 ,05424 3.178 ,07134 2.702 .OY141 2.304 .1149 1.963 ,1422 1.698 .1743 1.436 ,2125 1.194 .2589 0.9776 ,3161 ,7803 ,3693 .5995 .4874 .4327 .6302 .2788 ,1344 ,8852
I
=
~/IQ-
dF/dr
Calculated for counterions of equal valence, complete particle-diffusion control, one-dimensional geometry, and with the initial and boundary conditions t = 0, 0 5 z 5 20, C g = 0; and t 2 0, z > ~ 0 CA , = 0. (z~ = thickness of resin slab.) F = 0.2
~
0
0.5
1.0
DA/DB = 1 / 1 0
DA/DB= 114
0
0.5
1.0
D A / D R= l / z D.menslonless Space Coordinate x / x o ,
Fig. 3.-Calculated
intraresin concentration profiles.
since the simplifying assumptions of the theorynamely, constant individual diffusion coefficients, absence of co-ions in the resin, negligible swelling changes, absence of gradients of counterion activity coefficients-are not completely met and since the correction for convection conductivity is somewhat uncertain. In view of these facts, the complete quantitative agreement for conversion from the Na+ to the H+ form must be considered as fortuitous, The agreement for the (faster) reverse process is, indeed, not as good. The deviations, though minor, are considerably larger than the
analytical error and show definite trends; the predicted rate is too high, and the predicted concentration profiles are slightly too smooth. It would be highly speculative to attribute these deviations to specific physical causes. Possibly, a systematic experimental error is responsible; in the experiments, water is transferred into the disc stack (see farther below), and liquid films with high diffusion resistance might have formed between the individual discs, thus slowing down the exchange. This explanatioii is consistent with the observation that the difference between forward and reverse exchange rates is greater in the experiments with single discs (see Table 111) where, of course, no such retarding films can form. The agreement between experiments and theory is very gratifying. However, it should be kept in mind that the experiments were designed in such a way that the simplifying assumptions of the theory are approximately met. In particular, the following points were considered. The measurements were made with 0.1 N solutions because this concentration is low enough to keep the co-ion content small (at or below 1% of the counterion content, see Table I), and yet is high enough to guarantee complete particle-diff usion control. Furthermore, with H + and Na+ in the type of resin used, the resin swells or shrinks very little when being converted from one ionic form to the other (see Table I). Also, the ion-exchange selectivity coefficient is practically independent of ionic composition, so that possible gradients of ionic activity coefficients balance one another and thus have little effect on the ionic fluxes.6 In more complicated systems, much larger deviations between experiment, and theory must be expected. The measurements of the transient behavior of the co-ion and water contents of the resin during ion exchange were made with 1 M solutions, in order to obtain co-ion contents that are sufficiently high to be measured accurately. The purpose of these measurements was to test the following qualitative predictions.
Jan., 1962
THEORY O F pA4~TICLE-DIFFES1OKCOXTROLLED
As discussed ia detail in Part I, interdiffusion of counterionis in the resin gives rise to an electric potential gradient which slows down the faster counterion and speeds up the slower one and, in this way, enforces the equivalence of the ionic fluxes as required to preserve electroneutrality. I n other words, the unbalance of the purely diffusional fluxes of the two ions is automatically corrected by a superimposed electric transference of both counterions in the direction of diffusion of the slower counterion. The electric potential gradient, of course, also acts on any co-ions in the resin. The sign of the gradient is such that COions are transfemed in the direction in which the faster counterion diffuses. Hence, if a fast counterion enters the resin in exchange for a slower one which goes out into the solution, then co-ions are sucked into the resin by electric transference. Of course, the electric potential gradient decays as ion excliange approaches completion, and sorption equilibrium of the co-ion eventually is reestablished. The resulting effect thus is a temporary eo-ion accumulation in the resin. On the other hanid, in the reverse exchange where the faster couiiterion goes out into the solution, coions are sucked out of the resin, and the resulting effect is a temporary co-ion depletion of the resin. A similar argument can be advanced for the solvent. Electric transierence of counterions under the influence of an electric potential gradient is known to cause a cocurrent convection of the solvent; it is this effect which gives rise to “convection c o n d ~ c t i v i t y ” ~ -and ~ ~ to “anomalous osmosis.” l 4 In jnterdiffusion of counterions (in the absence of an electric current), the automatically superimposed electric transference of counterions is, as pointed out before, in the direction in which the slower couiiterion diffuses. The solvent thus is transferred in this direction. Accordingly, when a fast counterion enters the resin in exchange for a slower one which goes out into the solution, solvent is sucked out of the resin until ion exchange approaches completion, the electric potential gradient, decays, and sorption equilibrium is re-established. The resulting effect is a temporary solvent depletion of the resin. In the reverse exchange, on the other hand, the resulting effect is a temporary sdvent accumulation in the resin. Co-ion depletion thus is accompanied by solvent accumulation, and vice versa. The experimental results are shown in Fig, 2. For comparison, the equilibrium eo-ion and water contents of partially converted discs in equilibrium with mixed HCl-KaC1 solutions (total concentration 1 N ) are included in this figure. The experimental results are in complete qualitative agreement with the theoretical predictions. It is tempting to use the intra-resin diffusion coefficients of H+ and Na+, determined in resins in equilibrium with 0.1 M solutions, for predicting the ion-exchange rates with 1 M solutions also. (14) (a) R. Sohldgl, 2. phystk. C h e n . (Frankfurt), 3, 73 (1955); (b) it is irrelevant whether the electrio potential gradient is caused b y a n external voltage source (as in conductivity measurements) or by a diffusion process within the system in the absenoe of a n electric current (as a n anomalous osmosis and ion exchange): the individual ion has no means of knowing the cause of the electric potential aradient in its environment
ION
EXCIIAXGE
43
0.4
0
CON\’ERSlON Hi T O Na’ FORM. 0 , l b i X SOLCTIOS
8
COSVERSIOS %ini T O H ‘ FORM, O . l b 5 h SOLIITIOU
v
0.2
C O I V E R S I O S Na’ T O S r z + FORM, O . 2 5 N SOLVTIOS
SOLCTIOI
CONVERSION S r 2 + T O Sa+ FORM, 0 . 2 5 N
THEORETICAL CCR\’ES ARE C N E N AS SOLID LIKES 0
! I
1
I
I
1
1
1
1
Here, however the results are disappointing. The observed rates are listed in Table 111. The difference between forward and reverse exchange rates is as predicted, but the observed rates are throughout only little more than half as high as the predicted ones. The magnitude of this deriation is not surprising since the diffusion coefficients must be expected to be concentrationdependent, and since, contrary to the assumption of the theory, the internal co-ion concentration is high. The direction of the deviation was unexpected since, in most cases, the counterion diffusion coefficients are found to increase rather than decrease with increasing concentration of the external solution. l 6 However, a more detailed mathematical analysis shows that, indeed, the presence of substantial amounts of co-ions in the resin should delay ion exchange. The physical cause is that the electric driving force generated by diffusion of the faster counterion is spent in part for transferring co-ions rather than for accelerating the slower counterion. The results of the present study are interesting in still another respect. Results of self-diffusion measurements by Schlogl and Stein16 in a similar resin have been interpreted to indicate that (nonstationary) effusion out of a slab may occur with a higher effective diffusion coefficient than steadystate diffusion across the same slab. Such an effect may arise from particular pore geometries, I n the present study, no such effect was found. On the contrary, the rates of non-stationary ionexchange processes have been predicted correctly on the basis of diffusion coefficients determined by conductivity measurements under conditions which essentially correspond to a steady-state process. The deviations, where existing, are minor and in the opposite direction, Le., the non-stationary process was found to occur with slightly lower diffusion coefficients than the steady-state process. (15) R . Sohlogl, Z. Elektrochen., 67, 195 (1953); M. Tetenbaum and H. P. Gregor, J . P h w Chen., 58, 1156 (1954); N. Ishibashi, T. Seiyama and W. Sakai, J . Electrochem. Soc Japan, 23, 182 (1955); D. Richman a n d H. C. Thomas, J . Phgs. Chem., 60, 237 (1956). Decrease of counterion diffusion coefficients with increasing solution ooncentration WES observed only with strong-base anion exchangers b y I?. Nelson, J . Polgmer Sci., 40, 563 (1959). (16) R. Schlogl a n d B. Stein, Z. physzk. Chem. (Frankfurt), 13, 111 (1957); Z. Elektrochem, 62, 340 (1958); see also Erratum, zbzd., 63, 341 (1959).
44
H. R. BRONSTEIN, A. S. DWOREIN AND M. A. BREDIG
Conclusions The good agreement between experiments and theory shows that the latter is essentially sound and, within the limits given by the simplifying assumptions, is able to account quantitatively for observed ion-exchange rates and intra-resin concentration profiles. In particular, the predicted Characteristic differences in rates and concentration profiles between forward and reverse exchange of two given counterions have been verified experimentally. Since this is the point in which the new theory differs strikingly from the older ones, the superiority of the new theory is clearly established by experimental results. I n terms of physical forces, the new theory differs from the older ones by including the effect of intra-resin electric potential gradients. As qualitatively predicted, these gradients al8o were found to cause characteristic transient changes in the co-ion and water contents of the resin in the course of ion exchange. This additional confirmation of theoretical predictions is further evidence for the importance of the electric potential gradients and thus, indirectly, for the soundness of the theory. Nevertheless, one should not forget that even the new theory gives only limiting laws
voi. 66
for ideal systems and is likely to fail when its simplifying assumptions are not met. Note Added in Proof.-After this paper had been submitted, experimental results obtained by Fedoseeva, et al.,17 came to my attention. These authors have measured various forward and reverse exchange rates with a strongacid resin (KU-2) in bead form and also give individual counterion diffusion coefficients determined independently by a tracer technique. These results provide an additional possibility of testing the theory by comparing observed and yedicted rates. Such comparisons are shown in Fig, 4 for our cases. (The theoretical rates were calculated as described in Parts I and 11; since DH was not measured, it ~ 7, as found in aqueous soluwas assumed that D H / D NE tions and in most strong-acid resins.) The agreement is very satisfactory, except for the conversion from Na+ to Sr2+ form. This deviation, however, may arise in part from experimental errors, since the experimental points of the exchanges involving Sr*+scatter considerably. In any event, the predicted striking differences in rate between forward and reverse exchange are confirmed qualitatively by the experiments in Fig. 4. The same is true for additional experiments with Naf and Ca2+ and with H + and Sr*+, for which quantitative comparisons could not be made for lack of data.
Acknowledgments.-The help of D. Dere, F. D. Lozano, J. J. Parker, J. E. Fortado, J. Edvalson and E. J. Agazzi in carrying out the experiments, and of Mrs. E. B. Harris in programming the numerical calculations is gratefully acknowledged. (17) 0.P.Fedoseeva, E. P. Cherneva and N. N. Tunitskii, Z h w . Fir. Khim., 53, 936 and 1140 (1959).
THE ELECTRICAL CONDUCTIVITY OF SOLUTIONS OF METALS IN THEIR MOLTEN HALIDES. 111. CERIUM-CERIUM TRICHLORIDE' BY H. R. BRONSTEIN, A. S. DWORICIN AND M. A. BREDIG Chemistry Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee Received June 16, 1961
The specific conductivity of solutions of cerium metal in molten cerium trichloride has been redetermined by a method which avoids contact with ceramic material. The measurements were performed with two rigidly mounted parallel molybdenum electrodes immersed in the melt, which was contained in a molybdenum cup. The electrical conductivity of solutions of Cd metal in molten CdClz was redetermined employing a capillary cell technique. These solutions were used as standards to obtain the cell constant of the ['parallel electrode cell." The conductivity of the Ce-CeCl3 solutions a t 855" was found to rise from 1.20 for the pure salt to 5.35 ohm-' cm.-l a t saturation (9.0 mole % Ce). This increase is ascribed to the presence of mobile electrons which may be in equilibrium (Ce3+ e- Ft Ce2+) with divalent metal ions known to exist in other rare earth systems. Low conductivity results reported by previous investigators are attributed to a reaction between melt and ceramic container.
+
Introduction Several aspects of the behavior of solutions of cerium in cerium trichloride have been examined in recent years. Phase behavior2J and densitycomposition relationships4 as functions of temperature on the Ce-CeC18 system have been investigated by Senderoff and Mellors. These investigators also have examined electrical conductance4 and half-cell potentials5 for solutions of cerium in cerium trichloride. From these investigations, in assemblies which required contact of the metalsalt solution with ceramic materials, a strange dependence of electrical conductivity on metal (1) Work performed for the U.S. Atomic Energy Commission at the Oak Ridge National Lrtboratory, operated by the Union Carbide Corporation, Oak Ridge, Tennessee. (2) G.W.Mellors and S. Senderoff, J . Phye. Chem., 63, 1111 (1959). (3) D.Cubicciotti, J . Am. Chem. Soc., 71, 4119 (1949). (4) G. W.Mellors and S. Senderoff, J . Phys. Chem.. 64, 294 (1960). (5) S. Senderoff and G. W. Mellors, J . Electroehem. SOC.,106, 224 (1968).
concentration and the existence of a novel, monovalent rare earth ion (Ce+) have been r e p ~ r t e d . ~ J On the other hand, the recent discoveiy6-* of the solid, non-conductive dichloride and diiodide of Nd and of the metal-like diiodides of La, Ce and Pr does not tend to support the assumption that cerium dissolves in molten @eC&as Ce+. I n view of the apparently anomalous behavior of the Ce-CeC13 system, a re-examination of this system was undertaken in our general program of study of electrical conductivity in molten metalmetal halide systems. In a preliminary phase of this study, it was observed that Ce-CeCla solutions are quite stable in vessels of molybdenum, but that these solutions (6) L. F. Druding and J. D. Corbett, J . Am. Chem. Soc., 81, 5512 (1959). (7) L. F. Druding and J. D. Corbett, ibiil., 83,2462 (1961). (8) J, D. Corbett, L. F. Druding and C. B. Lindahl, J . Invrp. & Nuclear Chem.. 17, 178 (1961).
