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Mean Activity Coefficients for Electrolytes (120) A. Chiba, T. Ueki, T. Ashida, Y. Sasada, and M. Kakudo. Acta Crystallogr., Sect. 8,22, 863 (1967). (121) S. T. Rao, R. Srinivasan, and V. Vaiambal, lnd. J. Pure Appl. Phys., 6, 523 (1968). (122) M. S. Lehmann, T. F. Koetzle, and W. C. Hamilton, J. Cryst. Mol. Struct., 2, 225 (1972). (123) M. Ramanadham, S. K. Sikka, and R. Chidambaram, Acta Crystallogr., Sect. 8,28, 3000 (1972). (124) J. J. Verbist, M. S. Lehmann, T. F. Koetzle, and W. C. Hamilton, Acta Crystallogr., Sect. 8,28, 3006 (1972). (125) G. Kartha and A. de Vries. Nature (London), 192, 862 (1961). (126) T. F. Koetzle, M. N. Frey, M. S. Lehmann, and W. C. Hamilton, Acta Crysta/logr., Sect. 8,29, 2571 (1973). (127) K. Oda and H. Koyama, Acta Crystallogr., Sect. 8,26, 639 (1972). See this paper for other references. (128) M. S. Lehmann, T. F. Koetzle, and W. C. Hamilton, ht. J. Peptide Protein Res., 4, 229 (1972). (129) C. H. Wei, D. G. Doherty, and J. R. Einstein, Acta Crystallogr.. Sect. 8, 28, 907 (1972). (130) R. E. Stenkamp and L. H. Jensen, Acta Crystallogr., Sect. 8,29, 2872 (1973). (131) P. M. Cotrait and Y. Barrans, Acta Crystallogr., Sect. B, 30, 1018 (1974). (132) Y. Harada and Y. litaka, Acta Crystallogr., Sect. B, 30, 726 (1974). (133) P. M. Cotrait and J. P. Bideau. Acta Crystallogr., Sect. 8,30, 1024 (1974). (134) D. W. Smits and E. H. Wiebenga, Acta Crystallogr.. 6, 531 (1953). (135) J. A. Hamilton and L. K. Steinrauf, Acta Crystallogr., 23, 817 (1967). (136) T. Takigawa, T. Ashida, Y. Sasada. and M. Kakudo. Bull. Chem. SOC. Jpn., 39, 2369 (1966).
(137) J. Zussman, Acta Crystabgr., 4, 493 (1951). (138) J. Donohue and K. N. Trueblood, Acta Crystallogr., 5, 419 (1952). (139) J. Fridrichsons and A. McL. Mathieson, Acta Crystallogr., 15, 569 (1962). (140) R. L. Kayushina and B. K. Vainshtein, Kristallografiya, I O , 833 (1965). (141) T. Ueki, T. Ashida, M. Kakudo, Y. Sasada, Y. Katsube, Acta Crystallogr., Sect. 8,25, 1840 (1969). (142) T. Matsuzaki and Y. litaka, Acta Crystallogr., Sect. B, 27, 507 (1971). (143) J. J. Verbist, M. S. Lehmann, T. F. Koetzie. and W. C. Hamilton, Nature (London), 235, 328 (1972). (144) T. F. Koetzle, M. S. Lehmann, and W. C. Hamilton, Acta Crystallogr., Sect. B, 29, 231 (1973). (145) G. Kartha, T. Ashida, and M. Kakudo, Acta Crystallogr., Sect. 8,30, 1861 (1974). (146) E. Benedetti, M. R. Ciajolo, and A. Maisto, Acta Crystallogr., Sect. B, 30, 1783 (1974). (147) I. L. Karle, J. Am. Chem. SOC.,94, 81 (1972). (1481 G. Kartha. G. Ambadv. and P. V. Shankar, Nature (London), 247, 204 (1974). (149) G. N. Ramachandran and V. Sasisekharan, Adv. Protein Chem., 23, 283 (1968). (150) W. R. Krigbaum. R. J. Roe, and J. D. Woods, Acta Crystallogr.. Sect. 8, 24, 1364 (1968). (151) P. Ganis, G. Avitabiie, E. Benedetti, C. Pedone, and M. Goodman, Proc. Natl. Acad. Sci., U.S., 67, 426 (1970). (152) J. J. Stezowski and R. E. Hughes, private communication. (153) D. D. Jones, i. Bernal, M. N. Frey, and T. F. Koetzie, Acta Crystallogr., Sect. B,30, 1220 (1974). (154) S. S. Zimmerman and H. A. Scheraga, J. Am. Chem. SOC., to be submitted for publication. ~I
Mean Activity Coefficients for the Simple Electrolyte in Aqueous Mixtures of Polyelectrolyte and Simple Electrolyte. The Systems Potassium ChloridePotassium Poly (styrenesulfonate), Magnesium Chloride-Magnesium Poly (styrenesulfonate), and Calcium Chloride-Calcium Poly(styrenesulf0nate) Jan C. T. Kwak,” Mary C. O’Brlen, and David A. MacLean Depattment of Chemistry, Dalhousie University, Halifax, Nova Scotia, Canada (Received May 27, 1975) Publication costs assisted by the National Research Council of Canada
Mean molal activity coefficients of the simple electrolyte are reported in aqueous solutions of the K, Mg, or Ca salts of poly(styrenesu1fonic acid) (PSA) with added KC1, MgC12, or CaC12. PSA concentrations range from 0.005 to 0.07 m in KCl-KPSA and from 0.006 to 0.05 m in MgClz-Mg(PSA)Z and CaC12-Ca(PSA)2; PSA:Cl- ratios are varied from excess PSA to excess C1-. An electrochemical cell with a cation-exchange membrane as cation-selective electrode and an Ag-AgC1 electrode is used in all systems. In the divalent ion systems, liquid ion-exchange membrane electrodes for Mg2+ and Ca2+ are used as well. The results for log y + are compared to a “limiting law” derived by Manning, applying a correction for Debye-Huckel type interactions between the small ions. In the MgC12-Mg(PSA)2 and CaC12-Ca(PSA)Z systems good agreement of the experimental data with the corrected form of the limiting law is found over the complete concentration range studied. In the KC1-KPSA systems, experimental results for y~ in mixtures with a large excess of polyelectrolyte are significantly higher than the corrected limiting law values. This was also observed in the NaC1-NaPSA system studied earlier. The measurement or prediction of the activities of the simple ions in mixtures of polyelectrolytes and simple electrolytes is of importance for the calculation of electrical potentials across cell membranes and for the description of ionic transport across these membranes. I t is well known that very considerable deviations from ideality occur even in fairly dilute polyelectrolyte solutions. These deviations could appreciably alter for instance the permeability ratio
of various small ions calculated from resting potentials across membranes in living cells. This paper describes the results of an experimental program designed to measure the mean activity coefficient of the added electrolyte in various mixtures of polyelectrolyte and simple electrolyte, using thermodynamically well-defined electrochemical methods and polyions of which the structural parameters are known. Comparison of our results for the mean activity The Journal o f Physical Chemistry, Vol, 79, No. 22, 1975
Kwak, O’Brien,and MacLean
2382
coefficients to the values predicted theoretically for single ionic activity coefficients should lead to more reliable predictions of both mean and single ionic activity coefficients in the mixtures studied, and this in turn should lead to prediction methods for more complex mixtures. In a previous paper1 we described the determination of the mean activity coefficient of NaCl in mixtures of sodium poly(styrenesu1fonate) (NaPSA) with NaC1. The results were compared to a limiting law derived by Manning2 and were shown to be consistent with Manning’s equation in mixtures where the polyelectro1yte:simple electrolyte ratio was less than 8, if a correction for interactions between the small ions was applied. Wells3 had reached very similar conclusions in an extensive study of the sodium dextran sulfate-NaC1 system. It is of significant practical importance to extend these measurements to systems containing divalent counterions. Condensation theor9 predicts a twofold increase in “condensation” of the divalent ion on the polyion, compared to univalent counterions. This results in a strong lowering of the activity coefficient of the divalent ion but, on the other hand, in a lower effective charge of the polyion and consequently weaker Debye-Huckel type interactions between the polyion and the remaining small ions. It should be stated again5 that no “special” interactions between the divalent ion and the polyion have to exist to lead to very low values for osmotic coefficients or for counterion activity coefficients. Counterion condensation theory and Manning’s limiting laws use purely electrostatic interactions only, and as a result the limiting laws arrived at only depend on the charge density of the polyion and on the valencies of the counterions and coions. Thus, compliance or noncompliance of experimental results with the limiting laws might be used as an indicator for the existence of special, noncoulombic interactions, once the limiting laws have been well confirmed in a number of systems. It is useful briefly to summarize Manning’s limiting laws for the general case of an aqueous mixture of the type M”+P-,+ MZ+”+Yz--,-, where P- is one fixed ion (charge 1-) on the polymer backbone and Yz- the coion (charge z-). m, is the added simple electrolyte concentration, m p the polyelectrolyte concentration (moles of MZ+P-,+ per liter), and X the po1yion:coion concentration ratio, X = z+mp/v+ms.The total counterion concentration mM equals m p v-m,, and the coion concentration m y equals v-m,. The Debye-Huckel screening parameter is given by
+
+ K~
+
= (411e2/DkT)2 nlz12= Xz+2[mp u+m,(l
+ r)] (1)
1
with X = 4 n e 2 / D k T and r = u+/v-. The critical value of the charge density parameter2 ((( = el PI /DkT; I PI is the charge in a unit length of the polyelectrolyte cylinder, D the dielectric constant, k Boltzmann’s constant, and T the temperature), i.e., the value of ( above which counterions will condense on the polyion, is ( = l/z+. When ( > l / z + , counterions will condense leaving an effective polyion concentration mp/( and a noncondensed counterion concentration (m,/z+() u+m,. The effective value of X will be X / z + ( . The general form of Manning’s eq 20 of ref 2 (this is also eq 9 of ref 5 ) is
+
In y L= -1/z(Xz+mpK-2z12 A. T h e case (
(2)
< l / z + . Equation 2 yields
In y+ = -3/2(x[(X/t+) + r ( 1 + r1l-l
(3)
In y- = - f / z t X r z [ ( X / z + )+ r ( 1 + r1l-l
(4)
The Journal of Physical Chemistry, Vol. 79, No. 22, 1975
+ r)-1 In y+ + ( 1 + r)-l In yIn y+ = -1&Xr[(X/z+) + r ( 1 + r)-l]
and with In y+ = r ( l
(5)
Similarly, we find for the osmotic coefficient 4, starting from eq 17 of ref 2
4 = 1 - l / & X [ ( X / z + )+ ( 1 + r1l-l
--
The limits for X lyte) are In y+ 1/2tZ
+.
(6)
-
-
(Le., trace of Yz- in pure polyelectroIn y-l/2(z+r2, and 4 1-
-1/2&+,
B. T h e Case ( > l / z + . Now we can write y+ = y + ( l / z + , X l z + t ) [ ( m , / z + ( ) -I v+ms](mp+ v+m,)-l and y- = y 4 / z+, X / z + ( ) , where y l ( l / z + ,X / z + ( ) is the value of y 1 from case A with ( = l / z + and X = X / z + ( . We arrive a t In y+ = - ( % X / z + ( ) [ ( X / z + ( )+ z + r ( l + r)I-l + In [ ( X / z + ( )+ z+rl(X + z+r)-l
+
In y- = -(1/2r2X/z+[)[(X/z+() z + r ( l + r)]-l In y+ = - ( % r X / z + ( ) [ ( X / z + ( + ) z + r ( l + r)l-I + r ( 1 r)-l In [ ( X / z +f ) z+r]( X
+
+
(7) (8)
+ z+r)-I (9)
With similar reasoning, from eq 17 of ref 2
4 = [ ( K X / z + O+ z + ( l + r ) ] [ X+ z + ( l
-
+ r1l-l
(10)
Again, in the limit X m, we find the relations In y+ ---* - I , - In z+( and 4 l/z(z+()-l and the coion trace activity coefficient In y-l/zr2. Similar relations were derived by Devore.6 The relations for X m in case A, In y+ = 4p-1, and in case B, y+ = Equations 3 and 4 are independent of z + and and eq 7 and 8 show that for the divalent counterion systems studied in this paper (i.e., z+ = 2, r = %), the ion atmosphere of the polyion influences the univalent coion much less than it does in the case of systems with univalent counterions. This leads to limiting law values of y+ a t a given X in divalent counterion systems which are in fact higher than in the case of univalent counterion systems, even though the activity coefficient of a divalent counterion is much lower than the activity coefficient of a univalent counterion. Osmotic coefficients of pure polyelectrolyte solutions with divalent counterions have been measured by a number of authors, reviewed in ref 5 and 7. Generally, the values reported for 4 are fairly close to the predicted value of %(.In particular, with salts of poly(styrenesulfonate), Kozak and Dolars found for Mg and Ca salts of PSA to be in close agreement with the predicted value of 0.09. Reddy, Marinsky, and Sarkarg found the low-concentration limit of 4 for Ca, Sr, Zn, and Cd salts of poly(styrenesu1fonic acid) to be 0.11. 4 of the Mg salt was determined at higher concentrations only and found to be somewhat higher than 4 for the Ca salt. Activity coefficients of the divalent ion have been measured for Zn2+,Cd2+, and Pb2+ salts of PSAlO and for the Mg2+ salts of poly(vinylsu1fonic acid) and DNA.” In this paper we report measurements of the mean activity coefficients of KCl in KCl-KPSA mixtures, of MgClz in MgC12-Mg(PSA)2 mixtures, and of CaC12 in CaC12Ca(PSA)2 mixtures, using an electrochemical technique described earlier.1 In this notation PSA denotes the polymer segment carrying one unit of negative charge. We will compare our results with Manning’s limiting law (eq 9), applying a correction for Debye-Huckel type interactions between the added small ions following the method the
-
.
Mean Activity Coefficients for Electrolytes 0.0
-0.10
-log -0.20
-0.40
2383
0 rnp=0.005 A rnp=O.O1O A rnp=0.020 0 rnp=0.030 rnp=0.050 mp = 0.067
I
1
I
I
I
2
3
4
-X
5
v2
yi(KCI) vs. %”’ ( X = m+SA/rn~I)in KCI-KPSA mixtures: solid line, eq 9; vertical bars, log y+(cor) for mixtures with the same X(eq 11). Figure 1. log
W e l l ~ . ~AJsimilar ~ correction for the interaction between the small ions was applied by Ueda and Kobatake13 to reduce their data for y+ (NaCl) in the NaC1-NaPSA system a t various mp and X values to one curve. Although these authors did not compare their results to the limiting law, their Figure 3 can be compared to Figure 2 of ref 1 or Figure 1 of this paper.
Experimental Section Sodium poly(styrenesu1fonate) was kindly supplied by the Dow Chemical Co., Midland, Mich. (designation SC1585, average molecular weight given as 500000). This was the same batch as was used in previous activity1 and conductance14 measurements. Purification procedures, conversion to the K+, Mg2+, or Ca2+ form, and concentration determinations of the final stock solutions are described in ref 14. “Ultrapure” KC1 (Ventron Co., Beverly, Mass.) and analytical grade MgC12 and CaC12 were used without further purification. The concentrations of MgC12 and CaClz stock solutions were determined to f0.1% by complexometric titration. For each system, the same stock solution was used for preparing the polyelectrolyte mixtures and for preparing the reference solutions used to calibrate the electrodes. All polyelectrolyte-simple electrolyte mixtures were made up by weight from the polyelectrolyte stock solution and the simple electrolyte stock solution. Equilibrium water with a specific conductivity between 1.0 and 1.2 X ohm-l cm-’ a t 25OC was used. Generally, a series of solutions with fixed m p and a number of X values was prepared, the activities were determined, each solution was diluted by weight to a new value of m p (thus keeping X constant), activities were determined, etc. The AgCl electrodes, the membrane cell, and the potential measurement system have been described before.’ The C-322 ion-exchange membrane (American Machine and Foundry Co., Stamford, Conn.) was brought into the desired counterion form by equilibration of the H+ form with KC1 + KOH, MgC12, or CaC12. Reproducibilities of potentials measured with the membrane cell were slightly better for the Na+ and K+ systems than for the Mg2+ and Ca2+
systems, but even with the divalent ion systems they are well within f 0 . 5 mV. In case of the Mg2+ and Ca2+ systems, all activities were measured both with the ion-exchange membrane in the appropriate ionic form as a cation-selective electrode and with an ion-selective liquid membrane electrode (Orion Research Inc., Cambridge, Mass.), using the “divalent cation” exchange solution for Mg2+ and the 92-20-02 exchange solution for Ca2+. Nernstian response of the calibration curve and repeatability with these electrodes were generally less satisfactory than with the ion-exchange membrane. Typically, with CaC12 or MgC12 solution concentrations ranging from 5 X to 2 X 10-1 M, the cation-exchange membrane cell gave Nernst slopes within 1 mV from the theoretical value of 88.8 mV/ decade of activity and potential variations over a number of measurement and calibration series of less than 2 mV. With the same solutions, the liquid membrane electrodes gave Nernst slopes 1-4 mV lower than the theoretical value, and potential drift over a number of measurement and calibration series could be as much as 5 mV. The general procedure followed with both types of measurements was to measure the polyelectrolyte mixture and calibration points, and if drift of the calibration points occurred, this procedure was repeated until constant values were obtained. Still, the repeatability and internal consistency of the results obtained with the liquid membrane electrodes was generally less than was obtained with the ion-exchange membrane cell. Taking this into account, the agreement between the two types of activity measurements was reasonable (see Table 11), but in solutions with higher polyelectrolyte concentrations the liquid membrane electrodes consistently gave values for log yh which were 0.01-0.02 lower than the values determined with the membrane cell, All values reported in Table I and Figures 1-3 were measured with the membrane cell. Activity coefficients for KCl and CaClz solutions were interpolated from literature ~ a 1 u e s . lIn~ the case of MgCI2, activity coefficients at concentrations below 0.1 m were estimated using the Debye-Huckel limiting law slope and the literature values at and above 0.1 m as a guide. This is of course an unsatisfactory method; however, the values adopted resulted in a linear calibration plot of potential vs. activity, with a slope comparable to the slope obtained for CaC12 solutions. We estimate that the MgC12 activity coefficients used have an uncertainty of about f0.01 in y*. Thus, the y*(MgClp) values reported for the polyelectrolyte mixtures are internally consistent but could all be altered slightly if better activity coefficients in pure MgClz solutions became available.
Results and Discussion Table I lists results for log y*(KCl) in 47 KCI-KPSA mixtures with m p varying from 0.005 to 0.067 m and X from 2 1 to 0.3, for log yi(MgC12) in 32 M g c l ~ - M g ( P s A ) ~ mixtures with m p ( = Y ~ ~ P s A )varying from 0.003 to 0.025 m and X from 16 to 0.3, and for log y*(CaCIz) in 40 CaC12Ca(PSA)2 mixtures with m p varying from 0.003 to 0.025 m and X from 16 to 0.06.16 Figures 1-3 show all data points obtained with the ion-exchange membrane as the cation electrode. Table I1 gives a few representative comparisons of the determination of log 75 with the AMF C-322 ionexchange membrane or with the Orion liquid membrane electrode. Other measurement series follow the same trend. Typically, the scatter in the data obtained with the liquid membrane electrode is somewhat larger. The two measureThe Journal of Physical Chemistry, Vol. 79, NO. 22, 1975
Kwak, O'Brien, and MacLean
2384
TABLE 11: Comparison of Mean Activity Coefficients Determined with a Cation-Exchange Membrane (CEM) or with a Liquid Membrane Electrode (LME) as Cation-Selective Electrode
A rnp=0.006 A mp=0.009
0 mp=0.015 0 mp=0.025
-0.10
~
1Y
-0.30
-0.40
0
6 A
0
0 0
3
4
I u 1
2
x%
Figure 2. log y+(MgClp)vs. p" (X = ~ S A / ~ C inI MgC12-Mg(PSA)p ) mixtures: solid line, eq 9; vertical bars, log yh(c6r) for mixtures with the same X(eq 11). 0.C A mp=0.006 A mp=0.009 0
-0.IC
mp=0.025
2.53 16.3 0.358 0.36 2.53 9.0 0.338 0.38 2.53 1.0 0.271 0.30 0.91 16.3 0.331 0.35 0.91 9.0 0.314 0.32 0.91 1.0 0.225 0.24 0.92 0.3 0.229 0.24 0.30 4.0 0.255 0.26 0.30 1.0 0.191 0.18 0.30 0.3 0.182 0.18 a m , given as mol/kg of water.
-0.40
A
O
I
2
+x %
B
O
B
3
4
I
Figure 3. log yk(CaCl2)vs. p'2(X = I-??PSA/~C~)in CaC12-Ca(PSA)2 mixtures: solid line, eq 9; vertical bars, log ?+(cor) for mixtures with the same X(eq 11). ments show a reasonable agreement in case of the series with lower polyion concentrations. A decreased selectivity of the liquid ion exchanger in the case of higher polyion concentrations would lead to the observed lower values of yh, but we have no independent confirmation that this is indeed the cause of the moderate discrepancies found. Table I1 also gives a good indication of the general agreement found between the Mg and Ca systems: our measurements show that in the concentration range studied there is no significant difference between the activity coefficients of the two ions. Only 3 out of the 32 data points for which a comparison between the two systems can be made show a difference outside the estimated limits of error. A similar agreement exists between Na and K systems.l The data obtained for each system show a good internal consistency. The Journal of Physical Chemistry, Vol. 79, No. 22, 1975
~~
2.54 16.0 0.357 0.37 2.55 9.0 0.341 0.36 2.53 1.0 0.288 0.30 0.90 16.0 0.331 0.35 0.90 9.0 0.313 0.33 0.90 1.0 0.233 0.23 0.91 0.3 0.246 0.25 0.30 4.0 0.237 0.24 0.30 1.0 0.184 0.19 0.30 0.3 0.190 0.19
* X = m p s 4 - / m c , - . CPrecision of log -y+(CEM) estimated as *0.007. in MgClz-Mg(PSA)Z is based on estimated activity coefficients in standard MgClz solutions (see text).
ri-
This can be seen especially from the agreement between different dilution series, independently prepared with approximately the same mp and X ,in the KCl-KPSA and the CaC12--Ca(PSA)2 systems (Table I). Typically, the log ya values for two different solutions with the same mp and X are less than 0.005 unit apart. Our data for KC1-KPSA and NaCl-NaPSAl can be compared to the results of Boyd17 for y*(NaCl) in NaCl-NaPSA mixtures calculated from isopiestic data using the McKay-Perring method. Boyd reported values of ya(NaC1) a t X = 0.63, 2, and 5.45 and a total concentration m = m, 0.5mp of 0.1 and 1 m. At low X ,his results for m = 0.1 are in reasonable agreement with our data, but in systems with excess polyelectrolyte his values for y* become very much lower than what is indicated by our results. For instance, at Y , = 0.24 ( Y , = 0.5mpl (0.5mp + m,)) and m = 0.1, Boyd found log y+(NaCl) = -0.17, a t Y , = 0.48 and m = 0.1, log y+ = -0.27, and at Y , = 0.75 and m = 0.1, log y* = -0.40. Comparable estimates from Figure 2 of ref 1 or from Figure 1 of this paper are log y* = -0.18 (X= 0.63, mp = 0.039), log y+ = -0.24 (X = 2, mp = 0.065), and log ya = -0.26 (X = 5.45, mp = 0.085). Boyd's values at high X are much lower even than the limiting law's predictions and would lead to very low values for y- if we accept the approximate value of y+ either calculated from the limiting law (i.e., y+ = 1.21&) or obtained from single-ion activity m e a s ~ r e m e n t s . ~ ~ J ~ J ~ The solid lines in Figures 1-3 are calculated from eq 9 using a value of 2.79 for the charge density parameter (based on 99% sulfonate substitution per monomer unit14). The limiting laws take into account the Debye-Huckel type interactions between the polyion (partially neutralized by the condensed counterions) and remaining small ions. These interactions are responsible for the nonlogarithmic part of the right-hand side of eq 7 and 8. We notice that for the case of univalent counterions and coions these DebyeHuckel interactions lower y+ and y- equally, whereas for the case of divalent counterions and univalent coions the effect is 4 times stronger for the counterion than for the coion. On the other hand, the limiting laws do not account for the Debye-Huckel type interactions between the re-
+
- 0.30
~
Mean Activity Coefficients for Electrolytes
2385
maining small ions. Empirical corrections for these interactions have been applied by plotting log ydmeasured) vs. the total counterion concentration for a series of points at a given X1 and by subtracting log yko (y*O is the activity coefficient of the pure added electrolyte at its concentration in the mixture, m,) from the measured value of log y+. For reasons of convenience, we will adopt the latter method, due to W e l l ~ . ~We J ~write log y*(cor) = log ydmeasd)
- log yho
(11)
At a given X a range of values for log ya(cor) is obtained from points with different polyelectrolyte concentrations, indicated by a vertical bar in Figures 1-3. The values used for log yk0 are also listed in Table I.16 The spread in log y+(cor) values obtained at a given X, indicated by the length of the vertical bar, can be attributed to the estimated uncertainty in log yh of f0.005 in the KC1-KPSA system and of f0.007 in the Mg and Ca systems for each of the measurements points. Again, results for the KC1-KPSA system are virtually identical with those for the NaC1NaPSA system. At low values of X the corrected values of log y,(KCl) are very close to the values predicted by eq 9 using a value of [ based on the known structure of the polyion. However, in mixtures with a large excess of polyelectrolyte, the corrected values of log y* (the correction term is small in these mixtures) are appreciably higher than the values predicted by the limiting law. At high X, there is no significant dependence of log y* on m p , and the tentative conclusion drawn from our previous data for the NaC1NaPSA system that a t high X the positive deviation of log y+ from the limiting law increases with increasing m p is not substantiated. A different situation exists in the case of divalent counterions (Figures 2 and 3). Again, the data points clearly show the influence of the interactions between the small ions on log y+. But here log y*(cor) virtually equals the limiting law value over the complete range of X. The difference in log ydcor) between the Ca and Mg systems is within experimental error for all values of X.It is possible to speculate that the discrepancy in the univalent counterion systems at high X between the limiting law and log y+(cor) occurs because the limiting law overestimates the decrease in y- caused by the polyion field. This would be noticed less with divalent counterions because the increased counterion condensation leaves the polyion with a lower effective charge. There is general agreement that counterion condensation theory correctly describes the fraction of condensed counter ion^.^^^ This seems largely borne out by the dependence of the equivalent conductivity on the charge density on the polyion.20-21Thus, the major deviations from the limiting law should be caused by deviations from the nonlogarithmic term of the right-hand side of eq 7 and 8. Such deviations would be less noticeable in y- especially for 2:l electrolytes (r = $) when compared to 1:l electrolytes ( r = 1). For instance, in the limiting case X -, In y- for a 2:l electrolyte approaches -0.125, whereas In y- of a 1:l electrolyte approaches -0.5. If this reasoning is correct, significant deviations from the limiting law for log y* should be found in systems with univalent counterions and divalent coions. We may conclude that in the concentration range of polyelectrolyte studied (below 0.1 m ) fairly accurate predictions of log y* of the added electrolyte can be made using Manning’s limiting law and a simple correction term for the small ion interactions. These predictions hold for all
po1yion:coion ratios in the case of a 2:l added electrolyte (divalent counterions) and at X < 4 in the case of a 1:1 added electrolyte. A completely a priori prediction of log y+ of the added electrolyte for a mixture of known X and m p would be obtained by simply adding log yko (at m,) to the limiting-law value of log y+ at X (eq 11). In the case of a 1:l added electrolyte at relatively high X values, an empirical estimate obtained from Figure 2 of ref 1 or Figure 1 of this paper would be necessary. Recently, Boyd17 found that values of log y+(NaCl) in NaC1-NaPSA mixtures of constant stoichiometric ionic strength, calculated from isopiestic data, did not conform to the linear form of Harned’s rule.22 Also, both the linear and the quadratic proportionality terms were found to vary very strongly with the ionic strength of the mixture. Applying a similar analysis to our data for log y* of the added electrolyte in the NaC1-NaPSA and KC1-KPSA systems yields relationships between log y+ and Y , at different ionic strength which are linear taking into account reasonable limits of experimental error, but because the polyelectrolyte influences the added electrolyte almost equally at all polyelectrolyte concentrations studied (for a given value of Y,), the proportionality factor a 1 2 increases very strongly upon lowering the total ionic strength. A similar effect was of course noticed by Boyd. For this reason, we feel that Harned’s rule is not very useful in polyelectrolyte mixtures of relatively low concentration. Other prediction methods for the counterion activity coefficient or for the mean activity of the added electrolyte have been based mainly on various forms of the “additivity ru1e”.23-25Manning2i5i7has shown that this rule cannot be interpreted as indicating the absence of interactions between the polyion and the added electrolyte. Application of the rule to mixtures of finite ionic strength leads to ambiguities in the correction for the activity coefficient of the pure added ele~trolyte.’,~~ In the Mg and Ca systems studied here the additivity rule in the form given by Alexandr o w i ~ z(ref ~ ~ 1, eq 1) gives a reasonable representation of the data, using osmotic coefficients of the pure polyelectrolyte determined by Reddy, Marinsky, and Sarkar.g In fact, this form of the additivity rule yields expected values of log y* which are quite close to the corrected limiting law values in all systems studied, except in the Mg system if we employ 4p = 0.15, i.e., assuming a constant difference between 4p of Mg(PSA)2 and Ca(PSA12. I n the Mg system, Reddy et al. only reported 4pat higher concentrations, and if we assume that in this system as well 4pwill drop to the limiting-law value at low mp,8 again good agreement between the corrected limiting law and the additivity rule is obtained. Acknowledgments. The authors are grateful to Dr. G. S. Manning for his continued encouragements and comments and to the Dow Chemical Co., Midland, Mich., for providing NaPSA. This research was supported by the National Research Council of Canada.
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Supplementary Material Available. Table I, a listing of mean activity coefficients, will appear following these pages in the microfilm edition of this volume of the journal. Photocopies of the supplementary material from this paper only or microfiche (105 X 148 mm, 24X reduction, negatives) containing all of the supplementary material for the papers in this issue may be obtained from the Business Office, Books and Journals Division, American Chemical SoThe Journal of Physical Chemistry, Vol. 79, No. 22, 1975
H. R. lhle and C. H. Wu
2386
ciety, 1155 16th St., N.W., Washington, D.C.20036. Remit check or money order for $4.00 for photocopy or $2.50 for microfiche, referring to code number JPC-75-2381. References and Notes (1) (2) (3) (4) (5) (6) (7) (8) (9) (10)
J. C. T. Kwak, J. Phys. Chem., 77, 2790 (1973). G. S. Manning, J. Chem. Phys.. 51, 924 (1969). J. D. Wells, Roc. R. SOC.London, Ser. 13, 183, 399 (1973). F. Oosawa, “Polyelectrolytes”, Marcel Dekker, New York, N.Y., 1971. G. S. Manning in “Polyelectrolytes”, E. Slegny, Ed.. D. Reidel Publishing Co., Dordrecht, Holland, 1974, p 9. D. I. Devore, Ph.D. Thesis, Rutgers University, 1973. G. S. Manning, Annu. Rev. Phys. Chem., 23, 117 (1972). D. Kozak and D. Dolar, 2.Phys. Chem. (Frankfurt am Main). 76, 93 (1971). M. Reddy, J. A. Marinsky, and A. Sarkar, J. Phys. Chem. 74, 3891 (1970). S. Oman and D. Dolar. Z. Phys. Chem. (Frankfurt am Main), 56, 13
(11) (12) (13) (14) (15) (16) (17) (18) (19) (20) (21) (22) (23) (24) (25)
(1967). J. W. Lyonsand L. Kotin. J. Am. Chem. SOC.,87, 1670 (1965). J. D. Wells, Biopolymers, 12, 223 (1973). T. Ueda and Y. Kobatake, J. Phys. Chem., 77, 2995 (1973). J. C. T. Kwak and R. C. Hayes, J. Phys. Chem., 79,265 (1975). R. A. Robinson and R. H. Stokes. “Electrolyte Solutions”, Butterworths, London, 1959. Supplementary material. G. E. Boyd in “Polyelectrolytes“, E. SBlegny, Ed., D. Reidel Publishing Co., Dordrecht, Holland, 1974, p 135. S. Oman and D. Dolar, 2. Phys. Chem. (Frankfurt am Main), 56, 1 (1967). R. Fernandez-Prini, E. Baumgartner, S. Liberman, and A. E. Lagos, J. Phys. Chem., 73, 1420 (1969). G. S. Manning, Biopolymers, 9, 1543 (1970). J. C. T. Kwak and A. J. Johnston, Can. J. Chem., 53, 792 (1975). H. S. Harned and B. B. Owen, “The Physical Chemistry of Electrolyte Solutions”, 3rd ed. Reinhold, New York, N.Y., 1958, p 603. R. A. Mock and C. A. Marshall, and J. Polym. Sei., 13, 263 (1954). M. Nagasawa, M. Izumi, and I. Kagawa, J. Polym. Sci., 37,375 (1959). 2.Alexandrowicz, J. Polym. Sci., 56, 115 (1962).
Rayleigh Distillation Experiments with Respect to the Separation of Deuterium from Dilute Solutions in Lithium H. R. lhle and C. H. Wu” institut 1: Nuklearchemie, institut fur Chemie der Kernforschungsanlage Juiich GmbH, Juiich. Federal Republic of Germany (Received May 27, 1975) Publication costs assisted by Kernforschungsanlage Julich
The ratio of the atom fraction of deuterium in the gas phase, X D , ~ to , that in the liquid, XDJ, R = xD,g/XD,I, was determined by Rayleigh distillation of dilute solutions of deuterium in lithium in the temperature range from 1000 to 1600 K. With an initial atom fraction X D , ~= enrichment of deuterium in the gas phase was observed a t temperatures above 1240 K. The dependence of R on temperature can be expressed by the equation In R = -[(8.47 f 0.45) X 103/T] (6.94 f 0.55). This result agrees well with the R values calculated from the measured vapor pressures of the gas species LiZD, LiD, and D2 over dilute solutions of deuterium in lithium. The thermodynamic supposition that deuterium can be removed from highly dilute solutions in lithium by distillation a t high temperatures is shown by the experimental results to be correct. The same effect is expected for solutions of tritium in lithium.
+
Introduction Our recent investigation of the thermodynamics of dilute solutions of deuterium in lithium’ has led us to the conclusion that a t temperatures above -1240 K the atom fraction of deuterium in the gas phase X D , ~should exceed that in the liquid x1),1 under equilibrium conditions. The unexpectedly high density of deuterium in the saturated vapor arises from the existence of the molecules LizD and LID. The thermodynamic supposition that deuterium can be removed from liquid lithium by distillation should therefore exist. To prove the validity of this assumption, Rayleigh distillation experiments on lithium containing deuterium in low concentrations have been carried out. Experimental Section The ratio R = x ~ ) , ~ / x Dwas , ~ determined in the temperature range -1000-1600 K. R is related to the quantities which are determined experimentally by The Journal of Physical Chemistry. Vol. 79, No. 22, 1975
where N is the number of moles of solution and X D , ~is the atom fraction of deuterium in the solution. Subscript 1 denotes the state prior to distillation; subscript 2, the state thereafter. Application of eq 1 requires that the liquid mixtures exhibit “ideal” solution behavior and maintain uniform composition during distillation. Solutions of deuterium in lithium were prepared by a technique described previ~usly.’*~ The atom fraction X D , ~was determined from the known relation between composition of the liquid and the equilibrium pressures, which were measured by a quadrupole mass spectrometer. The amounts of sample vaporized were determined gravimetrically. The apparatus used is shown in Figure 1. A molybdenum Knudsen cell, 13 mm in diameter, 20 mm high with a 0.6mm diameter knife edge orifice, was loaded to about half its inner volume with solutions of -10 ppm of deuterium
