1978
The Journal of Physical Chemistry, Vol. 83,
No. 15, 1979
values are available for a compound. Even here, the unavailability of accurate heat capacity data for conversion of 82 "C values to 25 "C may impair the results. from AHf(g) values are often quite good, but some values (see n-hexadecane, Table I) have large uncertainties, and are quite out of line with the probable values. u H , ( g ) derived from Turner's acetic acid values are within 0.5-0.6 kcal/mol of the experimental gaseous values converted to 25 "C, in the two cases (cyclohexene, cycloheptatriene) for which data are available. AHH,(g) values derived from AHH,(HOAc) are almost always in better agreement with AHH,(g) from other sources, than are the uncorrected HH,(HOAc) ~ a l u e s . ~ * ~ , ' J ~ It is worth noting that conversion of AHH,(HOAc) to AHHz(g)often requires a correction of 1 kcal/mol or more for cyclic alkenes, and as much as 2.5 kcal/mol for the 1-alkenes. These are differences which should not be ignored, nor assumed to be negligible. Heats of Formation. The gaseous heat of formation of trans-2,2,5,5-tetramethyl-3-hexene (trans-1,2-di-tert-butylethylene) (-39.56 f 0.64 kcal/mol) has recently been determined.g This and the value of A",(g) can be combined to give AHf(g) for the hydrogenation product, 2,2,5,5-tetramethylhexane (46.41 f 0.66 kcal/mol), which, to our knowledge, has not previously been reported. The similarity in AH8(HOAc) (Table I) for these two compounds has previously been observed in other solvents.1° Conflicting values for AHf(g) of 1,3,5,7-~yclooctatetraene (+71.13 f 0.33, +69.6 f 1.4 kcal/mol) have been reported.13 We may use mH,(g) for this compound (Table I), and AHf(g)for the hydrogenation product, cyclooctane (-29.73 f 0.28 kcal/mol13), to calculate a new value of AHf(g) for cyclooctatetraene, +69.37 f 0.31 kcal/mol. Endo Vs. Exo Double Bonds. Turner16 measured AHH,(HOAc) of 1-ethylcyclopentene, ethylidenecyclopentane, 1-ethylcyclohexene, and ethylidenecyclohexane,
Y. M. Joshi and J. C. T. Kwak
and found the endo double bond is hydrogenated 1.2-1.3 kcal/mol less exothermically in both five- and six-membered rings. The greater stability of the endo double bond in the five-membered ring is also implied by the results in cy~lohexane,~ although the difference appears to be only 0.7 kcal/mol. These differences are essentially unchanged when the calculated gaseous values are considered. On the basis of the values of AHf(g),9the enthalpy difference is 0.39 f 0.33 kcal/mol in the C5 rings, and 0.47 f 0.32 kcal/mol in the C6 ring, again favoring the endo double bonds. Although the exact endo-exo enthalpy differences are somewhat in doubt, it is probable that Turner's conclusions were correct. Acknowledgment. This study was supported by The Robert A. Welch Foundation (Grant No. E-136).
References and Notes (1) J. L. Jensen, Prog. Phys. Org. Chem., 12, 189 (1976). (2) J. 5.Conant and G. 9. Kistiakowsky, Chem. Rev., 20, 181 (1937). (3) R. 5.Turner, W. R. Meador, and R. E. Winkler, J . Am. Chem. Soc., 79, 4116 (1957). (4) E. Bretschneider and D. W. Rogers, Mikrochim. Acta, 480 (1970). (5) D. W. Rogers and F. J. McLafferty, Tetrahedron, 27, 3765 (1971). (6) D. W. Rogers and S. Skanupong, J. Phys. Chem., 78, 2569 (1974). (7) R. B. Williams, J . Am. Chem. Soc., 64 1395 (1942). (8) H. A. Skinner and A. Snelson, Trans. Faraday SOC.,55, 404 (1959). (9) R. Fuchs and L. A. Peacock, Can. J. Chem., in press. (10) P. P. S.Saluja, T. M. Young, R. F. Rodewald, F. H. Fuchs, D. Kohli, and R. Fuchs, J . Am. Chem. Soc., 99, 2949 (1977). (1 1) L. A. Peacock and R. Fuchs, J. Am. Chem. Soc., 99, 5524 (1977). (12) R. Fuchs and L. A. Peacock, Can. J . Chem., 56, 2493 (1978). (13) J. D. Cox and G. Pilcher, "Thermochemistry of Organic and Organometallic Compounds", Academic Press, New York, 1970. (14) M. Mansson, P. Sellers, G. Stridh, and S. Summer, J. Chem. Thermodyn., 9, 91 (1977). (15) D. W. Rogers, 0. A. Dagdagan, and N. L. Allinger, J . Am. Chem. Soc., 101, 671 (1979). (16) R. 5.Turner and R. H. Garner, J . Am. Chem. SOC.,79, 253 (1957); 80. 1424 11958). (17) R. B . Turner, D.'E.Nettleton, Jr., and M. Perelman, J . Am. Chem. Soc., 80, 1430 (1958).
Mean and Single Ion Activity Coefficients in Aqueous Mixtures of Sodium Chloride and Sodium Pectate, Sodium Pectinate, and Sodium Carboxymethylcellulose Y. M. Joshi and Jan C. T. Kwak" Department of Chemistry, Dalhousie University, Halifax, Nova Scotia, Canada, B3H 4J3 (Received November 8, 1978; Revised Manuscript Received April 18, 1979) Publication costs assisted by the National Science and Engineering Research Council of Canada
Mean ion, counterion, and co-ion activity coefficients in aqueous solutions of sodium pectate ( E = 1-63),sodium carboxymethylcellulose (6 = 1-19),and sodium pectinates (6 = 1.20, 0.97, 0.82, and 0.25) with added sodium chloride are reported. Measurements were carried out by an electrochemical cell method with Na-glass, Ag/ AgC1, and calomel electrodes. Two different polyelectrolyte concentrations (mp = 0.0100,0.0250 mol/kg of HzO) were studied and at a given polyelectrolyte concentration the ratio of polyelectrolyteto salt concentration was varied from 0.1 to 5.7. In all systems studied, the co-ion activity coefficient is considerably larger than the counterion activity coefficient. The results are compared to the predictions of Manning's limiting law for activities and of new limiting laws developed by Iwasa et al. The reported difference between co-ion and counterion activity coefficients even for polyions with low charge densities is consistent with the prediction of new limiting laws.
Activity measurements are of considerable importance for the understanding of polyion-mobile ion interactions and for examining various theoretical treatments presented so far for these interactions. Although a large number of activity measurements are now available, in most cases 0022-3654/79/2083-1978$01 .OO/O
only polyelectrolytes of charge density considerably greater than unity were used and often only counterion activities were mea~ured.l-~ For a detailed study of polyelectrolyte systems the determination of both mean and single ion activities and a careful study of the influence of the polyion
0 1979 American Chemical
Society
The Journal of Physical Chemistry, Vol. 83, No. 15, 1979
Mean and Single Ion Activity Coefficients in Polyelectrolytes
charge density is required. The structure and the flexibility of polyions may also influence the result^.^ The aim of this paper is to report a systematic study involving measurements of both mean and counterion activities in polyelectrolytes of a well-defined backbone structure having varying charge density, [, ranging from 0.25 to 1.63. 4 = e2/DbkT,where e is the proton charge, D the dielectric constant, k the Bolzmann constant, T the temperature, and b the average linear charge separation on the polyion. These activity data will be compared with Manning’s and new limiting laws in a later section. The calculation of activity coefficients in Manning’s limiting laws is based on two essential features: (1)the counterion condensation phenomenon and (2) the use of Debye-Huckel linearization of the cylindrical PoissonBoltzmann equation5 According to the first, a critical charge density parameter tcritis defined for a polyelectrolyte as [ = lZ-l( and if the actual value of charge density parameter for a polyion is less than its critical value, no condensation of counterions on the polyions occurs. In this case the interactions of counterions and co-ions with polyions are treated according to Debye-Huckel approximation, which results in the following equation for univalent counterions and co-ions:
y+ = y- = y+ = exp[-1/[X(X + 2)-’] (1) for [ 5 1 where y+,y-,and yh are the counterion, co-ion, and mean ion activity coefficients and X = mp/m,. mp and m, are polyelectrolyte and added salt concentrations, respectively. In essence, the use of the Debye-Huckel approximation in Manning’s limiting laws for polyelectrolytes with [ 5 1 results in the same values of counterion and co-ion activity coefficients, for the simplest case of a uni-univalent added salt. On the other hand, if the charge density parameter for a polyelectrolyte is greater than its critical value, condensation of counterions on the polyions occurs, resulting in [,.+ = tcrit.In this case, interactions of co-ions and remaining counterions with polyions of reduced charge density are described with the Debye-Huckel approximation, leading to the following equations for uni-univalent salt: y+ = ([-lX + 1)(X + exp[-f/z[-lX/(f-lX + 2)] (2) y- = e~p[-1/~[-~X/([-lX+ 2)]
(3)
In essence, for polyelectrolytes with [ > Ecrit, the inequality in counterion and co-ion activity coefficients is caused by the fact that a fraction of counterions are now bound to the polyion, the free ions still having the same interactions as given by eq 1. Calculations of Iwasa and Kwak,6 based on the cluster expansion method, predict that even for polyelectrolytes with [ < 1 a considerable difference in counterion and co-ion activity coefficients will exist. Recently, Iwasa et al.’ have used the first higher term approximation to derive the so-called new limiting laws, which can be expressed as [51 Y+ =
exp[(-0.5
+ 0.3906[(X/(X + 2) - l.O)){[X/(X+ 2))] (4)
y- = exp((4.5 + 0.3906[(X/(X
+ 2) + l.O))([X/(X+ 2))]
Notice that the factor [ in the term 0.3906[(X/(X 1.0) was mistakenly omitted in ref 7.
+
(5) 2) f
1979
[ Z l y+ =
[([-lX
+ l)/(X + I)] exp[([-lX/([-lX + 2)) x (-0.5 + 0.3906([-1X/([-1X + 2) - l.O)}]
+
(6)
y- = exp[([-lX/([-lX 2)) x (-0.5 + 0.3906([-1X/([-1X + 2) + l.O))] (7) In essence, the interactions of polyions with counterions and co-ions are asymmetric and therefore even in the absence of counterion condensation, [ 5 1.0, inequality in y+ and y-exists. In assessing the validity of these higher order limiting laws, a number of factors need to be taken into account. First of all they depend on lnet being 1,Le., on the condensation criterion, since, if [ were larger than one, higher cluster terms not considered in the calculations would tend to diverge. Secondly, the neglect of cluster terms involving more than one polyion may be a severe restriction. Although in the new limiting laws it has been assumed that such terms will not contribute appreciably in systems with added salt, this has not yet been proven. On the contrary, recent work by Manning8 has led this author to the conclusion that the higher term contribution in eq 4-7 should be modified because of the influence of cluster terms containing two polyions, ultimately resulting in the deletion of the factor X/(X + 2) in the (X/(X + 2) f 1)term. Thus Manning’s proposed new formula of the first higher cluster term results in new limiting law y+ values which are again equal to the Manning limiting law (DH) values, but with the difference between y+ and yremaining the same as in the new limiting law formulation6>’for all values of [. Finally, there are of course many other still higher order terms which have not been evaluated. In the light of these considerations, the new limiting laws should be seen as a first attempt to evaluate the influence of higher cluster terms, and modifications either along the line indicated by Manning8 or due to yet higher cluster terms may need to be incorporated. However, the most important conclusion drawn from the higher cluster term calculation, i.e., the inequality of y+ and y- for all values of [ including [ I1, remains unchanged. It is this difference on which we will concentrate in the discussion of our results, since any theoretical calculation based on the Debye-Huckel approximation will predict that y+ and y-are equal when [ C 1. In all above equations, the yaoterm originating from mobile ion-mobile ion interactions has been omitted, as it is more convenient to correct the experimental data for comparison with the limiting laws. Nagasawa et ala9measured counterion and coion activity coefficients in sodium poly(vinylsu1fate)-NaC1 mixtures. The observed co-ion activity coefficients are close to unity and fit new limiting laws well,7whereas counterion activity coefficients are slightly lower than new limiting law predictions and therefore remain much below the Manning’s limiting law values. Ueda and Kobatake’slO co-ion activity coefficients in NaClsodium poly(styrenesulfonate) show better agreement with new limiting laws than Manning’s limiting laws. Counterion activity coefficients on the other hand were found to be dependent on poly(styrenesulfonate) concentration and vary from close to Manning’s limiting law curve to the new limiting law curve.’ Other reports on such measurements show similar behavior, with co-ion activity coefficients close to unity and low values of counterion activity coefficients falling between the two limiting laws.11-16 Single ion activities are nonthermodynamic quantities and their exact determination is not possible. However, if we were to assume that single ion activities do portray reality to a certain extentz8 the large difference in counterion and co-ion activity
1980
Y. M. Joshi and J. C. T. Kwak
The Journal of Physical Chemistry, Vol. 83, No. 75, 7979
TABLE I: Various Structural Parameters for Pectate? Pectinates,a and Carboxymethylcellulose
polymer sodium pectate sodium pectinate sodium carboxymethylcellulose
degree of degree of esterifi- substication, % tution 1 27 41 50.3 84.7
0.99 0.73 0.59 0.497 0.153 0.86
E
Kl
1.63 1.20 0.97 0.82 0.25 1.19
32000 43 000 53 000 62 000 21 000 M-
N*
a The values of degree of esterification and number average molecular weight for pectate and pectinates were determined for us by Dr. R. Kohn. The method of their determination is described in ref 1 9 and 20.
coefficients observed consistently by various workers is striking. The use of single ion activity coefficients is certainly necessary to examine the predicted equality or inequality of counterion and co-ion activity coefficients in polyelectrolytes of low charge density by the two types of equations presented and hence in determining the applicability of Debye-Huckel approximation or first higher cluster term approximation to polyelectrolyte solutions. Most of the counterion and co-ion activity measurements mentioned above were carried out on polyelectrolytes of charge density parameters considerably higher than unity. Therefore our mean and counterion activity data in polyelectrolytes of low charge density would be of considerable interest in examining the current problem in greater detail. By using poly(ga1acturonate)s of various degrees of esterification, we can study the influence of the charge density parameter for a single, uniform polyion structure, a structure moreover known to possess considerable rigidity, and thus one of the better polyions to conform to the rigid rod model requirements, at least when the degree of esterification is not too high.
Experimental Section Carboxymethylcellulose was obtained from Koch-Light Laboratories, Colnbrook, U.K. Purification was carried out by an ion-exchange method with Dowex 50W-X8 cation-exchange resin and Dowex ANGA 542 anion-exchange resin. The resulting solutions were neutralized with NaOH to pH 7.2. Concentration of the stock solution was determined by ion exchange of diluted stock solution on cation-exchange columns followed by titration with NaOH. Accuracy of the concentration determination is estimated at f0.570.In all the calculations a value of 1.19 is used for the charge density parameter, t , which corresponds to a degree of substitution of 0.86 determined earlier." Various ionic forms of pectate and pectinates of five different degrees of substitution (listed in Table I) were kindly supplied by Dr. R. Kohn of the Institute of Chemistry, Slovak Academy of Sciences, Bratislava, Czechoslovakia. Pectate and pectinates which were not in sodium or hydrogen forms were ion exchanged on cation-exchange columns to obtain pectic and pectinic acids. Purification and concentration determinations were carried out according to the procedures described above. All of these solutions were less viscous than carboxymethylcellulose (especially in hydrogen form) and the accuracy in their concentration determination is estimated to be within f0.3%, slightly better than that of carboxymethylcellulose. Analytical grade NaCl was used without further purification. All the solutions were prepared by weight from the stock solutions of NaCl and sodium polyelectrolytes. All concentrations listed in Table I1 are molalities.
M* M+ .4
1
2
1
0
3
X5 Figure 1. Corrected activity coefficients, y,in NaCI-sodium pectate mixtures: E(pectate) = 1.63. Experimental data: circles, mp = 0.01; triangles, m,, = 0.025; lower open symbols, y+;shaded symbols, yt; upper open symbols, y-. Theoretical curves: N, new limiting laws; M, Manning limiting laws; f,+, and - indicating theoretical curves for y*,y+,and y-,respectively.
Na-glass, Ag/AgCl, and calomel electrodes were used for the potential measurements, which have been described earlier.z2>z3 Calibration curves were determined both before and after every series of measurements on polymer solution and were seen to be within f0.4 mV for Na-glass-Ag/AgCl and within f0.7 mV for Na-glass-calomel electrode combinations. All the potential measurements were carried out in stirred solutions. Activity coefficients of the reference NaCl solutions were taken from Robinson and Stokes.18 All measurements were done in 150-mL jacketed glass vessels. The solution temperature was maintained at 25.0 f 0.1 "C. The volume of the solution for every measurement was at least 80 mL and it was observed that the diffusion of KC1 from the calomel electrode did not cause any problem in the potential measurements, even at concentrations as low as 3 X m. The measurement error is estimated to be f0.007 and fO.O1O in NaCl and Na+ activity coefficients, respectively.
Results and Discussion Mean and counterion activity coefficients measured in NaC1-Na polyelectrolyte mixtures are listed in Table II.31 Activity coefficients were corrected for mobile ion-mobile ion interactions and are plotted against X1lzin Figures 1-6. Also shown in these figures are the theoretical curves obtained from Manning's limiting law eq 1or 2 and 3, and from new limiting law eq 4 and 5 or 6 and 7, depending on whether 4 is less than or greater than unity. It was mentioned in earlier sections that experimental data if compared to the limiting laws must be corrected for mobile ion-mobile ion interactions. The correction procedure, employed by Wellsz1and other workers,22-26can be stated mathematically as Y i C = yaex/7*" (8) where y&exand yac are the experimental and corrected activity coefficients and y*" is the activity coefficient of added salt. Iwasa and Kwak2' have shown that separating out mobile ion-mobile ion interactions is justified by the cluster expansion method. Their equations, however, indicate that the correction term y+" should be taken at the total free mobile ion strength rather than at added salt concentrations, Le., y*" should be taken at the pure salt ionic strength (m, l/zmpt-l) if t > 1 and a t the ionic
+
Mean and Single Ion Activity Coefficients in Polyelectrolytes
The Journal of Physical Chemistty, Vol. 83, No. 15, 1979
1981
1.0
I
.9 .8
N.
Y .7
MMa
M+ N+
.4
'
I
3
2
1
0
.4
1 0
2
1
3
X4
X v2 Figure 2. Corrected activity coefficient, y , in NaCI-sodium pectinate mixtures as a function of [(pectinate) = 1.20. Designations for the experimental points and the theoretical curves are same as in Figure
1.
Flgure 5. Corrected activity coefficient, y, in NaCI-sodium pectinate mixtures as a function of [(pectinate) = 0.82. Designations for the experimental points and the theoretical curves are same as in Figure 1, With [ 5 1 the theoretical curves M+, M-, and M+ become identical.
I
I
N.
1.1
1.1
*
1.0
.9 .8
Ni
.7
M-
.6
Mi M+
1
Y
.6
. N+
.5 * .4
.51
i d 3
0
2
1
Xb
.4
0
.8
Y .7
h:; '8
I
M ---...
N+
.5 .4
1
0
1
x h2
1
2
3
Figure 6. Corrected activity coefficient, y, in NaCI-sodium pectinate mixtures as a function of [(pectinate) = 0.25. Designations for the experimental points and the theoretical curves are same as in Figure 1. With [ 5 1 the theoretical curves M+, M-, and M* become identical.
+
.9 .
~
Xh
Figure 3. Corrected activity coefficient, y, in NaCI-sodium carboxymethylcellulose mixtures as a function of [(CMC) = 1.19. Designations for the experimental points and the theoretical curves are same as in Figure 1.
1.0
,
-
2
3
Figure 4. Corrected activity coefficient, 7 ,in NaCI-sodium pectinate mixtures as a function of ((pectinate) = 0.97. Designations for the experimental points and the theoretical curves are same as in Figure 1. With ( I1 theoretical curves M+, M-, and M* become identical.
strength (m, l/gn,) if < 1. The two correction procedures of course become identical at low X. At high X or in salt-free systems the total ionic strength principle for the correction term becomes unrealistic, but in intermediate cases arguments can be put forward in favor of either method. In all our previous work on colligative properties of salt added polyelectrolyte s o l ~ t i o n swe~ have ~ ~ ~used *~~~~~ the Wells procedure, and in this paper we will also adopt this method. It should be noted here that at the relatively low polyion concentrations used in this study both correction methods lead to y*c and Y + ~ y-c , values which are virtually independent of the polyion concentration, with the possible exception of the C; = 0.25 pectinate system. In the case of divalent counterionsZ8the Wells method leads to yACcurves which are not polyelectrolyte concentration dependent, where a marked concentration dependence would be introduced by using the total ionic strength correction method. Clearly, if the charge density is high and X is low, the contribution of free counterions obtained from the dissociation of polyelectrolyte to the total free mobile ion concentration would be small and hence the change in yao and consequently in ykc due to
1982
The Journal of Physical Chemistry, Vol. 83, No. 15, 1979
additional term 1/2mpt-1would be insignificant. However, the additional terms ‘/2mpt-1 or lfZmpwould cause significant changes in yhcif the polyelectrolyte charge density is low or X is high. For example, in our previous investigations in NaC1-sodium dextran sulfate mixtures25at mp = 0.029 and at X = 0.5, the total ionic strength correction procedure yields a y+cvalue which is higher by 0.007 than the value obtained with the new correction procedure used here. This difference is of the same magnitude as that of our estimated experimental error. At X = 1.0,4.0, and 9.0 the ykcobtained would differ by 0.008, 0.015, and 0.018. A similar behavior is observed at other mp’s although a slight increase in the difference with an increase in mp can be noticed. Dextran sulfate is a highly charged polyelectrolyte, 4 = 2.8, and therefore only a relatively small fraction of the counterions is dissociated in the solution. For low charge density polyelectrolytes, this difference becomes larger, i.e., in sodium pectinate, = 0.97, at mp = 0.025 and at X = 0.48, 0.96, 2.0, and 4.2 the difference between the two procedures is 0.016, 0.025, 0.032, and 0.042, respectively. Since these differences are large, the choice of yho becomes an important factor, and further theoretical guidance is needed to arrive at a correct procedure. As an additional check on the reliability of the measurements, the data obtained for the t = 1.63 pectate were compared to data obtained for a commercial polypectate with 4 = 1.61. This pectate was obtained from the Nutritional Biochemical Corp., Cleveland, Ohio. For both of these pectates mean and counterion activity coefficients at various values of X were measured. For example, at mp = 0.01 and at X = 0.32, 1.0, 3.1, and 5.8, the yfeXvalues for 4 = 1.63 polymer are 0.787, 0.759, 0.709, and 0.690 as compared to 0.780, 0.751, 0.709, and 0.684 for the commercial polymer with 4 = 1.61. These values are in excellent agreement with each other. A similar behavior is observed if y+eXvalues in these polymers are compared at rnp = 0.025. Counterion activity coefficients in these polymers are also in reasonable agreement with each other, although the agreement is not as good as that observed for y+eXvalues, probably reflecting the increased repeatability problems observed in single ion activity measurements as compared to mean ionic activity measurements. For example, at mp = 0.01 and X = 0.32, 1.0, 3.1, and 5.8, values for [ = 1.63 are 0.745, 0.664, 0.552, and 0.508 as compared to 0.746, 0.682, 0.591, and 0.530 for the commercial sample. Our activity coefficients in sodium pectate can be compared with the measurements of Ander et aLZ6 in carrageenan of a similar charge density, [ = 1.60. At X = 1.0,2.0, 3.0, and 4.0 y*c values of Ander et al. are 0.83, 0.79, 0.74, and 0.73 as compared to our values of 0.842, 0.775,0.755, and 0.732. At the same X, y+cvalues of Ander et al. are 0.75, 0.69, 0.63, and 0.62 as compared to our values of 0.736,0.634, 0.587, and 0.564. The two mean ion activity measurements are in good agreement with each other. Counterion activity coefficients on the other hand show noticeable deviations. It should be mentioned here that y+cand y+cvalues mentioned above were corrected according to the Wells procedure and only the values with similar mp)s have been compared. Our results in sodium carboxymethylcellulose can be compared with those of Rinaudo et a1.,l1 also in carboxymethylcellulose but of slightly higher charge density, 4 = 1.38 against our value of 1.19. At X = 5.0, 2.5, 1.7, 1.3, and 1.0 our yhCvalues (E = 1.19) are 0.819, 0.831,0.842, 0.860, and 0.872 compared to Rinaudo et al. values of 0.797, 0.795, 0.819, 0.824, and 0.836, respectively. According to Manning’s limiting laws, we note that an increase in 4 from 1.19 to 1.38 corresponds
Y.
M. Joshi and J. C. T. Kwak
to a lowering in y*c of about 0.025-0.030. Although Manning’s limiting laws may not provide absolute predictions of the experimental data, the lowering in y+ estimated for the increase in 4 from 1.19 to 1.38 would be of reasonable magnitude. The experimental data of Rinaudo et al. are lower than our data by about 0.025 over the entire X range. This indicates an excellent agreement between these two investigations. Moreover, unlike in pectate a relatively small lowering of y*c with increasing X is observed in both works, but the counterion activity coefficients observed by Rinaudo et al. are much higher than our data. Corrected activity coefficients in polyelectrolytes of six different charge densities are shown in Figures 1-6. These data measured at two different mp’s do not show a significant mp dependence. Also shown in these figures are the Manning (eq 1-3) and new limiting law (eq 4-7) curves. It has previously been observed that Manning’s limiting laws provide a reasonable description of y+ data at low X values.21-26At low X the MLL and NLL predictions for y+ are not very different. Our y+ data in Figures 1-5 show a reasonably good agreement with NLL and MLL at X < 2. As X increases, the disparity between experimental and theoretical values increases, and the yICvalues tend to be closer to NLL for the higher 5 systems and closer to MLL for the lower 4 systems. Unlike the other systems the y+ data in Figure 6 ( 4 = 0.25) are lower than both limiting laws, and generally as 4 decreases from 0.97 to 0.82 to 0.25 the yh upward shift observed experimentally is less than what is predicted by both limiting laws. This point will be discussed later. A comparison of co-ion and counterion activity coefficients with the limiting laws can now be made. Although single ion activities are nonthermodynamic quantities and their exact determination is impossible, even a rough estimate of them is important for studying polyion-mobile ion interactionsOz8An interesting feature of the work presented here is that a large inequality in y+ and y-exists for all polyelectrolytes studied. For polyelectrolytes with 4 > 1, Manning’s limiting laws incorporate counterion condensation and as a result of this predict y+ < y-. However, the magnitude of the difference in y+ and ypredicted by Manning’s limiting laws is much smaller than that observed in the actual data. For polyelectrolytes with 4 < 1,Manning limiting laws predict the same values for y+ and y-whereas data still show a large difference. New limiting laws, in agreement with the experimental data, predict a large inequality in y+ and y-. If single ion activities are taken seriously, it is tempting to conclude that although in high 4 systems counterion condensation plays an important role in causing an inequality in y+ and y-, the interactions of polyions with mobile ions play an equally important role. In other words, the first higher term introduces a strongly asymmetrical term in the polyion-mobile ion interaction, and this is in accordance with what is observed experimentally. The question regarding the applicability of the Debye-Huckel approximation for polyelectrolyte solutions thus seems to depend on the meaning assigned to single ion activities. Since single ion activities are nonthermodynamic quantities, it is inappropriate to say that the Debye-Huckel approximation, leading to Manning’s limiting laws, is incorrect. After all Manning’s limiting laws do provide a reasonable description of y h data of univalent counterion systems21-z2B2”26 and an almost exact prediction of divalent counterions ~ y s t e m s In . ~addition, ~ ~ ~ ~ inclusion of the first higher cluster term leads to predicted y+ values higher than what is predicted from the DH approximation, but
Mean and Single Ion Activity Coefficients in Polyelectrolytes
the data are not clearly in better agreement with one or the other. Moreover, a reconsideration of the higher terms contribution may lead to the conclusion that even with the first higher term included, y+ in the intermediate X range will be either the same as the DH predictioqs or higher but not necessarily as high as is calculated from the new limiting laws. The important observation is that the difference between y+cand y-c in polyelectrolyte systems, with f' values of the same polyion backbone ranging from 0.82 to 1.63, is very close to what is predicted by the new limiting laws, and quite at variance with predictions based on the DH approximation. Our data show that in this respect the value 4 = 1 at least for colligative properties is not as singular a point as would be inferred from the predictions of the linearized Poisson-Bolzmann equation. Again this behavior is anticipated by the new limiting laws. It was mentioned earlier that y+ and y+ data for sodium pectinate with f' = 0.25 are fairly low compared to the limiting laws. This polymer has an average charge separation of about 28 A and because of this repulsion between neighboring charges may not be strong enough to prevent some folding of the polymer. This would result in a lower apparent f' and consequently lower yh and y+ values, as is observed in our results. The values of 4 required to fit these data range from 0.5 to 0.65 which correspond to a folding factor of 112 to 2/5. It has been shown previously that even polyelectrolytes of high charge density coil considerably at high ionic strength. For instance, Harrington30 has reported recently that DNA coils by a factor of 113 as the concentration changes from 0.05 to 0.6 M. It may be that in the case of pectinate with f' = 0.25 repulsion between neighboring charges is not strong enough even at low ionic strength to prevent some folding of the polymer molecules. Odijk and Mande14 find that In y*,In y+,and In y-are all lowered to the same extent by chain flexibility effects. Given the fact that for our case o f f ' = 0.25 only one in every six glucosides has a charged carboxylic group, a lowering of In y+,In y-,and In ?* by about a factor 2 is not incompatible with the sample calculation of Odijk and Mandel.
Acknowledgment. The authors express their gratitude to Dr. R. Kohn for preparing and characterizing the po-
The Journal of Physical Chemistry, Vol. 83, No. 15, 1979
1983
lypectate and polypectinate samples. One of the authors (Y.M.J.) is grateful to the Killam trust for a Postgraduate Fellowship (1976-1979). We thank Drs. K. Iwasa and G. S. Manning for extensive comments and discussions leading up to the final version of this paper.
Supplementary Material Available: Table I1 containing experimental mean and counterion activity coefficients in NaCl-sodium polyelectrolyte mixtures (3 pages). Ordering information is available on any current masthead page. References and Notes (1) A. Katchaisky, Z. Alexandrowicz, and 0. Kedem in "Chemical Physics of Ionic Solutions", B. E. Conway and R. G. Barradas, Ed., Wiley, New York, 1966, p 295. (2) N. Ise, Adv. folym. Sci., 7, 536 (1971). (3) G. S. Manning, Annu. Rev. fhys. Chem., 23, 117 (1972). (4) T. Odijk and M. Mandel, fhysica, 93A, 298 (1978). (5) G. S. Manning, J . Chem. fhys., 51, 934 (1969). (6) K. Iwasa and J. C. T. Kwak, J . fhys. Chem., 81, 408 (1977). (7) K. Iwasa, D. A. McQuarrie, and J. C. T. Kwak, J. fhys. Chem., 82, 1979 (1978). (8) G. S. Manning, private communication. (9) M. Nagasawa, M. Izumi, and I. Kagawa, J . folym. Sci., 13, 563 (1954). (10) T. Ueda and Y. Kobatake, J . fhys. Chem., 77, 2995 (1973). (11) M. Rinaudo and M. Milas, Chem. fhys. Lett., 41, 456 (1976). (12) T. J. Podlas and P. Ander, Macromolecules, 3, 154 (1970). (13) I. Kagawa and K. Katsuura, J . folym. Sci., 17, 365 (1955). (14) I. Kagawa and K. Katsuura, J . folym. Sci., 9, 405 (1952). (15) J. W. Lyons and L. Kotin, J . Am. Chem. Soc., 87, 1670 (1965). (16) M. Nagasawa, M. Izumi, and I.Kagawa, J . Polym. Sci., 37, 375 (1959). (17) J. C. T. Kwak and A. J. Johnston, Can. J . Chem., 53, 792 (1975). (18) R. A. Robinson and R. H. Stokes, "Electrolyte Solutions", 2nd ed, Butterworths, London, 1959 (19) R. Kohn and I.Furda, Collect. Czech. Chem. Commun., 32, 1925 (1967). (20) R. Kohn and 0. Luknar, Collect. Czech. Chem. Commun., 40, 959 ( 1975). (21) J. D. Wells, f r o c . R . SOC. London, Ser. B , 183, 399 (1973). 1221, J. - C. T. Kwak. J . fhvs. Chem.. 77. 2790 (19731. (23) J. C. T. Kwak,'M. C. O'Brien, and D. A. MacLean,'J. fhys. Chem., 79, 2381 (1975). (24) G. E. Boyd, J . Solution Chem., 6, 95 (1977). (25) Y. M. Joshi and J. C. T. Kwak. Bloohys. Chem., 8, 191 (1978). (26) M. Kowblansky, M. Tomasula, and P. Ander, J. fhys. Cbem., 82, 1491 (1978). (27) K. Iwasa and J. C. T. Kwak, J . fhys. Chem., 80, 215 (1976). (28) H. S. Frank, J. fhys. Chem., 67, 1554 (1963). (29) J. C. T. Kwak and R. W. P.Nelson, J . fhys. Chem., 82, 2388 (1976). (30) R. E. Harrington, Biopolymers, 17, 1919 (1978). (31) See paragraph at end of text regarding supplementary material. \-
