J . Phys. Chem. 1986, 90, 4673-4676 quadrupole moment as described in the last section. There is little concrete justification for this assumption, except that pressureinduced charge rearrangement may be occurring as it does in the other ha10gens.l~ Under such circumstances it would not be unusual to expect 8 to depend on pressure. In summary, this first attempt to calculate the high-pressure properties of solid F2 has shown that good agreement with experiment is obtained on the basis of a description of the interactions that assumes the influence of charge transfer and rearrangement in the condensed phase is negligible. In fact the low-frequency libron mode predicted in this work at all pressures has now been observed in Raman scattering' at all measured pressures. The lone exception to this good agreement is the poor prediction of the pressure dependence of the librons. Whether that is due to a weakness in our description of the orientational part of the potential or is a consequence of pressure-induced charge transfer remains to be seen. It is clear that further progress in the theory
4673
depends on the availability of high-pressure X-ray diffraction, infrared absorption, and ab initio potential data for the condensed phase and the isolated dimer.
Note Added in Proof. A reexamination of the Raman data, cited by ref 1, shows that the identification of the heretofore missing fourth libron mode at low frequency is not unambiguously established. Acknowledgment. This work was supported by the U S . Department of Energy (No. DE-AC02-84ER45050). We thank D. Schiferl, R. Hanson, S. Kincade, and D. Pinnick for supplying the results of Raman scattering measurements prior to publication. We also thank the Colorado State University Computer Center and the Institute for Computation Studies for generously supplying their computer facilities. Registry No. F2,7782-41-4.
Entropies of Dllutlon of Strong Polyelectrolyte Solutions G. Vesnaver* and J. Skerjanc Department of Chemistry, Edvard Kardelj University, Ljubljana, Yugoslavia (Received: January 17, 1986; In Final Form: April 15, 1986)
The free enthalpies of dilution of aqueous alkali metal poly(styrenesu1fonate)solutions have been determined at 0, 25, and 40 "C from osmotic coefficient measurements. From the values of enthalpies of dilution determined previously in our laboratory, the corresponding entropies of dilution have been calculated. An attempt has been made to correlate these values with the changes of water structure around the counterions. It has been found that the entropy of dilution increases when going from Li+ through Na+ and Kf to the Csf counterion and when the temperature is decreased.
Introduction Recent studies of enthalpies of mixing' and temperature dependence of enthalpies of dilution2s3of polyelectrolyte solutions containing monovalent counterions have led to the suggestion that these quantities are partly determined by the solute-solvent interactions which are reflected in the changes of water structure around the counterions. Since the entropy may be regarded as a measure of the degree of disorder in the given state of the system, it seems that these structural changes would reveal themselves more directly in the entropy of dilution values than in the corresponding enthalpy of dilution data. In the present paper we report a comprehensive study of free enthalpies, enthalpies, and entropies of dilution of alkali metal poly(styrenesu1fonate) solutions at different temperatures. The results are discussed with reference to those predicted by the electrostatic theory of polyelectrolyte solutions based on the cell model4and in terms of changes of water structure in the immediate neighborhood of counterions. Experimental Section Alkali metal salts of poly(styrenesu1fonic acid) were prepared from sodium poly(styrenesu1fonate) with a molecular weight of about 70.000 and a degree of sulfonation 1.0, supplied by Polysciences, Inc. (Warrington, PA). Aqueous solutions of lithium (LiPSS), sodium (NaPSS), potassium (KPSS), and cesium (1) Skerjanc, J.; Pavlin, M. J . Phys. Chem. 1977, 81, 1166. (2) Hales, P. W.; Pass, G . Eur. Polym.,l. 1981, 17, 657. (3) Vesnaver, G.; Rudei, M.; Pohar, C.; Skerjanc, J. J . Phys. Chem. 1984,
88, 2411. (4) (a) Fuoss, R. M.; Katchalsky, A.; Lifson, S. Proc. N a f l .Acad. Sci. U.S.A.1951,37, 579. (b) Alfrey, T., Jr.; Berg, P. W.; Morawetz, H. J . Polym. Sci. 1951, 7, 543.
0022-3654/86/2090-4673$01.50/0
(CsPSS) poly(styrenesu1fonates) were prepared by the usual dialysis and ion-exchange techniques.] Their osmotic coefficients were measured a t 25 and 40 OC by Knauer vapor pressure osmometer to which a Honeywell Elektronik 19 lab recorder was connected. The instrument was calibrated by means of aqueous KCl solutions whose osmotic coefficients are known.5 The osmotic coefficient of polyelectrolyte solution, cp, was obtained from the relation Q=
2 m d m
where m is the monomolality of the polyeelectrolyte solution at which the same resistance change was measured as with the reference KCl solution of molality mo and osmotic coefficient cpO.
Results The concentration dependence of the osmotic coefficient of LiPSS, NaPSS, KPSS, and CsPSS at 25 and 40 "C is shown in Figures 1 and 2. It can be seen that at both temperatures the osmotic coefficient does not depend much on the nature of the counterion. Its values for NaPSS, KPSS, and CsPSS are practically the same while those for LiPSS are slightly higher. Another important feature that follows from our measurements is the observation that in the measured temperature range the osmotic coefficient of each salt remains practically unchanged. Comparison of its values measured at 25 and 40 "C with those reported at 0 OC6 shows a reasonably good agreement in the overlapping concentration range, indicating that between 0 and 40 "C the (5) Robinson, R. A.; Stokes, R. H. Electrolyte Solutions,2nd ed.; Butterworths: London, 1959; p 481. (6) Kozak, D.; Kristan, J.; Dolar, D. Z . Physik. Chem. (FrankfurflMain) 1971, 76, 85.
0 1986 American Chemical Society
4674
The Journal of Physical Chemistry, Vol. 90, No. 19, 1986
I
tz25-C
0 0
NaPSS KPSS
Vesnaver and Skerjanc where v is the number of ionic groups on the polyion, y+ and yi0 are the polyelectrolyte mean activity coefficients at the initial and final concentrations, and cp and (ao are the corresponding osmotic coefficients. According to the Gibbs-Duhem equation the ratio of mean activity coefficients can be expressed as
and since v
>> 1 it
follows from eq 2 and 3 that
+ R T 1mo
m
AGD(m+mo) = R T In m o / m
0.21
l S I 3
4
I
f
,
I
,
I
10
15
%
I
I
/
05
-log m
Figure 1. Osmotic coefficient of alkali metal poly(styrenesu1fonate) solutions at 25 O C as a function of monomolality, m,compared with theoretical values (eq 15) for the parameter X as indicated; Xeff = 1.4XS,, = 3.968.
1
(4)
The corresponding expression for the free enthalpy of dilution of a hypothetical uncharged polyelectrolyte solution can be obtained from eq 4 by assuming that in the measured range of low concentrations the osmotic coefficient of uncharged polyelectrolyte is equal to 1. Then, for low concentrations AGDo(M+mo) = R T In m o / m
(5)
and by combining eq 1, 4, and 5 we obtain
t = 40-c
0 0
(1 - p) d In m
NmPSS
R T L r ( 1 - cp) d In m = AGDc(m-+mo) + AGD"(m+mo)
KPSS
(6)
0 CSPSS
where the term on the left can be determined directly from the experimental data. The application of Gibbs-Helmholtz equation to eq 5 shows that the AHD0term should be equal to zero and therefore the measured enthalpy of dilution, AH,, can be expressed according to eq 1 as
L-
---
AHD(m-mo)
1.5
1.0
0.5
-log m
Figure 2. Osmotic coefficient of alkali metal poly(styrenesu1fonate) solutions at 40 O C as a function of monomolality, m,compared with theoretical values (eq 15) for the parameter X as indicated; Xelf = 1.4XS,, = 4.046.
dependence of osmotic coefficient of alkali metal poly(s:yrenesulfonates) on temperature is very small.
Discussion The free enthalpy of dilution, AGD, enthalpy of dilution, AHD, and entropy of dilution, ASD, of a polyelectrolyte solution are defined as the changes of these quantities accompanying the dilution of a solution which contains one monomole of the solute from monomolality m to some reference monomolality mo. They can be formally expressed as AGD(m+mo) = AGDo(m-+mo) + AGDC(m+mo) + AGD"(m+mo) AHD(m-.mo) = AHDo(m-mo) 4- AHDc(m+mo)
AHDn(m+mo)
ASD(m--mo) =
ASDo(m-+mo)+ ASDC(m+mO)+ ASD"(m-.mo) (1)
where AGDo,AHD', and ASDoare the free enthalpy, enthalpy, and entropy of dilution of a hypothetical uncharged polyelectrolyte solution from the same initial to the same final concentration and AGDC, AHDc, and A S D c and AGD", AHD", and ASD" represent contributions due to coulombic and noncoulombic interactions, respectively, caused by the charged species present in the real polyelectrolyte solutions. It can be easily shown that the free enthalpy of dilution of alkali metal poly(styrenesu1fonate) solutions is
= AHDc(m-+mo)
+ AHDn(m+mo)
(7)
With known experimental data for the free enthalpy of dilution (eq 4) and enthalpy of dilution, the corresponding entropy of dilution simply follows from A S D ( ~ - + ~=O(1 ) / 'I?(AHD(m+mo)
- AGdm-mo))
(8)
The A S D O term can be obtained from eq 5
ASDo(m-*mo) = -R In mo/m
(9)
and from eq 1, 4, 8, and 9 it follows that
where both terms on the left are obtained from experimental data. Finally, the coulombic contributions AGDc, AHDc,and ASD' can be estimated on the basis of some electrostatic theory of polyelectrolyte solutions, and then, the corresponding noncoulombic contributions AGD", AHD", and ASD" follow directly from eq 6, 7, and 10. The experimental data for AGD, AH,, and ASD at 25 and 40 "C are summarized in Table I. The initial and final concentrations are m = 0.16 and mo = 0.016 monomoles/kg of H,O, respectively. The AGD values were obtained from eq 4 by graphic integration of the term s(1 - cp) d In m. At 25 and 40 OC this integration was carried out by using the cp vs. m relation reported in this work (Figures 1 and 2) while at 0 OC the corresponding data were taken from literature.6 The AH, values were taken from our previous paper,3 and the corresponding ASD values were calculated from eq 8. As can be seen from Table I the AGD values do not change much with the temperature or type of the counterion while the corresponding changes of its enthalpic and entropic contributions are significant. Their dependence on the temperature and nature of the counterion may be regarded as a reflection of the state of solvation of the counterions. The large positive entropy of dilution increases in the order Cs+ > K+ > Na+ > Li+ and this, along with the corresponding increasingly endothermic AHDvalues, may be according to Frank and Wen' a reflection of the struc(7) Frank, H. S.; Wen, W. Y . Discuss. Faraday SOC.1957, 24, 133.
The Journal of Physical Chemistry, Vol. 90, No. 19, 1986 4675
Entropies of Dilution of Strong Polyelectrolytes ture-breaking abilities of the counterions. Thus, the observed effects can be explained by assuming that at higher concentrations there is some overlapping of regions of structure-broken water around the counterions which upon dilution is reduced. Therefore, at high dilution the structure-breaking abilities of counterions become fully developed, leading to increased disruption of water structure accompanied by an increase of entropy and enthalpy. The Cs+ counterions which are considered to be the strongest structure breakers should give the largest positive contributions to the entropy and enthalpy of dilution, while for other counterions these contributions should decrease in the same order as their structure-breaking abilities, that is from K+ through Na+ to Li+. The described contributions should also be larger at lower temperatures where water is more structured. Comparison of results given in Table I clearly supports such explanation. The coulombic contributions AGDC,AHDc, and hSDC were estimated from the electrostatic polyelectrolyte theory based on the cell model4 which has been successfully used for the interpretation of various thermodynamic properties of polyelectrolyte solutions.* For poly(styrenesu1fonates) with monovalent counterions this theory, which takes into account only coulombic polyion-counterions interactions, predicts that the electrostatic free enthalpy of solution per monomole of polyelectrolyte, G', is9J0
In
(1
- X)2
- p2
1-82
-
x
1
(11)
TABLE I: Values of AGb AH,, and A S D for LiPSS, NaPSS, KPSS, and CsPSS at 273.15, 298.15, and 313.15 K and the Corresponding AGDo, AHDo,and ASDO Values for a Hypothetical Uncharged Polyelectrolyte Solution" LiPSS
J monomole-' NaPSS KPSS CsPSS
273.15 298.15 313.15
AGDo, J monomole-' -5229 -5708 -5995
-1333 -1428 -1500
-1344 -1403 -1474
T, K
AHDO, J monomole-'
LiPSS
273.15 298.15 313.15
0 0 0
-90 -432 -600
T, K
J monomole-' K-'
273.15 298.15 313.15
19.14 19.14 19.14
T, K
ASDO,
X = eo2/ekTb
(12)
103 - -1 -1 In In c,
T, K 273.15 298.15 313.15
x=
The osmotic coefficient, cp, and the electrostatic enthalpy of the solution Hc, obtained from eq 11, arelOJ1 1 -p2 =
y
(1
105 -260 -450
225 -160 -353
345 -70 -260
AS,, J monomole-' K-' LiPSS NaPSS KPSS CsPSS 4.55 3.34 2.87
5.30 3.83 3.27
5.48 4.17 3.58
5.64 4.47 3.88
A~D', J monomole-' 3980 4378 4624
AGD", J monomole-' LiPSS NaPSS KPSS CsPSS -84 -98 -129
-95 -74 -103
-24 -74 -103
54 -74 -103
~~
raZbN, 2
and the constant /3 is related to X and y by 1 -p* 1 + coth By
P
AH,, J monomole-] NaPSS KPSS CsPSS
~
T, K
(
-1195 -1403 -1474
TABLE 11: Computed Coulombic Contributions of Dilution, AGDc, AHD~, and ASD' and the Corresponding Non-Coulombic Contributions ACD", AHD",and A S D for ~ LiPSS, NaPSS, KPSS, and CsPSS at 273.15, 298.15, and 313.15 K"
273.15 298.15 313.15
2
-1273 -1403 -1474
'Initial concentration m = 0.16, and final concentration m = 0.016 monomoles/kg of H 2 0 .
where the charging parameter X and the concentration parameter y are defined as
y =
AGD,
-
(15)
T, K 273.15 298.15 313.15
AHD', J monomole-' -293 -464 -512
ASD', J monomole-' K-' -15.64 -16.24 -16.59
A H D ~J , monomole-' LiPSS NaPSS KPSS CsPSS 203 32 -28
398 204 122
518 304 219
638 394 312
ASSn, J monomole" K-] LiPSS NaPSS KPSS CsPSS 1.05 0.44 0.32
1.80 0.93 0.72
1.98 1.27 1.03
2.14 1.57 1.33
"Initial concentration m = 0.16, and final concentration m = 0.016 monomoles/kg of H20.
d In V and the electrostatic entropy of solution, Se, follows from Se = (l/T)(Hc - Ge) (17) All symbols in eq 11 through 16 have the same meaning as defined By taking a = 0.8 nm, b = 0.252 nm, and E = 87.896, 78.358 and 73.151 at 0,25, and 40 OC,I2we calculated the corresponding osmotic coefficients of alkali metal poly(styrenesu1fonates) (eq 15). A reasonably good agreement with experiment was obtained at all three temperatures (Figures 1 and 2 and ref 6) by using for the charging parameter X effective values 1.4 times higher than the corresponding structural values (eq 12). At 0, 25, and 40 OC (8) Dolar, D. In Polyelectrolytes; Selegny, E., Mandel, M., Straws, U. P., Eds.; Reidel: Dordrecht, Holland, 1974; p 97. (9) Skerjanc, J. J . Phys. Chem. 1973, 77, 2225. (10) Lifson, S.; Katchalsky, A. J . Polym. Sci. 1954, 13, 43. (1 1) Skerjanc, J.; Dolar, D.; LeskovSek, D. Z . Phys. Chem. (Frankfurt am Main) 1967, 56, 207. (12) Owen, B. B.; Miller, R. C.; Milner, C. E.; Cogan, H. L. J . Phys. Chem. 1961, 65, 2065.
these Xeffvaluesare 3.861, 3.968, and 4.046 and they were used in further computations of Ge, He, and Se (eq 11, 16, and 17) together with the following values of the parameters characteristic for water:12 d In c/d In T = -1.257, -1.368, and -1.434 and d In V/d In T = -0.0161, 0.0767, and 0.120 at 0, 25, and 40 "C, respectively. Then, the AGD', AHDe, and AS,' values for the dilution of polyelectrolyte solution from the initial monomolality m = 0.16 to the final monomolality mo = 0.016 were calculated. By assuming that these theoretical values may be equated with the coulombic contributions AGD', AH,', and ASD', we obtained the corresponding noncoulombic contributions AGDn, AH,., and ASD. from eq 6, 7, and 10 (see Table 11). Several features of Table I1 are of interest. The noncoulombic contributions to the free enthalpy of dilution seem to be very small and practically independent of the temperature and the type of the counterion. On the other hand, the noncoulombic contributions to the enthalpy and entropy of dilution show quite different behavior. The AH,'' values are endothermic and of the same order of magnitude as the corresponding exothermic AH,' values. They increase in the order CsPSS > KPSS > NaPSS > LiPSS and with decreasing temperature. Similarly, the AS,'' values are positive and increase substantially from LiPSS through NaPSS
4676
J. Phys. Chem. 1986, 90, 4676-4678
and KPSS to CsPSS. The dependence of AGD", AHD",and ASD" on the temperature and the nature of the counterion is consistent with the one observed with the experimental quantities of dilution AGD, AHD, and ASD (Table I). It appears therefore that it is also associated with the already described structural effects of the counterions. In conclusion, it has to be emphasized that due to the known deficiencies of the cell model all values of coulombic and noncoulombic contributions reported in Table I1 should be considered only as first approximations. We believe, however, that the relative order of noncoulombic contributions with respect to the type of
the counterion and their trends with temperature clearly indicate the importance of counterion-solvent interactions in determining enthalpies and entropies of dilution of aqueous polyelectrolyte solutions.
Acknowledgment. The partial financial support of the Yugoslav American Joint Fund for Scientific Cooperation (Pr. No. 8509373) and the Research Community of Slovenia is gratefully acknowledged. Registry No. LiPSS, 9016-91-5; NaPSS, 9080-79-9; KPSS, 901 199-8; CSPSS, 37286-93-4; HZO, 7732-18-5.
Coexistence Curve of Methanol- Isooctane A. C. Ploplis, P. S. Wardwell, and D. T. Jacobs* Department of Physics, The College of Wooster, Wooster, Ohio 44691 (Received: January 22, 1986; In Final Form: March 26, 1986)
The coexistencecurve of the binary fluid mixture methanol and isooctane (2,2,4-trimethylpentane)was determined by precisely measuring the refractive index both above and below the upper critical consolute point. Thirty-five two-phase data points were obtained on three compositions to determine the location of the critical point: critical temperature = 42.38 O C and critical composition = 33.2% by mass methanol. The simple scaling relationship n,,- nL = Bt@determined the critical exponent p = 0.323 i 0.009 and the amplitude B = 0.090 0.001, where t = (T, - Q/T,is the reduced temperature. Correction-to-scaling terms were not needed to describe the data taken over 3 decades in reduced temperature.
*
Introduction The substantial recent interest in critical phenomena in a multitude of physical systems has been built upon concepts of scaling and universality that developed from studying second-order phase transitions in fluid systems. Wilson' provided a comprehensive framework in renormalization group theory that has been used by many others to predict relationships among exponents, the values exponents should have when they belong to different universality classes, and relationships among the amplitudes of thermodynamic phenomena. Although renormalization group theory does not predict the location of the critical point, all systems behave similarly once they are close to their own critical point. The predictions for the exponents have been well verified in liquid-gas and in binary fluid mixtures, both of which belong to the same universality class (three-dimensional Ising model). The amplitude predictions from two-scalefactor universality have been confirmed in a few systems but violated in others. Several reviews the present status of experiments and theory. In order to determine amplitudes and exponents, it is essential to be near the critical point. The coexistence curve provides the location of the critical point and is the first experiment that must be conducted on a system. The shape of the coexistence curve also provides information on the critical exponent @ and amplitude Bo, since, for simple scaling, the difference in a generalized order parameter, x, between the upper and lower phases goes as a power law in reduced temperature t ( T , - T ) / T c Ax xu - x L = Bot@ (simple scaling) The critical exponent, 8, is predicted to be 0.325 f 0.0015 from renormalization group theory6 applied to a 3-D Ising model and (1) Wilson, K. G. Phys. Rev. B: Solid Stare 1971, 4, 3174. (2) Green, M. S.; Domb, C., Eds. Phase Transition and Critical Phenomena; Academic: New York, 1972; Vol. 1 and subsequent volumes. ( 3 ) Greer, S . C.; Moldover, M. R. Annu. Rev. Phys. Chem. 1981,32,233. (4) Kumar, A.; Krishnamur, H. R.; Gopal, E. S . R. Phys. Rep. 1983, 98, 57. (5) Beysens, D. In Phase Transitions, Cargese 1980; Levy, M.,LeGuillou, J. C., Zinn-Justin, J., Eds.; Plenum: New York, 1981; p 25. (6) LeGuillou, J. C.; Zinn-Justin, J. Phys. Reu. B: Condens. Mutter 1980, 21, 3976.
0.328 f 0.003 from series expansion.' In liquid-gas systems the order parameter is the d e n ~ i t y ;for ~ . ~binary fluid mixtures the proper order parameter is still uncertain although many favor volume fraction since this gives3v4q8a more symmetric Ising-like coexistence curve. By volume fraction we mean the volume of one component divided by the actual volume of the phase it occupies. Others have defined volume fraction as the volume of one component divided by the sum of the components' volumes in that p h a ~ e . ~The . ~ latter definition is equivalent to the former only if there is no volume change on mixing. The experimental study of binary liquid mixtures has several advantages over liquid-gas systems. Many binary liquid mixtures have a critical point at atmospheric pressure, a critical temperature close to room temperature, and (usually) small gravity effects. Also, simple scaling appears to hold for relatively large reduced temperature^.^,^ Since both liquid-gas and liquid-liquid systems belong to the same universality class, the information gained by studying one can be applied to the other. The coexistence curve of a binary liquid mixture can be measured by several technique^.^ The two that are relevant to this work are the following: (1) A set of different composition vials are put in a a water bath with the height of the menisci determing the degree of miscibility and, hence, the coexistence c ~ r v e . ~(2) J~ One sample is prepared and the refractive index can be used to determine the composition of each phase at various temperatures."J2 The last method is the one utilized in the investigation reported here. It has an advantage over the first method because the refractive index is a precise, nonintrusive probe that can measure the properties of a single sample of fixed composition. Investigating one sample also avoids the problem of preparing multiple (7) Nickel, B. G. In Phase Transitions, Cargese 1980; Levy, M., LeGuillou, J. C., Zinn-Justin, J., Eds.; Plenum: New York, 1981; p 291. (8) Greer, S . C. Phys. Rev. A 1976, 14, 1770. (9) Reeder, J.; Block, T. E.; Knobler, C. M. J . Chem. Thermodyn. 1976, 8 , 133. (10) Ngubane, S . B.; Jacobs, D. T. Am. J . Phys. 1986, 54, 542. (11) Jacobs, D. T.; Anthony, D. J.; Mockler, R. C.; O'Sullivan, W. J. Chem. Phys. 1977, 20, 219. (12) Jacobs, D. T. J. Phys. Chem. 1982, 86, 1895.
0022-3654/86/2090-4676$01.50/00 1986 American Chemical Society
