J . Phys. Chem. 1986, 90,4163-4166 ment reaction is actually a composite process. The individual steps are
CH3 + C6H6
methylcyclohexadienyl
+ toluene
AH
-
In terms of elementary reactions, the rate constant is k8(CH3 CsH6 H + toluene) = k,(k,/(kb + k,))
+
If we assume that k,/kb is equal to the ratio of rate constants for the decomposition of sec-butyl radical to propylene and the 2butenes,I5 then
k,
= 2.9 X lo9 exp(-5600/T)
4163
L/(mol s)
Unfortunately, there are no low-temperature measurements on the rate constants of the reaction that can be used for comparison. The rate parameters are however larger than for methyl additions to olefins. Registry No. C6HSCHa,108-88-3; H, 12385-13-6. (15) Tsang, W. J . Am. Chem. Soc. 1985, 107, 2872.
CONDENSED PHASES AND MACROMOLECULES Dielectric Constant near the Liquid-Liquid Critical Point in Perfluoromethylcyclohexane Carbon Tetrachloride
+
R. H. Cohn and S. C. Greer* Department of Chemistry and Biochemistry, The University of Maryland a t College Park, College Park, Maryland 20742 (Received: February 12, 1986; In Final Form: April 3, 1986) We have measured the static dielectric constant, e, as a function of temperature for a mixture of perfluoromethylcyclohexane + carbon tetrachloride which was near the critical composition. The measurements extend over the temperature range 6.6 X lo4 < f < 5.5 X lo-*, where t is the reduced temperature ( T - T,)/T,, Tis the temperature, and T, is the critical temperature (302.223 K). The data are consistent with the theoretical prediction of a temperature-dependent term due to critical fluctuations which has a critical exponent (1 - a ) , where a is 0.1 1. For most other critical liquid mixtures for which t(t) has been studied, t ( t ) shows a decrease near T,, below an extrapolation of the “background” behavior. For this mixture, however, the critical fluctuations cause an increase in e(t) near T, which is very small, barely greater than the resolution of the experiment. It is possible that the increase reflects the behavior of the density, rather than the inherent anomaly in t.
-
We note that the effect of the (1 - a ) term is the addition or subtraction of a term, the magnitude of which decreases as t The large fluctuations which develop in a fluid near a critical 0; thus, a positive sign for A3 will cause an apparent decrease in point affect the thermodynamic properties, including the static t(t) below “background” as T, is approached and a negative A3 dielectric constant, e. Thermodynamic1g2 and m i c r o s ~ o p i c ~ ~ ~ will cause an increase in t(t) near T,. The thermodynamic artheories have been developed for the behavior of the dielectric guments of Sengers et al.’ and Mistura2 predict that A3 should constant near a liquid-gas or liquid-liquid critical point. The have the opposite sign from that of dT,/dl?, where E is the electric expected functional form is field. The two measurements which have been reported for dT,/dE* (for 2,2,4-trimethylpentane + nitrobenzene9 and for t/p = Al A2t + A3t(l-4 A4t(1-rr+A) ... (1) aniline cyclohexaneI0) both find dT,/dE2 to be negative, which where p is the density, t = ( T - Tc)/ T, is the reduced temperature, requires A3 to be positive, which leads to a decrease in t near T,. Tis the temperature, T, is the critical temperature, a is the critical Shakhparonov” has argued that the effect of fluctuations in a exponent for the heat capacity5 (at constant volume at a liquidfluid is to decrease its dielectric constant; Lomova and Shakhvapor critical point and at constant pressure and composition at paronov12 add that while this decrease is expected near an upper a liquid-liquid critical point), and A is the correction-to-scaling critical point, an increase would be expected near a lower critical e ~ p o n e n t . ~The . ~ most recent calculations find a = 0.1 1 and A point. = 0.50.s At a liquid-gas critical point, measurements o f t are There is a school of thought that the anomaly in the dielectric made at constant density. For a liquid-liquid critical point, constant should have the same critical exponent as does that in measurements of t are made at constant pressure and the density the electrical re~istivity.’~Measurements of the resistivity at on the left side of eq 1 is not a constant; in fact, the density also liquid-liquid critical points in nonaqueous mixtures14 frequently has a (1 - a) critical term and must be included in eq 1 in order yield the exponent 28 = 0.65, where 8 is the exponent describing to assess unambiguously the behavior of c(t) . l the coexistence curve,5 rather than (1 - a ) . In two cases, the mixtures isobutyric acid + water and phenol + water, the resistivity (1) Sengers, J. V.; Bedeaux, D.; Mazur, P.; Greer, S. C. Physica A: has a ( 1 - a ) a n ~ m a l y . ’ ~ (Amsterdam) 1980, 104A. 573.
Introduction
+
+
+
(2) Mistura, L. J . Chem. Phys. 1973, 59, 4563. (3) Stell, G.; Hoye, J . Phys. Rev. Lett. 1974, 33, 1268. (4) Goulon, J.; Greffe, J.-L.; Oxtoby, D. W. J . Chem. Phys. 1979, 70, 4742. (5) Stanley, H. E. Introduction to Phase Transitions and Critical Phenomena; Oxford: New York, 1971. (6) Wegner, F. Phys. Rev. B.: Solid State 1972, 5, 4529. (7) Ley-Koo, M.; Green, M. S. Phys. Rev. A 1977, 16, 2483. (8) LeGoulou, J. C.; Zinn-Justin, J. Phys. Rev. E Condens. Matter 1980, 21, 3976; J . Phys., Lett. 1985, 46, L-137.
0022-3654/86/2090-4163.$01.50/0
+
(9) Debye, P.; Kleboth, K. J . Chem. Phys. 1965, 42, 3155. (10) Beaglehole, D. J. Chem. Phys. 1981, 74, 5251. (1 1) Shakhparonov, M. I. Russ. J . Phys. Chem. (Engl. Transl.) 1960.34, 706
(12) Lomova, N. N.; Shakhparonov, M. I. Proc. Acad. Sci. USSR 1957, 134, 899.
(13) Kumar, N.; Jayannavar, A. M. J . Phys. C 1981, 14, L785. (14) Shetty, C.;Gunasekaran,M. K.; Vani, V.; Gopal, E. S. R. Pramana 1983, 21, 71 and references therein.
0 1986 American Chemical Society
4164
The Journal of Physical Chemistry, Vol. 90, No. 17, 1986
Experimental measurements of ~ ( tmust ) be made with attention to three potential problems. First, for both liquid-gas and liquid-liquid critical points, attention must be given to possible density gradients due to the field of gravity acting on the very compressible near-critical fluid.I6 Second, for liquid-liquid critical mixtures, the density must be included in eq 1, as discussed above. Third, for liquid-liquid critical mixtures, ionic conductivity can give rise to a low-frequency dispersion (the Maxwell-Wagner effect), which can cause a "spurious" increase in t ( t ) as T, is neared.I7 In only one case has a critical anomaly been detected experimentally at a liquid-gas critical point;18 carbon monoxide has been found to show an increase in t near T,, the functional form of which is consistent with eq 1. There are no reports in the literature of measurements of t( t ) near lower liquid-liquid critical points. Early work on t ( t ) near upper liquid-liquid critical points ignored problems due to gravity, the superposition of the critical anomaly in the liquid density, and ionic conductivity.' However, both the density and the dielectric constant were measured for the system nitrobenzene + hexane;'%21the presence of an anomalous decrease in t near T, (which is much larger than the anomaly in the density) is clear from Piekara's plots,1g but the original data were not published, precluding quantitative comparison with eq 1 . Several other recent studies of t ( t ) at liquid-liquid critical points also show decreases in t near T, but lack complementary density data: nitrobenzene isooctane,z2 nitrobenzene 2,2,4-trimethylpentane,20nitrobenzene cycloheptane,zOnitrobenzene heptane,Iz and nitrobenzene octane.lZ For the systems polystyrene cyclohexane and benzonitrile isooctane, measurements of both the dielectric c o n ~ t a n t ' and ~ . ~ the ~ d e n ~ i t yhave ~ ~ ,been ~ ~ made; for both systems, the data show a decrease in t ( t ) near T, which is consistent with eq 1, but cannot differentiate between a (1 a) anomaly and a 2@anomaly. In the most definitive experiment to date, Thoen, Kindt, and Van Dae124observe an anomalous decrease in t ( t ) for nitroethane cyclohexane: they cite unpublished results that the density anomaly is negligible and obtain a critical exponent for t(t) of ( 1 - a) = 0.89 f 0.02, which clearly excludes an exponent 2@= 0.65. In two previous cases has an increase in t near T, been observed-the systems methanol heptanez7 and methanol + cyc10hexane.l~ For methanol heptane, the data analysis shows a better fit to the data for a 2@ exponent than for a (1 - a) exponent, except when only the data near T, are considered, in which case (1 - a ) is better. For methanol cyclohexane, the data do not allow a distinction to be made between the two exponents. There are problems which make the results for both these systems questionable. First, the density anomaly is not accounted for in the analysis for either system. We can know whether the density will increase or decrease near T, from the sign of dT,/dP, where P is the pressure.26,28 For methanol + cyclohexanez9and for methanol heptane,30 dT,/dP is positive. Positive values of dT,/dP indicate that an increase is to be expected in the density as t 0 in both cases. This increase in density could cause the apparent increase in dielectric constant. Second, as Shetty et al.
+
+
+ +
+
+
+
+
+
+
+
-
+
(15) Anderson, E. M.; Greer, S. C . Phys. Rev. A 1984, 30, 3129. (16) Moldover, M. R.; Sengers, J. V.; Gammon, R. W.; Hocken, R. J. Reo. Mod.Phys. 1979, 51, 79. (17) Thoen, J.; Kindt, R.; Van Dael, W. Phys. Left. A 1980, 76A, 445. (18) Pestak, M. W.; Chan, M. H. W. Phys. Rev. Lett. 1981, 46, 943. (19) Piekara, A . Phys. Reu. 1932, 42, 448. (20) Konecki, M. Chem. Phys. Left. 1978, 57, 90. (21) Ziejewska, Z.; Piotrowska-Szczepaniak, J.; Ziolo, J. Acto Phys. Pol., 4 1979, 56,347 is in disagreement with ref 19 and 20 in that these workers report an increase in c ( t ) near T,. (22) Merabet, M.; Bose, T.K. Phys. Reu. A 1982, 25, 2281. (23) Jacobs, D. T.; Greer, S. C. Phys. Rev. A 1981, 24, 2075. (24) Thoen, J.; Kindt, R.; Van Dad, W. Phys. Left. A 1981, 87A, 7 3 . (25) Greer, S. C.; Jacobs, D. T. J . Phys. Chem. 1980,84, 2888. (26) Miller, B. C.; Clerke, E. A,; Greer, S. C. J . Phys. Chem. 1983, 87, 1063. (27) Balakrishnan, J.; Gunasekaran, M. K.; Gopal, E. S.R. Indian J . Pure Appl. Phys. 1984, 22,286. See also: Balakrishnan, J.; Gunasekaran, M. K.; Gopal, E. S. R. Chem. Phys. Left. 1982,88, 305. (28) Wheeler, J. C.; Griffths, R. B. Phys. Reu. A 1970, 2, 1047. (29) Timmermans, J. J . Chim. Phys. Phys.-Chim. Biol. 1923, 20, 491. (30) Sivaraman, A.; Tiwari, M. K.; Jyothi, S.; Gopal, E. S.R. Ber. Bunsenges. Phys. Chem. 1980, 84, 196.
Cohn and Greer point in the data analysis for both systems, the coefficient of the correction-to-scaling term is found to be larger than that of the leading critical term, contrary to the expectation that the coefficients of higher order terms should decrease in magnitude. This problem, along with the result for methanol heptane that the exponent changes with the range of the data, could be related to an unrecognized Maxwell-Wagner contribution." These two mixtures are quite conductive, and the measurements were made at a relatively low frequency of 100 kHz, so a Maxwell-Wagner dispersion, which would cause an apparent increase in t ( t ) , is possible. In summary, experimental studies of the dielectric constant near a fluid critical point support eq 1 and show an increase in t ( t ) near T, (which requires that A3 in eq 1 be negative) for the liquid-gas critical point and a decrease in c ( t ) near T, (A3positive) for every liquid-liquid critical point for which the data are reliable. Theoretical treatments suggest a decrease in c ( t ) near T, for all the systems studied experimentally. We report here new measurements of the dielectric constant as a function of temperature near the liquid-liquid critical point carbon tetrachloride. The of perfluoromethylcyclohexane temperature range is 6.6 X lo6 < t < 5.5 X lo-*. The resolution of the t data is 3/105. The data show a very small (barely greater than the experimental resolution) increase in t ( t ) near T,, the functional form of which is consistent with eq 1 . The density has been measured for this system by Thompson and Rice:' who found no anomaly in the density within their resolution of 1/105 and < t
