J. P h p . Chem. 1980, 84, 3508-3516
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transition. The critical exponents of this transition agree fairly well with a simple model of stirred percolation. The values of the critical concentration 4c vary from 0.10 to 0.26, and it is difficult to relate those values to a precise microscopic property of the suspension of droplets. The effect of additional cosurfactant in the course of titration is very strong. It increases steeply the percolation threshold in the case of microemulsion A, B, C, made with pure water, and its decreases the background conductivity in the case of microemulsion B’, made with salted water. (c) In the high-concentration range, the conductivity presents a second transition: the inversion of microemulsion from water in oil to oil in water. The water phase becoming continuous, the increase of the conductivity is steep in this region. 0
0.1
0.2
0.15
0
QW
Effect of additlonal cosurfactant: the electrical conductlvlty at the ower W i n the case of addition of a cosurfactant concentration of 10- (a),compared to the case of no addition of surfactant (b). The slope of the plot remains unchanged, whlle the critical concentration 4,’ increases from 0.078 to 0.092. Flgure 7.
P
background conductivity Gb which is principally modified. This phenomenological analysis of the effect of cosurfactant is difficult to relate to corresponding microscopic properties. As for the discussion of the absolute value of 4c,several different microscopic properties may affect this threshold, for instance, mutual interactions, polydispersity, and nonsphericity of the droplets.
VII. Conclusion We have presented here electrical conductivity measurements of microemulsions in a very wide range of concentrations. The main features of these measurements may be understood with simple models. (a) In the low-dilution limit, the conductivity reflects the mean charge of droplets. This charge is proportional to the number of soap molecules in the droplets. (b) In the medium range of concentration, the steep increase of conductivity is described as a percolation
Acknowledgment. We are indebted to P. G. de Gennes and C. Taupin for stimulating discussions. This work was supported by a grant from the DBlBgation GBnBrale 6 la Recherche Scientifique et Technique.
References and Notes (1) M. Dvoldtzky, M. Guyot, M. Lagues, J. P. Lepesant, R. Ober, C. Sauterey, and C. Taupln, J. Chem. Phys., 69, 3279 (1978). (2) M. Lagues, R. Ober, and C. Taupin, J . Phys. (Paris),Lett., 3QL, 487 (1978). (3) T. P. Hoar and J. H. Schulman, Nature (Paris), 152, 102 (1943). (4) H. Schulman and J. B. Montagne, Ann. N. Y . Acad. Scl., 92, 366 (1961). (5) L. M. Prince, Surfactant Sci. Ser. 0, Part I, Chapter 3 (1976). (6) E. Ruckenstein and J. C. Chi, J . Chem. SOC.,Faraday Trans. 2, 71, 1690 (1975). (7) A. Skoullos and D. Guillon, J . Phys. (Paris),Lett., 38L, 137 (1977). (8) A. Graciaa, J. Lachalse, A. Martinez, M. Bourrel, and C. Chambu, C. R. Hebd. Seances Acad. Sci., Ser. B, 282, 547 (1976). (9) C, Taupln, J. P. Cotton, and R. Ober, J. Appl. Crysfallogr., 11, 613 (1978). (10) C. Taupin and R. Ober, to be submitted for publication. (11) B. Lagourette, J. Peyrelasse, C. Boned, and M. Clausse, Nature (London), 281, 60 (1979). (12) H. F. Eicke and A. Denss, “Solution Chemistry of Surfactants”,Vol. 2, K. L. Mittal, Ed., Plenum Press, 1979, p 699. (13) M. J. Stephen, Phys. Rev. B, 17, 4444 (1978). (14) P. S. Clarke, J. W. Orton, and A. J. Guest, Phys. Rev. 8 , 18, 1813 (1978). (15) M. Lagues, J. Phys. Lett., 40, L331 (1979). (16) J. P. Straley, Phys. Rev. B, 15, 5733 (1977). (17) J. L. Finney, Nature (London), 200, 309 (1977).
Energy-Partitioning Tunneling Model and Prediction of “LOW”A Factors for Intramollecular Hydrogen Transfer Reactions Robert J. Le Roy’ Theoretical Chemistry Department, Oxford University, Oxford OX 1 3TG, England (Received: May 19, 1980: In Final Form: August 12, 1980)
A recently proposed model for intermolecularhydrogen-atom abstraction reactions in solids,which allows for finite reaction rates at T = 0 K, is generalized for the case of any hydrogen-atom transfer process which is initiated by the vibrational stretching of the R-H (or R-D) bond being broken. I& application to data for the isomerization of 2,4,6-tri-tert-butylphenyl radicals yields an effective one-dimensionalbarrier to reaction which quantitatively accounts for H-isotope rate constants for T = 28-247 K and D-isotope rate constants for T = 123-293 K,over which temperature interval the calculated tunneling factors for the H-atom transfer range from to lo2. It similarly accounts for the T = 113-320 K rate constants for the isomerization of octamethyloctahydroanthracen-9-yl radicals. This generalized model incorporates a simple dynamical model for the isomerization reaction frequency factor which provides a plausible microscopic rationalization for the “low” A factors (Le., 0 the H-atom barrier will be distinctly smaller than those seen by the heavier isotopes. Within the present model, Eo = 0 corresponds to the stretching momentum of the donor bond being orthogonal to the reaction path. Thus, the model predicts that nonlinear transition-state geometries would tend to minimize the high-temperature primary kinetic isotope effect. This depmdence of the high-temperature isotope effect on the transition-state geometry is a well-studied phenomenon (for recent examples, see ref 29) which has been previously explained in the context of transition-state the0ry.3~3~~ However, the traditional e x p l a n a t i ~ n ~is ~v~l based on the isotope dependence of the zero-point energy associated with the “symmetric stretch” motion orthogonu2 to the reaction path, and hence depends on the properties of the potential energy surface along a second degree of
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J. Phys. Chem. 1980, 84, 3516-3521
freedom. In contrast, the present model yields the same prediction solely from consideration of the motion along the effective one-dimensional reaction path.
Acknowledgment. I am very grateful to the Theoretical Chemistry Department a t the University of Oxford both for its warm hospitality and for its generous provision of computer time during the course of this work. I am also grateful to Drs. K. U. Ingold, D. J. Le Roy, and F. Williams for their comments on the manuscript.
References and Notes (1) John Simon Guggenhelm Foundation Fellow 1979-80. GuelphWaterloo Centre for Graduate Work in Chemistry, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada. (2) R. J. Le Roy, H. Murai, and F. Williams, J. Am. Chem. Soc., 102, 2325 (1980). (3) E. M. Mortensen and K. S. Pltzer, Spec. Pub/.-Chem. Soc., No. 16, 57 (1962). (4) M. D. Harmony, Chem. Soc. Rev., 1, 211 (1972). (5) R. P. Bell, “The Proton In Chemistry”, 2nd ed, Chapman and Hall, London, 1973. (6) E. Caldin and V. Gold, “Proton-Transfer Reactions”, Chapman and Hall, London, 1975. (7) V. I.Goldanskii, Dokl. Akad. Nauk. SSSR, 127, 1037 (1959). (8) G. Brunton, D. Griller, L. R. C. Barclay, and K. U. Ingold, J . Am. Chem. Soc., 96, 6803 (1976). (9) G. Brunton, J. A. Gray, D. Griller, L. R. C. Barclay, and K. U. Ingold, J . Am. Chem. Soc., 100, 4197 (1978). (10) L. Endrenyi and D. J. Le Roy, J . Phys. Chem., 70, 4081 (1966). (11) K. J. Mintz and D. J. Le Roy, Can. J . Chem., 51, 3534 (1973). (12) P. H. Crlbb, S. Nordholm, and N. S. Hush, Chem. Phys., 29, 43 (1978). (13) T. Shimanouchi, Nati. Stand. Ref. Data Ser. ( U . S . Net/. Bur. Stand.), 6 (1967); 11 (1967); 17 (1968). (14) This harmonic approximation implies both that A and v are Independent of vand that e, = (2v+ l)eo = ( v + ‘/,)hv and E, = (2v l)Eo.
+
(15) Wlth the exception of the three rate constants obtained by “method B” reported in ref 8, whlch those authors identified as unreliable. (16) C. Eckart, Phys. Rev., 35, 1303 (1930). (17) R. J. Le Roy, K. A. Quickert, and D. J. Le Roy, Trans. Faraday SOC., 66, 2997 (1970). (18) R. J. Le Roy, E. D. Sprague, and F. Williams, J. Phys. Chem., 76, 546 (1972). (19) An annotated Fortran listing of the computer program used in these calculations may be obtained from the author on request. (20) R. 0. Weast, Ed., “Handbook of Chemistry and Physlcs”, Chemical Rubber Publlshlng Co., Cleveland, OH, any recent edltlon. (21) Though In view of the different potential shapes, not the values of the barrier width parameter a (22) This correlation does not occur for rate constants at high temperatures where the effect of tunnellng Is minimal and eq 6 collapses to the Arrhenlus-like form: k( T ) = Ae4Vi-Eo)’kBT. (23) Withln the present model, v = 8.8 X loi3 s-I is an upper bound to the range of physically acceptable values of AH. However, the dashed curves in the lower segment of Figure 2 rernaln perfectly flat for A values well beyond 10”: (24) These resuks of Figures 2-4 contradict the Implicit suggestion of ref 8 and 9 that unique values of AH, V1, and a 1couki be obtained from Eo = 0 fRs to the H-isotope data alone. The degree of agreement between the barrier parameters of ref 9 and those obtained here (Table I) therefore merely reflects the previous author’s good physical intuition in their choice of the lnltlal trial A H values used in their analysis. (25) And applied in its A = u (or S = 1) form in ref 2. (26) As for reaction 1, the asymmetric barrlers for reaction 7 were assumed to correspond to ( V , - VI) = 20 kcal mol-I. (27) K. W. Watklns and L. A. Ostreko, J. Phys. Chem., 73, 2080 (1969). (28) Thls implicitly assumes that eo - E, >> kBT, in order that F(E)F= 1. However, for the same geometric reasons ralsed in connection with reactions 1 and 7, it seems most llkely that €deo
