1997
J. Phys. Chem. 1980, 84, 1997
Assuming that all radical pairs are equally reactive, the probabilities P, P, and E’ (eq A4-A6) of formation of RR,
P = (P{/4)[1 + 2 w / ( l + u ) + wZ1”/(l + 21’2v)1 p = (P0’/2)[1 - w21’2/(1 + 2%)1
(A4)
E“’ = (/3{/4)[1 - 2w/(l + u ) + ~ ~ ” ~+/ 21/2u)] ( 1 (A6) RR’, and R’R’ are obtained as integrals of the type JO”h&) q(t) dt, where q(t) and the related q’(t) and q”(t) are the probabilities that a radical pair that is (R. Re) initially is (Re R.), (R. R’.), and (R’. R’.) at time t [see ref 1 for q(t),
q’(t), and q”(t)]. Since P, P’, and P’ are proportional to the molar amounts of RR, RR’, and R’R’ formed, eq A4A6, together with the definitions of r, n, and s, give the required equations, eq 2-4.
References and Notes (1)J. F. Garst, J . Am. Chem. SOC.,97, 5062 (1975). (2) R. M. Noyes, frog. React. Kinet., 1, 129 (1961). (3) K. R. Naqvi, K. J. Mork, and S. Waldenstrerm, “Diffusion-Controlled Kinetics: Equivalence of the Particle Pair Approach of Noyes and the Concentration Gradient Approach of Collins and Kimball,” manuscript, Institute of Physics, University of Trondhem (NLHT), N-7000 Trondheim, Norway, 1979.
COMMlJNICATIONS TO THE EDITOR Comment on “Plastlc Phases In Globular Phosphorus Compounds. A New Structural Criterion for Plastic Behavior”
Sir: Postel and Reiss’ have proposed a new structural criterion for pliastic behavior according to which the ratio R , defined as R = dm/DM where d, is the minimum distance between molecular centers within the crystal lattice, and D M is the diameter of the sphere that will just circumscribe the freely rotating molecule, should be higher than 0.80 for plastic phases. After Ghelfenstein2 pointed out that this criterion discloses the highL temperature reorientrtltional behavior of solid benzene3 (R = 0.81) that both Timmermans’ criteria4 fail to detect, this empirical rule was tested on a series of heterocyclic compounds which are known to exhibit plastic behavior, that lis furan, thiophen (both of which have two plastic phases), and p-dioxane. Table I shows the results of our investigations: First, it can be seen that Postel and Riess’ criterion easily distinguishes the reorientational phase of p-dioxane from the rigid one, ailthough the value R = 0.81 for phase I is disappointingly low for a crystal that flows under its own weight.6 A value higher than 0.90 as iin cyclohexane was expected. Furthermore, this criterion does not reveal any plasticity for the reorientationally disordered phases of furan and thiophen for which Timmermans’ thermodynamic criterion (ASmeIc, I5 eu) is fulfilled9J6(a high-pressure experiment on thiophen gave A S 3.4 eu for the entropy of melting of phase I1 a t 300 K and 400 MPa”), or almost fulfilled (AS = 5.17 eu for the melting of phase I of thiophen18). As benzene, p-dioxane, furan, and thiophen molecules can all be assimilated to oblate tops, the only geometrical differences should arise from the kind of motions the molecules undergo, either rotating in their plane (or pseudo-plane for p-dioxane), or tumbling over around an axis (or axes) lying in the same plane. This contention is supported by examining the molecular packing of furan I:8J0 a tumbling over of furan molecules seems highly unlikely. So, if geometrical properties are indeed leading parameters determining plasticity, it can be suggested that ful-
-
0022-3654/80/20841997$0l .OO/O
TABLE I: Values of Postel and Riess’ Ratio R for Different Solids Exhibiting Plastic Phases compd p-dioxane I p-dioxane I1 furan I furan HPa thiophen I thiophen I1
plastic behavior
+ -5
f7
c9 +11-13
+”-I3
d,
5.726 5.026 4.76’ 4.441° 4.8514915 4.6315
DM
7.10 7.10 6.80 6.80 6.80 6.80
0.81 0.71 0.70 0.65 0.71 0.68
High pressure phase.
fillment of Postel and Riess’ criterion indicates that a quasi-spherical volume is described during reorientational molecular motions and that reorientations around more than one axes should be looked for. Thus, this empirical rule would test dynamic as well as static geometrical properties within the crystal lattice.
References and Notes (1)M. Postel and J. G. Riess, J . fhys. Chem., 81, 2634 (1977). (2) M. Ghelfenstein, prlvate communication. (3) E. R. Andrew and R. G. Eades, Roc. R. Soc. London, Ser. A , 218, 537 (1953). (4) J. Timmermans, J . Chim. fhys., 35, 331 (1936). (5) F. Fried, Mol. Cryst. Li9. Cryst., 13, 279 (1971). (6) G. Clec’h, These de Sp6cialit6, Universit6 de Paris VI, 1974. (7)F. Fried and 6.Lassler, J . Chim. fhys., 1, 75 (1966). (8) R. Fourme, Acta Crystalbgr., Sect. B, 28, 2984,(1972). (9) P. Figuiere, High Temp. High Pressures, 7, 395 (1975). (10)R. Fourme, Thbse de Doctorat, Paris, 1970. (11)W. E. Sanford and R. K. Boyd, Can. J . Chem., 54, 2773 (1976). (12) F. Frled, These de Doctorat, Nice, No. A.O. 9740, 1974. (13)J. E. Anderson, Mol. Cryst. Liq. Cryst., 11, 343 (1970). (14) S.C. Abrahams and W. N. Lipscomb, Acta Crystallogr.,5, 93 (1952).
(15) D. Andr6, These de Doctorat, Universit6 de Paris-Sud, Orsay, No. 1557, 1975. (16) G. 6.Guthrie, Jr., D. W, Scott, W. N. Hubbard, C. Katz, J. P. McCullough, M. E. Gross, K. 0.Williamson, and G. Waddington, J . Am. Chem. SOC.,74, 4462,(1952). (17) D. Andr6, R. Fourme, P. FlgulBre, M. Ghelfenstein, D. Labarre, and H. Szwarc, unpublished results. (18) G. Waddington, J. W. Knowkon, D. W. Scott, G. D. Oliver, S. S. Todd,
W. N. Hubbard, J. C. Smith, and H. M. Huffman, J. Am. Chem. SOC., 71, 797 (1949). (19) Part of the Laboratoire Associ6 au CNRS No. 75. Laboratoire de Chimie Physique des Matgriaux Amorphes l9 Universit6 de Paris-Sud 9 1405 Orsay, France Received: June 29, 1978
0 1980 American Chemical gociety
Henrl Szwarc
