J. Phys. Chem. 1985,89, 3966-3967
3966
Low-Temperature Behavlor of the Classical Second Vlrlal Coefficlent for Potentials with a Smooth Mlnlmum A. J. M. Garrett Cavendish Laboratory, University of Cambridge, Cambridge CB3 OHE, U.K (Received: July 2, 1985)
An asymptotic formula, valid at low temperatures, is derived for the classical value of the second virial coefficient of an imperfect gas in which the interaction potential has a smooth, negative minimum. The formula is checked against the known result for the Lennard-Jones 12,6 potential.
in (4) in powers of multiply this and (3), and integrate with respect to x . All integrals are over the form XP exp(-x2); this is was chosen. why the scaling of r by a factor proportional to It is readily shown that even/odd powers q of x in the product If of (3) and (4) are associated with even/odd powers of we extend the lower limit of x integration, -(1/2U{@)1/zro, to -a, all odd powers of x vanish by symmetry on integrating; this procedure introduces a proportional error O(exp(-'/@ro2U,,")), which is "exponentially small" a t low temperatures ( p 5 ($U,")-'). Subject to this approximation, only alternate powers appear in the result. of B1jZ We now derive a mathematical result facilitating the expansion of the last exponential in (4) in powers of B1/2 or of x . Define
The second virial coefficient B2 is given classically as a functional of the interaction potential U(r)for spherically symmetrical particles by the integral transform'
Bz(@)= 2 r J m d r ?[l
- exp(-@v)];
p = (k7'l-I
(1)
We assume that U(r) is such as to cause (1) to converge (which implies U(-) = 0) and is bounded below by a single minimum U = Uo(> (@V,(2P))-i’(2P) where UJ2P)refers to either of two adjacent turning points), contributions of the form (13) can be added for each minimum. Clearly the most negativk. minimum is dominant. Unfortunately the exponentially small error discussed below (4) may cause inaccuracy in this dominant term greater than all contributions from other minima. Similarly there is no point in including the effects of any positive minima, which are exponentially small compared to the (neglected) contribution from the tail of the potential at large r. The above analysis ignore quantum effects, which separate naturally into exchange statistical effects and dynamical wave effects; both depend on the form of U(r).3 The wave effects can be dealt with by regarding (1) as the leading term in an expansion of Bz, believed to be asymptotic, in ascending powers of ti2? The coefficient of each power is an integral similar to (1) which may similarly be approximated at low temperature, akhough the series is not useful below temperatures kT = 8’ fi2/mrO2. We now use these results to derive more simply the known low-temperature result for the Lennard-Jones 12,6 potential:
-
U(r) = M / r I 2 - N / r 6
(14)
where M and N are positive. This potential is of the type assumed, with
ra = (2M/h91/6, U, = -N2/4M
(15)
rozUo” = 1 8 p / M
(16)
and which is nonzero, so that p = 1. It follows from (12) that
Bo
-
- ‘ / ( 2 ~ ) ~ / ~ ( 2 M / hi/2(M/2/3N2) 9
exp(/3N2/4M)
(17)
in accordance with the low-temperature asymptotic approximation of the exact e x p r e s ~ i o n . ~ . ~
and find similarly that
B2 = -27rpI’(
3967
t)..[
-I/(Zp)
L/3Uo(2P)] (2P)!
exp(-/3Uo)
X
(3) Lieb, E. H.J . Math. Phys. 1967, 8, 43, formulas (1.2)-(1.7). (4) Reference 1, p 48. (5) Eptein, L. F.; Roe, G. M. J . Chem. Phys. 1951, l g , 1320. (6) Garrett, A. J. M. J . Phys. A . 1980, 13, 379.
Photochemical Processes in 2-Fluoroethanol in Solid Neon M. Rasanen, AT& T Bell Laboratories, Murray Hiil, New Jersey 07974
J. Murto, Department of Physical Chemistry, University of Helsinki, Meritullinkatu 1 C SF-001 70, Helsinki, Finland
and V. E. Bondybey* AT& T Bell Laboratories, Murray Hill, New Jersey 07974 (Received: July 8. 1985)
FTIR study of 2-fluoroethanol in solid neon shows that, contrary to some previous reports, only the Gg’ and Tt conformers can be trapped. Infrared irradiation in either OH or CH stretching region results in a very efficient conversion of the initially present Gg’ form into the higher energy Tt conformers. The overall isomerization rate is essentially unchanged by deuteration of the OH group.
Introduction Studies of rotational isomers of simple hydrocarbons represent Saturated hydrocarbons a field of considerable current (1) Giinthard, Hs.H.J . Mol. Strucr. 1984, 113, 141. (2) Barnes, A. J. J. Mol. Srruct. 1984, 113, 161.
0022-3654/85/2089-3967$01.50/0
and similar unbranched chain compounds are characterized by a potential with a 3-fold barrier with respect to rotation around the carbon-carbon single bond. Studies of this rotational isomerism were greatly assisted by the observations that (a) the in(3) Lotta, T.; Murto, J.; RisBnen, M.;Aspiala, A. J . Mol. Srrucr. 1984, 114, 333.
0 1985 American Chemical Society
