Consider molecular rotations of a system comprised of N heteronuclear diatomic molecules. The rotation eigenstates of on

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Consider Molecular Rotations Of A System Comprised Of N Heteronuclear Diatomic Molecules The Rotation Eigenstates Of On 1 (54.48 KiB) Viewed 27 times

Consider molecular rotations of a system comprised of N heteronuclear diatomic molecules. The rotation eigenstates of one molecule are described by two quantum numbers and m. Here, is a non-negative integer, and m is an integer that satisfies -l≤msl. The rotational energy em of an eigenstate is given by Elm = e(l + 1)q, 1 assuming that the bond distance is constant. Here q is a positive constant. Define the number of molecules of the eigenstate characterized by quantum numbers and m as Nem, and the average of Nem in thermal equilibrium as (Nem). k and T denote the Boltzmann constant and absolute temperature, respectively. Answer the following questions. (1) Express the internal energy E in terms of Nem and m- (2) Express the number of cases W where Nem is specified for all possible values of e and m in terms of N! and Nem! (3) Define the free energy F as F = E-TS. Here S is the entropy. Express the free energy F where Nem is specified for all possible values off and m in terms of N!. Nem Nem!, Eem, k, and T. (4) The free energy F defined in (3) must be at a minimum in thermal equilibrium. Consider the change in free energy F when a molecule of an eigenstate characterized by quantum numbers and m is transferred to another eigenstate and m², and show that characterized by quantum numbers (Ne'm') (Nem) if Nem and Ne'm' are sufficiently larger than 1. = = exp[- 12/17 (²²m² - € Because equation 2 holds for any pair off and m, one can deduce a relation (Nem) == exp(-). where Z is called a partition function. (3