Partition function rotational motion

Here, represents the contribution of all other internal motions of the molecule to the molecular partition function (rotations, vibrations, electronic and nuclear spin motions). For atomic liquids, this term can be taken as being equal to 1. 

For translational, rotational and vibrational motion the partition function Ccin be calculated using standard results obtained by solving the Schrodinger equation ... 

The above treatment has made some assumptions, such as harmonic frequencies and sufficiently small energy spacing between the rotational levels. If a more elaborate treatment is required, the summation for the partition functions must be carried out explicitly. Many molecules also have internal rotations with quite small barriers, hi the above they are assumed to be described by simple harmonic vibrations, which may be a poor approximation. Calculating the energy levels for a hindered rotor is somewhat complicated, and is rarely done. If the barrier is very low, the motion may be treated as a free rotor, in which case it contributes a constant factor of RT to the enthalpy and R/2 to the entropy. 

Partition functions are very important in estimating equilibrium constants and rate constants in elementary reaction steps. Therefore, we shall take a closer look at the partition functions of atoms and molecules. Motion, or translation, is the only degree of freedom that atoms have. Molecules also possess internal degrees of freedom, namely vibration and rotation. 

The function g is the partition function for the transition state, and Qr is the product of the partition functions for the reactant molecules. The partition function essentially counts the number of ways that thermal energy can be stored in the various modes (translation, rotation, vibration, etc.) of a system of molecules, and is directly related to the number of quantum states available at each energy. This is related to the freedom of motion in the various modes. From equations 6.5-7 and -16, we see that the entropy change is related to the ratio of the partition functions ... 

Rotational motion of a general (nonlinear) polyatomic molecule accounts for three degrees of freedom. The partition function in this case was given by Eq. 8.67. It is easy to verify that... 

For a temperature of 298.15 K, a pressure of 1 bar, and 1 mole of H2S, prepare a table of (1) the entropy (J/mol K), and separately the contributions from translation, rotation, each vibrational mode, and from electronically excited levels (2) specific heat at constant volume Cv (J/mol/K), and the separate contributions from each of the types of motions listed in (1) (3) the thermal internal energy E - Eo, and the separate contributions from each type of motion as before (4) the value of the molecular partition function q, and the separate contributions from each of the types of motions listed above (5) the specific heat at constant pressure (J/mol/K) (6) the thermal contribution to the enthalpy H-Ho (J/mol). 

The Boltzmann distribution of the populations of a collection of molecules at some temperature T was discussed in Section 8.3.2. This distribution, given by Eq. 8.46 or 8.88, was expressed in terms of the quantum mechanical energy levels and the partition function for a particular type of motion, for instance, translational, vibrational, or rotational motion. It is useful to express such population distributions in other forms, particularly to obtain an expression for the distribution of velocities. The velocity distribution function basically determines the (translational) energy available for overcoming a reaction barrier. It also determines the frequency of collisions, which directly contributes to the rate constant k. 

Hard-Sphere Collision Limit It is interesting to evaluate the behavior of Eq. 10.9 when both reactants A and B are atomic species. In this situation the only degrees-of-freedom contribution to the molecular partition functions are from translational motion, evaluated via Eq. 8.59. The atomic species partition functions have no vibration, rotational, or (for the sake of simplicity) electronic contributions. 

The internal motion partition function of the guest molecule is the same as that of an ideal gas. That is, the rotational, vibrational, nuclear, and electronic energies are not significantly affected by enclathration, as supported by spectroscopic results summarized by Davidson (1971) and Davidson and Ripmeester (1984). 

Note that the 1 /N term is assigned to the translational partition function, since all gases have translational motion, but only molecular gases have rotational and vibrational degrees of freedom. The electronic partition function is usually equal to one unless unpaired electrons are present in the atom or molecule. 

The partition functions can be factorized into contributions corresponding to the various forms of motion when they are uncoupled (see Appendix A.l), and it is advantageous to rewrite the expression for the rate constant in terms of partition function ratios for the translational, rotational, vibrational, and electronic motion ... 

We place the dividing surface between reactants and products at a critical separation rc and calculate the rate constant according to Eq. (6.8) (assuming, as in Section 4.1.2, that the atoms react with a probability of one when r = rc). In the relevant partition function for the activated complex, there are both translational degrees of freedom for the center-of-mass motion and rotational degrees of freedom... 

The harmonic approximation reduces to assuming the PES to be a hyperparaboloid in the vicinity of each of the local minima of the molecular potential energy. Under this assumption the thermodynamical quantities (and some other properties) can be obtained in the close form. Indeed, for the ideal gas of polyatomic molecules the partition function Q is a product of the partition functions corresponding to the translational, rotational, and vibrational motions of the nuclei and to that describing electronic degrees of freedom of an individual molecule ... 

The quantized energy ej can be of electronic, vibrational, rotational or translational type, readily calculated from the quantum laws of motion. In a macrosystem the sum over all the quantum states for the complete set of molecules, the sum over states defines the canonical partition function ... 

To obtain a more quantitative idea of these changes, let us consider individual partition functions for single-component, translational, rotational, and vibrational motions. They are given by (Secs. IX.8, 9, 10) ... 

The rotational partition function is dependent upon the equilibrium geometry. Assuming separation of rotational and vibrational motions. 

Neglecting rotational motion, and assuming that the vibrational modes are all harmonic, the classical limit of the vibrational partition function of a molecule... 

With respect to the third assumption, for a number of years, Tanaka has shown that the internal particle partition function of the guest molecules differ significantly in the cages, with restricted rotation and vibration, particularly for those molecules larger than methane. Indeed, some restriction on such motions is the basis, for example, in Raman spectroscopic determination of differing environments of the methane molecule, in the gas, in solution, and in the hydrate cages. ... 

Although the nuclear partition function is a product of the translational, rotational, and vibrational partition functions, the isotope effect is determined almost entirely by the latter, specifically by vibrational modes involving motion of isotopically different atoms. In the case of light atoms (i.e., protium vs. deuterium or tritium) at moderate temperatures, the isotope effect is dominated by ZERO-POINT energy differences. 

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