Molecular partition functions rotation

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. 

Table 6.2 Forms for translational, rotational, and vibrational contributions to the molecular partition function...

Some important systems, which certainly do not fulfill the assumptions of harmonic transition state theory are gas phase reactions. In the gas phase, there are zero-modes such as translation and rotation, and these lead to totally different configuration integrals than those obtained from a normal mode analysis. For these species one can in a simple manner modify the terms going into the HTST rate by incorporating the molecular partition functions [3,119]. 

We have thus reduced the problem from finding the ensemble partition function Q to finding the molecular partition function q. In order to make further progress, we assume that the molecular energy e can be expressed as a separable sum of electronic, translational, rotational, and vibrational terms, i.e.,... 

We see that the partition function of a molecule is the product of the contributions of the translational, rotational, vibrational, and electronic partition functions, which we can calculate separately, as discussed next. We will see in Section 8.5 that any thermodynamic quantity of interest can be derived from the molecular partition function. Thus it is important to be able to evaluate q. 

Here we use the label i to denote a molecular energy level, which may denote at once the specific translational (t), rotational (r), vibrational (u), and electronic (e) energy level of the molecule. From Eq. 8.46 and the definition of the molecular partition function q,... 

All calculations will be done for the standard pressure of 1 bar and, unless otherwise noted, at T = 298.15 K for one mole of gas. Table 8.1 lists the calculated molecular partition function, thermal energy (energy in excess of the ground-state energy), heat capacity, and entropy. The individual contributions from translation, rotation, each of the six vibrational modes, and from the first excited electronic energy level are included. 

The translational contribution to the molecular partition function, which is calculated using Eq. 8.59, clearly makes the largest contribution. (In obtaining this value, we also made use of the ideal gas law to calculate the volume V = 0.02479 m3 of a mole of gas at this temperature and pressure.) The rotational partition function is evaluated via Eq. 8.67, and the vibrational partition function for each mode is found via Eq. 8.71. Only the very... 

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). 

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. 

Each molecular partition function qt may in turn be factored into contributions from translational (g transX vibrational (qiyib), rotational (q Yib), and electronic (qi iec)... 

The information needed to evaluate the molecular partition functions qb (13.63), may in principle be obtained from experimental spectroscopic measurements or theoretical calculations on each molecule i. Each type of energy contribution to qt (translational, rotational, vibrational, electronic) in principle requires associated quantum energy levels... 

E17.10(b) The contributions of rotational and vibrational modes of motion to the molar Gibbs energy depend on the molecular partition functions... 

The molecular partition function has been factorized into contributions g vib > resulting from the vibrations of the localized molecule as a whole around its equilibrium position, q ot, resulting from rotations, and gint, resulting from intramolecular vibrations. The configurational entropy Scant results from the degeneracy due to the distribution of the... 

The partition function of the system Q is related to the molecular partition function of the individual molecules in the system. In our development of rate constants we make use of the molecular partition functions. The molecular partition function per unit volume for an ideal gas is the product of the translational, rotational, vibrational and electronic energy states in the molecule... 

In the molecular-motion contribution, the molecular partition function is the product of translational, rotational, vibrational, and electronic partition functions. If the molecule in solution is assumed to have the entire volume of the solution available to it, the ratio of gas-phase and solution-phase translational partition functions equals one. Likewise, the electronic partition function ratio will be one. It is unclear what one should use for the rotational partition function in solution, but if this is assumed to have the same form as that in the gas phase, the rotational partition function ratio (which involves the moments of inertia) will be very close to one, since structural changes from gas to solution are slight. Significant contributions to the vibrational partition function are made only by the low-frequency vibrational normal modes, and these modes sometimes show substantial changes in frequency on going from the gas phase to solution. If a vibrational calculation is done in the gas phase and in solutitm, one can calculate AG°oiv m, but the most common procedure is to omit it, assuming that its contribution is negligible. 

TST predicts the trend of decreasing Arrhenius pre-exponential factor with increasing reactant size and molecular complexity that is revealed by experimental measurements of rate coefficients, and that SCT explained away by invoking the steric factor. This trend arises in TST through the internal degrees of freedom, which are accounted for in the partition functions, and which are not present in the structureless point masses of SCT. If electronic, vibrational, rotational, and translational dfs are independent, the molecular partition function factors into electronic, vibrational, rotational, and translational contributions, i.e., Q = e v r t-The orders of magnitude of the partition functions per df are v l-10 per vibrational or internal rotational df, per overall... 

In particular, the microscopic formulation of the transition state theory, that is, by writing in terms of molecular partition functions, allows accounting for differences in translational, rotational and vibrational degrees of freedom between the species involved as reactant and the transition state. 

In equation 85, Q represents the molecular partition function for the transition state species and reactants A and B. It is constructed from contributions related to translational, rotational, and vibrational degrees of freedom as follows ... 

In brief, the distribution of a particular typo of energy, if it is separable, can be discussed without reference to the other types of energy each distribution function is determined only by its own partition function. It will be apparent also that the larger is the value of, say,/, the smaller is the fraction of all the molecules which populate any single rotational level. The magnitude of a molecular partition function, or any factorized part of it, is a measure of the total number of levels which can be appreci-ably populated at the given temperature. 

Statistical thermod5rnatnics enables us to express the entropy as a function of the canonical partition function Zc (relation [A2.39], see Appendix 2). This partition function is expressed by relation [A2.36], on the basis of the molecular partition functions. These molecular partition functions are expressed, in relation [A2.21], by the partition functions of translation, vibration and rotation. These are calculated on the basis of the molecule mass and relation [A2.26] for a perfect gas, the vibration frequencies (relation [A2.30]) of its bonds and of its moments of inertia (expression [A2.29]). These data are determined by stud5dng the spectra of the molecules - particularly the absorption spectra in the iirffared. Hence, at least for simple molecules, we are able to calculate an absolute value for the entropy - i.e. with no frame of reference, and in particular without the aid of Planck s hypothesis. 

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