Molecular partition functions vibration

The q terms are the molecular partition functions of the superscript species. For the transition state,, the vibration along which the reaction takes place is omitted in the partition function, q. ... 

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

The second product is over the 3N—6(3N—5) normal mode frequencies of the ideal gas harmonic molecule to which Equation 4.78 applies. Thus the product over vibrations Equation 4.90 is indeed the quantum mechanical contribution to the molecular partition function for the ideal gas. 

This result indicates that (s2/si)f (compare Equations4.78 and 4.93) is just the quantum effect on the molecular partition functions of the normal mode vibrations. This result has now been derived without the explicit use of the Teller-Redlich product rule. 

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.,... 

In a polyatomic molecule witli many vibrations, we simplify the vibrational partition function much as the original molecular partition function was simplified we assume that the total vibrational energy can be expressed as a sum of individual energies associated with each mode, in which case, for a non-linear molecule, we have... 

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. 

The partition function provides the bridge to calculating thermodynamic quantities of interest. Using the molecular partition function and formulas derived in this section, we will be able to calculate the internal energy E, the heat capacity Cp, and the entropy S of a gas from fundamental properties of the molecule, such as its mass, moments of inertia, and vibrational frequencies. Thus, if thermodynamic data are lacking for a species of interest, we usually know, or can estimate, these molecular constants, and we can calculate reasonably accurate thermodynamic quantities. In Section 8.6 we illustrate the practical application of the formulas derived here with a numerical example of the thermodynamic properties for the species CH3. 

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. 

The molecular partition functions, Q, can be related to molecular properties of reactants and products. The partition function expresses the probability of encountering a molecule, so that the ratio of partition functions for the products versus the reactants of a chemical reaction expresses the relative probability of encountering products versus reactants and, therefore, the equilibrium constant. The partition function can be written as a product of independent factors at the level of various approximations, each of which is related to the molecular mass, the principal moments of inertia, the normal vibration frequency, and the electronic energy levels, respectively. When the ratio of isotopic partition function is calculated, the electronic part of the partition function cancels, at the level of the Born-Oppenheimer approximation, an approximation stating that the motion of nuclei in ordinary molecular vibrations is slow relative to the motions of electrons. 

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