Vibrations partition function for

The classical value is attained by most molecules at temperatures above 300 K for die translation and rotation components, but for some molecules, those which have high heats of formation from die constituent atoms such as H2, die classical value for die vibrational component is only reached above room temperature. Consideration of the vibrational partition function for a diatomic gas leads to the relation... 

In the infinite sum each successive term is smaller than the previous by a constant factor ( -hujVT which is <1), and can therefore be expressed in a closed form. Only the vibrational frequency is needed for calculating the vibrational partition function for a harmonic oscillator, i.e. only the force constant and the atomic masses are required. 

Under most circumstances the equations given in Table 10.4 accurately calculate the thermodynamic properties of the ideal gas. The most serious approximations involve the replacement of the summation with an integral [equations (10.94) and (10.95)] in calculating the partition function for the rigid rotator, and the approximation that the rotational and vibrational partition functions for a gas can be represented by those for a rigid rotator and harmonic oscillator. In general, the errors introduced by these approximations are most serious for the diatomic molecule." Fortunately, it is for the diatomic molecule that corrections are most easily calculated. It is also for these molecules that spectroscopic information is often available to make the corrections for anharmonicity and nonrigid rotator effects. We will summarize the relationships... 

To make contact with atomic theories of the binding of interstitial hydrogen in silicon, and to extrapolate the solubility to lower temperatures, some thermodynamic analysis of these data is needed a convenient procedure is that of Johnson, etal. (1986). As we have seen in Section II. l,Eqs. (2) et seq., the equilibrium concentration of any interstitial species is determined by the concentration of possible sites for this species, the vibrational partition function for each occupied site, and the difference between the chemical potential p, of the hydrogen and the ground state energy E0 on this type of site. In equilibrium with external H2 gas, /x is accurately known from thermochemical tables for the latter. A convenient source is the... 

The vibrational partition function for each vibrational level / was given by Eq. 8.71 and E — Eq is given by Eq. 8.84. Substituting into Eq. 8.100 gives... 

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... 

The vibrational partition function for a molecule with s vibrational degrees of freedom is given by Eq. 8.71, noting that the contribution to q from each oscillator is multiplicative as seen in Section 8.4.3 ... 

The translational, rotational, and vibrational partition functions for the ideal gas, assuming a rigid rotator and a harmonic oscillator approximation, are as... 

The vibrational partition function for i degrees of freedom is given by vlb = (9)... 

Calculate the rate constant, at T = 300 K, according to transition-state theory. The vibrational partition functions for vibrations with wave numbers larger than 1000 cm-1 can be set to 1. 

We must now take into account two extreme situations. The first is where the adsorption in the site is very strong, that is, localized adsorption, and the partition function for the movement in the z-direction in the adsorption site is a vibration partition function for a molecule in a potential energy well [110]... 

The vibrational partition function for a quantised harmonic oscillator is... 

All of the ab initio calcnlations that include electron correlation to some extent clearly favor the concerted pathway for Reaction 4.1. All of these computations also identified a transition state with Q symmetry, indicating perfectly synchronons bond formation. One method for distinguishing a synchronous from an asynchronous transition state is by secondary kinetic isotope effects (KIEs). Isotopic snbstitution alters the frequencies for all vibrations in which that isotope is involved. This leads to a different vibrational partition function for each isotopicaUy labeled species. Bigeleisen and Mayer determined the ratio of partition functions for isotopicaUy labeled species. Incorporating this into the Eyring transition state theory results in the ratio of rates for the isotopicaUy labeled species (Eq. (d. ))." Computation of the vibrational frequencies is thus... 

Under these conditions the vibrational partition function for N independent molecules is given by... 

If now it is assumed that the rupture of the molecule that leads to reaction corresponds to the motion of one of the normal coordinates of the complex of frequency v = Vc and further that Vc kT/hy then the individual vibrational partition function for this coordinate is 9 = (1 — =... 

In order to illustrate the consequences of equation (70), it will be assumed that the partition functions for the reactants and the complex can be expressed as products of the appropriate numbers of translational, rotational and vibrational partition functions. For simplicity we shall also neglect factors associated with nuclear spin and electronic excitation. If = total number of atoms in a molecule of species i and = 0 for nonlinear molecules, 1 for linear molecules, and 3 for monatomic molecules, then the correct numbers of the various kinds of degrees of freedom are obtained in equation (70) by letting... 

To relate these thermodynamic quantities to molecular properties and interactions, we need to consider the statistical thermodynamics of ideal gases and ideal solutions. A detailed discussion is beyond the scope of this review. We note for completeness, however, that a full treatment of the free energy of solvation should include the changes in the rotational and vibrational partition functions for the solute as it passes from the gas phase into solution, AGjnt. ... 

Clyne and Stedman consider the discussion in terms of transition state theory to be more realistic since it predicts an increase with temperature of the apparent activation energy due to large increase with temperature of the vibrational partition function for the H-H-Cl bending mode. This increase occurs for a wide variety of models of the potential surface. 

A. Riganelli, F.V. Prudente, and A.J.C. Varandas, Evaluation of vibrational partition functions for... 

The rates and mechanisms of chemical reactions can be predicted, in principle, by the standard methods of statistical thermodynamics, in terms of the partition functions of reactants and the transition-state complex. However, the range of applicability of this transition-state (absolute rate) theory is severely limited by the fact, that an evaluation of the vibrational partition function for the transition state requires a detailed consideration of the whole PES for the reaction. Thus, a calculation of the absolute rate constants is possible only for relatively simple systems. This indicates a need for more approximate, empirical methods of treating chemical reactions and formulating the reactivity theory, which would allow... 

The vibrational partition function for polyatomic molecules can be factored with respect to each normal mode of vibration. Since each normal mode is described by a harmonic oscillator with frequency v = and vibrational temperature 0 = 1.44A. ... 

Expressions for the equation of state, de Broglie thermal wavelength, and rotational and vibrational partition functions for a diatomic molecule, provided by (28-87) and (28-89), as well as multiplication by Navo, allow one to determine the chemical potential on a molar basis. The final result for p, is consistent with its definition from classical thermodynamics ... 

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