Partition function table

This is an extremely simple expression for a partition function. Table 18.1 lists a few 6y values for some diatomic molecules. At temperatures well above 6y for each gas-phase molecule, the vibrational partition function is given simply by equation 18.18. At temperatures near 0y or lower, the more complete expression in equation 18.17 must be used. (But be careful For some of the molecules listed, the stable phase is not the gas phase at T < 0 )... 

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... 

By starting with this partition function and going through considerable mathematical manipulation, one arrives at the following equations for calculating the corrections to the rigid rotator and harmonic oscillator values calculated from Table 10,4, U... 

Table A4.6 gives the internal rotation contributions to the heat capacity, enthalpy and Gibbs free energy as a function of the rotational barrier V. It is convenient to tabulate the contributions in terms of VjRTagainst 1/rf, where f is the partition function for free rotation [see equation (10.141)]. For details of the calculation, see Section 10.7c.

In contrast to disproportionation, in the dimerization equilibrium at lower temperatures the reaction product is favored. The contributions coming from partition functions and from the exponential term are presented together with the temperatures and logarithms of equilibrium constants of the reaction (103) in Table XI. 

Table 3.5. The number of degrees of freedom in translation, rotation and vibrations of the reacting molecules and the transition state in the gas phase reaction of CO and O2 and the temperature dependence these modes contribute to the partition function. Note that one of the modes of the transition state complex is the reaction coordinate, so that only six vibrational modes are listed.

Table 10.4 lists the rate parameters for the elementary steps of the CO + NO reaction in the limit of zero coverage. Parameters such as those listed in Tab. 10.4 form the highly desirable input for modeling overall reaction mechanisms. In addition, elementary rate parameters can be compared to calculations on the basis of the theories outlined in Chapters 3 and 6. In this way the kinetic parameters of elementary reaction steps provide, through spectroscopy and computational chemistry, a link between the intramolecular properties of adsorbed reactants and their reactivity Statistical thermodynamics furnishes the theoretical framework to describe how equilibrium constants and reaction rate constants depend on the partition functions of vibration and rotation. Thus, spectroscopy studies of adsorbed reactants and intermediates provide the input for computing equilibrium constants, while calculations on the transition states of reaction pathways, starting from structurally, electronically and vibrationally well-characterized ground states, enable the prediction of kinetic parameters. 

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... 

Use the solar Fe I curve of growth in Fig. 3.12 to deduce the solar iron abundance, using 9i0n = 0.9 and given that Fe I has an ionization potential of 7.87 eV and that the partition function of Fe II is 42. Compare the result with the one in Table 3.4. 

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

Show that, for the bimolecular reaction A + B - P, where A and B are hard spheres, kTsr is given by the same result as jfcSCT, equation 6.4-17. A and B contain no internal modes, and the transition state is the configuration in which A and B are touching (at distance dAR between centers). The partition functions for the reactants contain only translational modes (one factor in Qr for each reactant), while the transition state has one translation mode and two rotational modes. The moment of inertia (/ in Table 6.2) of the transition state (the two spheres touching) is where p, is reduced mass (equation 6.4-6). 

Note that gtrans is given per degree of freedom, implying that the total translational partition function for an adsorbed molecule is given by (Qtrms)2- Also, the total partition functions for vibration and rotation are the products of terms for each individual vibration and rotation, respectively. Table 2.2 gives values for the partition functions for adsorbed atoms and molecules at 500 K. Vibrational partition functions are usually close to one, but rotational and translational partition functions have larger values. 

Table 2.2 Partition functions for selected moleculesb) and atoms at 500 K.

Figure 2.15 Microscopic pictures of the desorption of atoms and molecules via mobile and immobile transition states. If the transition state resembles the ground state, we expect a prefactor of desorption on the order of 1013 s. If the adsorbates are mobile in the transition state, the prefactor goes up by one or two orders of magnitude. In the case of desorbing molecules, free rotation in the transition state increases the prefactor even further. The prefactors are roughly characteristic of atoms such as C, N and O and molecules such as N2, CO, NO and 02. See also the partition functions in Table 2.2 and the prefactors for CO desorption in Table 2.3.

Table 4.1 gives the expressions for partition functions and approximate order of magnitude. 

Table 4.1 Partition functions evaluated in the rigid rotor harmonic oscillator approximation... 

The formulae given in Table 4.1 for the molecular partition functions enable us to write the partition function ratio qheavy/qiight or q2/qi where, by the usual convention, the subscript 2 refers to the heavy isotopomer and 1 refers to the light isotopomer if heavy and light are appropriate designations. Then, a ratio of such partition function ratios enables one to evaluate the isotope effect on a gas phase equilibrium constant, as pointed out above. Before continuing, it is appropriate to... 

Now look at the partition function for each normal mode of vibration (Table 4.1)... 

In Section 4.8, Equations 4.78,4.79 and Table 4.1 develop the connections between the harmonic oscillator rigid rotor partition function and isotope chemistry as expressed by the reduced partition function ratio, RPFR = (s/s ) f. RPFR is defined in Equation 4.79 as the product over oscillators of ratios of the function [u exp(—u/2)/ (1 - exp(u))]... 

The Horiuti group treats the temperature coefficient of the rate differently from the way it is usually treated in TST. They clearly identify E as the experimentally observed activation energy, but according to TST [cf. Eq. (5)] the (E — RT) quantity of Eq. (52) is the enthalpy of activation. The RT term in Eq. (5) arises because the assumption that the Arrhenius plot is linear is equivalent to the assumption that the preexponential factor A of the Arrhenius equation is constant, whereas, according to TST, A always contains the factor (kT/h). In addition, the partition function factors of Table I are also part of A, and most of them are functions of T. Since the Horiuti group takes this temperature dependency of the preexponential factor into account, the factor exp[(5/2)(vi -I- V2)] (where 5/2 is replaced by 3 for nonlinear molecules) arises. 

The general equation for L is Eq. (10), and the expressions to be used for C in that equation are listed in Table 1 in terms of partition functions. But the entropies of both the activated complex and the reactants, and therefore AS, can also be expressed in terms of partition functions. Therefore, C can be expressed in terms of the entropies of the activated complex and the reactants. As we shall see, it is possible to eliminate partition functions entirely. Also, in all but one case. Step 11, the entropy factor in C can be determined if one knows only the entropy of activation in those cases the entropy of a reactant or the activated complex is not needed. 

The expressions for C in terms of entropy instead of partition functions are given in Table II. The method of obtaining these expressions is illustrated in the following discussion by developing the appropriate expression for Step 1. 

Explicit expressions for these partition functions are listed in Table VI, together with the corresponding expressions for S and Cy. 

Occasionally, the rates of bimolecular reactions are observed to exhibit negative temperature dependencies, i.e., their rates decrease with increasing temperature. This counterintuitive situation can be explained via the transition state theory for reactions with no activation energy harriers that is, preexponential terms can exhibit negative temperature dependencies for polyatomic reactions as a consequence of partition function considerations (see, for example, Table 5.2 in Moore and Pearson, 1981). However, another plausible explanation involves the formation of a bound intermediate complex (Fontijn and Zellner, 1983 Mozurkewich and Benson, 1984). To... 

Table 11.9 Third-order polynomials relating rednced partition functions to inverse of squared absolute temperature x = 10 X (from Clayton and Kieffer, 1991).

Combining the results of Kieffer s model and of laboratory experiments, Clayton and Kieffer (1991) obtained a set of third-order equations relating the reduced partition functions of various minerals to the inverse of the squared absolute temperature (table 11.9) according to... 

Barrer (3) makes similar calculations for the entropies of occlusion of substances by zeolites and reaches the conclusion that the adsorbed material is devoid of translational freedom. However, he uses a volume, area or length of unity when considering the partition function for translation of the adsorbed molecules in the cases where they are assumed to be capable of translation in three, two or one dimensions. His entropies are given for the standard state of 6 = 0.5, and the volume, area or length associated with the space available to the adsorbed molecules should be of molecular dimensions, v = 125 X 10-24 cc., a = 25 X 10-16 cm.2 and l = 5 X 10-8 cm. When these values are introduced into his calculations the entropies in column four of Table II of his paper come much closer together, as is shown in Table I. The experimental values for different substances range from zero to —7 cals./deg. mole or entropy units, and so further examination is required in each case to decide... 

For strongly shocked Argon (see Table 1), calculations based on Eq 8 give a s that are 30 to 40% higher than the more accurate (partition function) calculations of Reynolds... 

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