Partition function applications

Lee, K. H., Lombardo, M., and Sandler, S. I. (1987). The Generalized van der Waals Partition Function Application to Square-Well Fluids. Fluid Phase Equilib. 21,177. 

This is connnonly known as the transition state theory approximation to the rate constant. Note that all one needs to do to evaluate (A3.11.187) is to detennine the partition function of the reagents and transition state, which is a problem in statistical mechanics rather than dynamics. This makes transition state theory a very usefiil approach for many applications. However, what is left out are two potentially important effects, tiiimelling and barrier recrossing, bodi of which lead to CRTs that differ from the sum of step frmctions assumed in (A3.11.1831. 

Next, Ah and Ad are written in terms of partition functions (see Section 5.2), which are in principle calculable from quantum mechanical results together with experimental vibrational frequencies. The application of this approach to mechanistic problems involves postulating alternative models of the transition state, estimating the appropriate molecular properties of the hypothetical transition state species, and calculating the corresponding k lko values for comparison with experiment.""- " "P... 

The theory introduced by Lennard-Jones and Devonshire13 17 for the study of liquids provides a powerful method for the quantitative evaluation of the partition function of a solute molecule within its cavity.51 Because the application of this method to the present problem has been described in detail,62 we shall restrict ourselves to its most essential features. 

Again, therefore, all thermodynamic properties of a system in quantum statistics can be derived from a knowledge of the partition function, and since this is the trace of an operator, we can choose any convenient representation in which to compute it. The most fruitful application of this method is probably to the theory of imperfect gases, and is well covered in the standard reference works.23... 

In the following, the MO applications will be demonstrated with two selected equilibrium reactions, most important in radical chemistry disproportionation and dimerization. The examples presented will concern MO approaches of different levels of sophistication ab initio calculations with the evaluation of partition functions, semiempirical treatments, and simple procedures employing the HMO method or perturbation theory. 

Thus, given sufEcient detailed knowledge of the internal energy levels of the molecules participating in a reaction, we can calculate the relevant partition functions, and then the equilibrium constant from Eq. (67). This approach is applicable in general Determine the partition function, then estimate the chemical potentials of the reacting species, and the equilibrium constant can be determined. A few examples will illustrate this approach. 

Most students are introduced to quantum mechanics with the study of the famous problem of the particle in a box. While this problem is introduced primarily for pedagogical reasons, it has nevertheless some important applications. In particular, it is the basis for the derivation of the translational partition function for a gas (Section 10.8.1) and is employed as a model for certain problems in solid-state physics. 

The various contributions to the energy of a molecule were specified in Eq. (47). However, the fact that the electronic partition function was assumed to be equal to one should not be overlooked. In effect, the electronic energy was assumed to be equal to zero, that is, that the molecule remains in its ground electronic state. In the application of statistical mechanics to high-temperature systems this approximation is not appropriate. In particular, in the analysis of plasmas the electronic contribution to the energy, and thus to the partition function, must be included. 

Simplified considerations are applicable to calculating ion energies classically and for also the partition functions r. 

The remark just made suggests that a natural place to begin our discussion of equilibrium equations is with the occupation of different charge states. Let a hydrogen in charge state i(i = +, 0, or - ) have possible minimum-energy positions in each unit cell, of volume O0, of the silicon lattice. (O0 contains two Si atoms, so our equations below will be applicable also to zincblende-type semiconductors.) To account for spin degener-ancies, vibrational excitations, etc., let us define the partition function... 

In a more sophisticated application, one calculates an abscissa log X, which is a theoretical value of log W/X taking into account all atmospheric effects except saturation, as a function of the desired abundance ratio M/H log X = log (M/H) + log gf + log T, where T is calculated for given excitation and ionization potentials, ionic partition functions and the model atmosphere. The abundance is then chosen to give the optimal fit for weak lines. The same curve can also be used (with due... 

The use of reduced isotopic partition function ratios to study kinetic isotope effects was first undertaken by Bigeleisen this work was corrected and elaborated by Bigeleisen and Wolfsberg. References are cited at the end of this chapter. Application of the equations developed above to specific chemical reactions will be found in Chapter 10, where other theoretical approaches will also be presented. 

Bigeleisen, J. and Ishida, T. Application of finite orthogonal polynomials to the thermal functions of harmonic oscillators. I. Reduced partition function of isotopic molecules, J. Chem. Phys. 48, 1311 (1968). Ishida, T., Spindel, W. and Bigeleisen, J. Theoretical analysis of chemical isotope fractionation by orthogonal polynomial methods, in Spindel, W., ed. Isotope Effects on Chemical Processes. Adv. Chem. Ser. 89, 192 (1969). 

Bigeleisen and Mayer (1947) simplified the reduced partition function by observing that vibrational frequency shifts caused by isotope substitution are relatively small (except when deuterium is substituted for normal hydrogen). When the dimensionless quantity hv/kr is of the order 5 or less (corresponding to a typical 1000 cm vibration at 288 K)—a condition applicable to most geochemical situations. 

Figure 11.30 shows the reduced partition function (1/r) X 1000 In / calculated by Kieffer (1982) by application of equation 11.61 and plotted against The adoption of this form of plot is dictated by the fact that the natural logarithm of the partition function ratio is asymptotically proportional to T at high tern-... 

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. 

A further complication associated with the application of molecular mechanics calculations to relative stabilities is that strain energy differences correspond to A(AH) between conformers with similar chromophores (electronic effects) and an innocent environment (counter ions and solvent molecules), whereas relative stabilities are based on A(AG). The entropy term, TAS, may be calculated by partition functions,... 

There are some situations, such as reactions in solution, where the partition function form is not immediately useful, whereas a thermodynamic formulation is more immediately applicable. 

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