Molecular partition function applications

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. 

Another application of intermediate coupling calculations has been to use the calculated results to reevaluate dissociation energies derived using the third-law method and mass- spectral data. Balasubramanian and Pitzer have shown how this can be accomplished in their calculations on Sn2 and Pb2 (90). This method requires the molecular partition function, which can be written... 

Adiabatic Treatment of Torsional Anharmonicity and Mode Coupling in Molecular Partition Functions and Statistical Rate Coefficients Application to Hydrogen Peroxide... 

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... 

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,... 

The traditional apparatus of statistical physics employed to construct models of physico-chemical processes is the method of calculating the partition function [17,19,26]. The alternative method of correlation functions or distribution functions [75] is more flexible. It is now the main method in the theory of the condensed state both for solid and liquid phases [76,77]. This method has also found an application for lattice systems [78,79]. A new variant of the method of correlation functions - the cluster approach was treated in the book [80]. The cluster approach provides a procedure for the self-consistent calculation of the complete set of probabilities of particle configurations on a cluster being considered. This makes it possible to take account of the local inhomogeneities of a lattice in the equilibrium and non-equilibrium states of a system of interacting particles. In this section the kinetic equations for wide atomic-molecular processes within the gas-solid systems were constructed. 

The above formula for Z, the NPT partition function, was first reported by Guggenheim [74], who wrote the expression down by analogy rather than based on a detailed derivation. While this form of the partition function is thought to be broadly valid and is widely applied (for example in molecular dynamics simulation [6]), it introduces the conceptual difficulty that the meaning of the discrete volumes Vi is not clear. Discrete energy states arise naturally from quantum statistics. Yet it is not necessarily obvious what discrete volumes to sum over in Equation (12.50). In fact for most applications it makes sense to replace the discrete sum with a continuous volume integral. Yet doing so results in a partition function that has units of volume, which is inappropriate for a partition function that formally should be unitless. 

In addition the reader may find tables with selection rules for the Resonance Raman and Hyper Raman Effect in the book of Weidlein et al. (1982). Special discussions about the basics of the application of group theory to molecular vibrations are given in the books of Herzberg (1945), Michl and Thulstrup (1986), Colthup et al. (1990) and Ferraro and Nakamoto (1994). Herzberg (1945) and Brandmiiller and Moser (1962) describe the calculation of thermodynamical functions (see also textbooks of physical chemistry). For the calculation of the rotational contribution of the partition function a symmetry number has to be taken into account. The following tables give this number in Q-... 

Recent years have seen the extensive application of computer simulation techniques to the study of condensed phases of matter. The two techniques of major importance are the Monte Carlo method and the method of molecular dynamics. Monte Carlo methods are ways of evaluating the partition function of a many-particle system through sampling the multidimensional integral that defines it, and can be used only for the study of equilibrium quantities such as thermodynamic properties and average local structure. Molecular dynamics methods solve Newton s classical equations of motion for a system of particles placed in a box with periodic boundary conditions, and can be used to study both equilibrium and nonequilibrium properties such as time correlation functions. 

Given the great success of perturbation theories in treating the properties of atomic liquids, a large effort has been devoted to extending the methods to deal with molecular systems. The first rigorous application of these theories to molecular fluids seems to have been made in 1951 by Barker, who expanded the partition function for a polar fluid about that for a fluid of isotropic molecules. 

Application of this procedure requires knowledge of the partition functions for each molecular conformer, and the energy differences between the conformers. Calculations have been reported for hydrazine, 1-chloropropane, I-bromopropane, and buta-l,3-diene. For each of these molecules, the calculation was simplified by assuming that the... 

Once we establish the complete partition function of a molecular gaseous species, we will consider one additional application of the partition function the chemical change. In the last chapter, a few exercises asked for a determination of the A(something) of a physical process, like the expansion of a monatomic gas. However, in chemistry we are often concerned with the change in the chemical identity of a species—a chemical reaction. It may surprise you to learn that the partition functions of each chemical species in a balanced chemical reaction can be used to determine a characteristic property of that reaction its equilibrium constant. 

The application of thermodjtnamics to chemical reactions enables equilibrium constants to be calculated from a knowledge of the macroscopic thermal, or microscopic molecular, properties of the reactants (A and B) and the products (C and D). Using statistical thermodynamics [20], the equilibrium constant for a reaction involving gas-phase species, can be expressed, in terms of the per unit volume partition functions, (q, /V), for the reactants and products, by... 

In the third approach, we use our qualitative understanding of the structure of fluids and intermolecular forces in formulating the partition function. To account for the molecular voliune, for example, we introduce in the partition function, Eq. 17.3.6, the so-called free volume. By understanding the limitations of the approximations made, this procedure provides avenues for the development of improved equations of state. We will place more emphasis on this approach because it is simpler and has found considerable application in the literature. 

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