Heat capacity derivation from partition functions

However, sufficiently accurate measurements of gaseous heat capacities down to the boiling point have usually not been made, and so the entropy contribution of the gaseous state, assuming ideal behavior, is derived from heat capacities obtained from partition functions as described in Chapter VI. If the heat capacity can be expressed as a function of the temperature over the required range, the entropy change can be derived analytically, as described in 20a. 

As seen earlier ( 12k), Aflo for a reaction may be evaluated from thermal measurements, including heat capacities at several temperatures. However, instead of using experimental heat capacity data to derive AHq from A/f values, the results may be obtained indirectly from partition functions (cf. 16c). The energy content of an ideal gas is independent of the pressure, at a given temperature hence, E — Eo in equation (16.8) may be replaced by E — JSo, so that... 

Expressions similar to those given above may be derived easily from partition functions in other ensembles.The choice of ensemble is very important in calculations of hydration entropy, enthalpy, and heat capacity, as discussed below. Many other quantities, including all free energies, are ensemble invariant, with the choice of ensemble affecting only system size dependence. For simplicity, the discussion here is therefore limited to the canonical ensemble except in such cases where a true ensemble dependence exists. 

The thermodynamic properties that we have considered so far, such as the internal energy, the pressure and the heat capacity are collectively known as the mechanical properties and can be routinely obtained from a Monte Carlo or molecular dynamics simulation. Other thermodynamic properties are difficult to determine accurately without resorting to special techniques. These are the so-called entropic or thermal properties the free energy, the chemical potential and the entropy itself. The difference between the mechanical emd thermal properties is that the mechanical properties are related to the derivative of the partition function whereas the thermal properties are directly related to the partition function itself. To illustrate the difference between these two classes of properties, let us consider the internal energy, U, and the Fielmholtz free energy, A. These are related to the partition function by ... 

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. 

The free energy of the system also includes entropic contributions arising from the internal fluctuations, which are expected to be different for the separate species and for the liganded complex. These can be estimated from normal-mode analyses by standard techniques,136,164 or by quasi-harmonic calculations that introduce approximate corrections for anharmonic effects 140,141 such approaches have been described in Chapt. IV.F. From the vibrational frequencies, the harmonic contribution to the thermodynamic properties can be calculated by using the multimode harmonic oscillator partition function and its derivatives. The expressions for the Helmholtz free energy, A, the energy, E, the heat capacity at constant volume, C , and the entropy are (without the zero-point correction)164... 

Calculation of Thermodynamic Functions from Molecular Properties The calculation methods for thermodynamic functions (entropy S, heat capacities Cp and Cy, enthalpy H, and therefore Gibbs free energy G) for polyatomic systems from molecular and spectroscopic data with statistical methods through calculation of partition functions and its derivative toward temperature are well established and described in reference books such as Herzberg s Molecular Spectra and Molecular Structure [59] or in the earlier work from Mayer and Mayer [7], who showed, probably for the first time in a comprehensive way, that all basic thermochemical properties can be calculated from the partition function Q and the Avagadro s number N. The calculation details are well described by Irikura [60] and are summarized here. Emphasis will be placed on calculations of internal rotations. 

For a linear molecule, the rotational contribution is only 2/2, not 3/2.) The origins of each part of the heat capacity have been labeled under each term. The derivative in the definition of Cy converts the product function from the vibrational partition function into a summation of terms. Notice what equation 18.52 implies Translations and rotations contribute the same amount to the heat capacity of a gaseous molecule, and vibrations also contribute.The greater the number of atoms N in the molecule, the more vibrations, and so the greater the vibrational contribution to Cy. Such observations are indeed made experimentally. An expression for Cp is left as an exercise. 

In the following we shall give enthalpy and heat capacity equations for some typical folding models. For each model the relative partition function is given as well as the enthalpy and the heat capacity functions resulting from the temperature derivatives according to equations 35 and 36. In this... 

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