Canonical partition function of translation

Similarly, for a binary mixture of two species A and B, we can write the canonical partition function of translation in the form ... 

Statistical thermod5rnatnics enables us to express the entropy as a function of the canonical partition function Zc (relation [A2.39], see Appendix 2). This partition function is expressed by relation [A2.36], on the basis of the molecular partition functions. These molecular partition functions are expressed, in relation [A2.21], by the partition functions of translation, vibration and rotation. These are calculated on the basis of the molecule mass and relation [A2.26] for a perfect gas, the vibration frequencies (relation [A2.30]) of its bonds and of its moments of inertia (expression [A2.29]). These data are determined by stud5dng the spectra of the molecules - particularly the absorption spectra in the iirffared. Hence, at least for simple molecules, we are able to calculate an absolute value for the entropy - i.e. with no frame of reference, and in particular without the aid of Planck s hypothesis. 

Of course, one is not really interested in classical mechanical calculations. Thus in normal practice the partition functions used in TST, as discussed in Chapter 4, are evaluated using quantum partition functions for harmonic frequencies (extension to anharmonicity is straightforward). On the other hand rotations and translations are handled classically both in TST and in VTST, which is a standard approximation except at very low temperatures. Later, by introducing canonical partition functions one can direct the discussion towards canonical variational transition state theory (CVTST) where the statistical mechanics involves ensembles defined in terms of temperature and volume. There is also a form of variational transition state theory based on microcanonical ensembles referred to by the symbol p,. Discussion of VTST based on microcanonical ensembles pVTST is beyond the scope of the discussion here. It is only mentioned that in pVTST the dividing surface is... 

The quantized energy ej can be of electronic, vibrational, rotational or translational type, readily calculated from the quantum laws of motion. In a macrosystem the sum over all the quantum states for the complete set of molecules, the sum over states defines the canonical partition function ... 

We began this chapter by considering the phase space volume and surface area which are related to the sum and density of states for a system at a given total energy E, that is, a microcanonical system. This is the system of major interest in this book. However, during the discussion of several topics it will be necessary to make use of the partition function, which is appropriate for constant-temperature, or canonical, systems. Because the partition functions for translations, rotations, and vibrations are derived in all undergraduate physical chemistry texts, we will not derive them here, but simply summarize the results. 

Microscopically speaking, the gas is defined by its canonical partition function taking into account the terms linked to the internal energy, of translation and intermolecular interactions. 

The translational canonical partition function with interactions can be written, on the basis of expression [A.3.38], taking account only of the z molecules that are near neighbors of each molecule /. ... 

By substituting expression [1.57] back into relation [1.53], the contribution of the translational motion to the canonical partition function... 

Note 1.2.- In expression [1.60], by comparison with the translational canonical partition function for a perfect gas (relation [A.3.26]), we can define the free volume of the molecules as ... 

Hence, instead of relation [1.61], the translational canonical partition function is written as ... 

The authors show that if we take account of relation [1.81], using the Bragg-Williams approximation Ae= 0 in g, see section 3.1.2) and Stirling s approximation [A.3.1], the translational canonical partition function [1.78]... 

As usual it will be assumed that the canonical partition function Q of the system may be written as the product of an internal partition function which does not depend on the configurational coordinates (but includes the contributions of the internal degrees of freedom and the kinetic energy of rotation and translation) and a configurational partition function... 

Microcanonical TST has found wide application in the case of unimolecular reactions at the limit of high pressure (see below). In this situation, the translational partition functions for the reactants and transition state species are identical, so that eqn (1.14) for the canonical rate constant simplifies to ... 

Ii configuration integral of the canonical translational partition function... 

As previously stated, the classical molecular partition function has units of kg s raised to some power, so a divisor with units must be included to make the argument of the logarithm dimensionless. If a divisor of lkgm s is used, values are obtained for the entropy and the Helmholtz energy that differ from the experimental values. However, when the classical canonical translational partition function is divided by h A and Stirling s approximation is used for ln(iV ), the same formulas are obtained as Chapter 26. For a dilute monatomic gas the corrected classical formula is... 

---