Approximate treatment, partition function

The above treatment has made some assumptions, such as harmonic frequencies and sufficiently small energy spacing between the rotational levels. If a more elaborate treatment is required, the summation for the partition functions must be carried out explicitly. Many molecules also have internal rotations with quite small barriers, hi the above they are assumed to be described by simple harmonic vibrations, which may be a poor approximation. Calculating the energy levels for a hindered rotor is somewhat complicated, and is rarely done. If the barrier is very low, the motion may be treated as a free rotor, in which case it contributes a constant factor of RT to the enthalpy and R/2 to the entropy. 

For the vibrational term qivib, a classical high-T continuum approximation is seldom valid, and evaluation of the discrete sum over states is therefore required over the quantum vibrational distribution. (As pointed out in Sidebar 5.13, accurate treatment of molecular vibrations is crucial for accurate assessment of entropic contributions to AGrxn.) A simple quantum mechanical model of molecular vibrations is provided by the harmonic oscillator approximation for each of the 3N — 6 normal modes of vibration of a nonlinear polyatomic molecule of N atoms (cf. Sidebar 3.8). In this case, the quantum partition function can be evaluated analytically as... 

The simplest QCE model incorporates environmental effects of cluster-cluster interactions by (1) approximate evaluation of the excluded-volume effect on the translational partition function >trans (neglected in Section 13.3.3) and (2) explicit inclusion of a correction A oenv) for environmental interactions in the electronic partition function qiQiec. Secondary environmental corrections on rotational and vibrational partition functions may also be considered, but are beyond the scope of the present treatment. 

The cell model is a commonly used way of reducing the complicated many-body problem of a polyelectrolyte solution to an effective one-particle theory [24-30]. The idea depicted in Fig. 1 is to partition the solution into subvolumes, each containing only a single macroion together with its counterions. Since each sub-volume is electrically neutral, the electric field will on average vanish on the cell surface. By virtue of this construction different sub-volumes are electrostatically decoupled to a first approximation. Hence, the partition function is factorized and the problem is reduced to a singleparticle problem, namely the treatment of one sub-volume, called cell . Its shape should reflect the symmetry of the polyelectrolyte. Reviews of the basic concepts can be found in [24-26]. 

It is helpful to contrast the view we adopt in this book with the perspective of Hill (1986). In that case, the normative example is some separable system such as the polyatomic ideal gas. Evaluation of a partition function for a small system is then the essential task of application of the model theory. Series expansions, such as a virial expansion, are exploited to evaluate corrections when necessary. Examples of that type fill out the concepts. In the present book, we establish and then exploit the potential distribution theorem. Evaluation of the same partition functions will still be required. But we won t stop with an assumption of separability. On the basis of the potential distribution theorem, we then formulate additional simplified low-dimensional partition function models to describe many-body effects. Quasi-chemical treatments are prototypes for those subsequent approximate models. Though the design of the subsequent calculation is often heuristic, the more basic development here focuses on theories for discovery of those model partition functions. These deeper theoretical tools are known in more esoteric settings, but haven t been used to fill out the picture we present here. 

Also plotted in Fig. 1.2 is the experimental heat capacity of the liquid (at omi-stant pressure) In simple cases, such as polyethylene, the heat capacity of the liquid state could be understood by introducing a heat capacity contribution for the excess volume (hole theory) and by assuming that the torsional skeletal vibration can be treated as a hindered rotator A more general treatment makes use of a separation of the partition function into the vibrational part (approximated for heat capacity by the spectrum of the solid), a conformational part (approximated by the usual conformational statistics) and an external or configurational part. 

Because fluctuations become large at the critical point, the simple, mean-field theory used here breaks down. Large fluctuations mean that the approximation, JijSiSj -> JijSi( ), used to simplify the partition function, Eq. 0-65), is no longer valid since the local value of the concentration is no longer approximately given by the average concentration. Even if these fluctuations are included as corrections to the mean-field approximation, the theory becomes quantitatively inaccurate near the critical point. A detailed theoretical treatment of these critical phenomena is outside the scope of this book (see for example Ref. 24). However, analysis of both simple mean-field theories plus their fluctuation corrections includes most of the important physics and provides a guide to when one must include more sophisticated treatments very close to the critical point. 

The same interpretation results from the above approximate treatment of radical recombination reactions, using the "diatomic" model, where the collision diameter d is related to the high temperature approximation of expression OOliV) for the rotational partition function of product molecule AB, being in a state which should be considered a.non-stationary transition state. 

Partitia functias. If in addition to the ideal gas assumption above, we suppose that the species can be approximated by a rigid rotor-harmonic oscillator treatment, then the molecular partition function of a species i may be separated into its molecular... 

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