The classical limit of statistical thermodynamics

In section 1.2, we introduced the quantum mechanical partition function in the T, V, N ensemble. In most applications of statistical thermodynamics to problems in chemistry and biochemistry, the classical limit of the quantum mechanical partition function is used. In this section, we present the so-called classical canonical partition function. 

The canonical partition function introduced in section 1.2 is defined as 

The classical analog of Q(T, V, N) for a system of N simple particles (i.e., spherical particles having no internal structure) is 

Here pi is the momentum vector of the zth particle (presumed to possess only translational degrees of freedom) and m is the mass of each particle. The total potential energy of the system at the specified configuration RN is denoted by U R ). 

Note that the expression (1.60) is not purely classical since it contains two corrections of quantum mechanical origin the Planck constant h and the M. Therefore, Q defined in (1.60) is actually the classical limit of the quantum mechanical partition function in (1.59). The purely classical partition function consists of the integral expression on the rhs of (1.60) without the factor (h3NM). This partition function fails to produce the correct form of the chemical potential or of the entropy of the system. 

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