Chapter 7
Classical Statistical Mechanics
This chapter focusses mainly upon the classical mechanical evaluation of the
partition function for a gas of structureless atoms, and begins with a discussion
of energy equipartition. The role of interatomic interactions is examined using
the grand partition function, as it enables a more convenient separation of the
roles played by kinetic and potential energy terms. Expressions are developed for
non-ideal contributions to the thermodynamic pressure, entropy, and the internal,
Helmholtz, and Gibbs energies of a gas in terms of virial coefficients and their first
temperature derivatives. A discussion of distribution functions leads to the classical
Liouville and Boltzmann equations, while a consideration of binary collision
dynamics enables development of the form for the monatomic Boltzmann collision
term. An introduction to nonequilibrium phenomena is provided by the derivation
of the traditional transport equations for an atomic gas.
7.1 Introduction
In this chapter we shall consider the classical statistical mechanics for a monatomic
gas. We have seen in Chap. 5 that the canonical partition function for a monatomic
gas made up of N atoms is given by
Z(N, T , V ) =
z N (T , V )
N !
,
in which z(V , T ) is the canonical partition for a single atom, given as
z(T , V ) =
2πmk B T
h 2
2
V .
(7.1.1)
© Springer Nature Switzerland AG 2021
F. R. W. McCourt, Statistical Thermodynamics for Pure and Applied Sciences,
https://doi.org/10.1007/978-3-030-52006-9_7
357
Classical Statistical Mechanics
This chapter focusses mainly upon the classical mechanical evaluation of the
partition function for a gas of structureless atoms, and begins with a discussion
of energy equipartition. The role of interatomic interactions is examined using
the grand partition function, as it enables a more convenient separation of the
roles played by kinetic and potential energy terms. Expressions are developed for
non-ideal contributions to the thermodynamic pressure, entropy, and the internal,
Helmholtz, and Gibbs energies of a gas in terms of virial coefficients and their first
temperature derivatives. A discussion of distribution functions leads to the classical
Liouville and Boltzmann equations, while a consideration of binary collision
dynamics enables development of the form for the monatomic Boltzmann collision
term. An introduction to nonequilibrium phenomena is provided by the derivation
of the traditional transport equations for an atomic gas.
7.1 Introduction
In this chapter we shall consider the classical statistical mechanics for a monatomic
gas. We have seen in Chap. 5 that the canonical partition function for a monatomic
gas made up of N atoms is given by
Z(N, T , V ) =
z N (T , V )
N !
,
in which z(V , T ) is the canonical partition for a single atom, given as
z(T , V ) =
2πmk B T
h 2
2
V .
(7.1.1)
© Springer Nature Switzerland AG 2021
F. R. W. McCourt, Statistical Thermodynamics for Pure and Applied Sciences,
https://doi.org/10.1007/978-3-030-52006-9_7
357