Chapter 5
Atomic Systems
This chapter begins with the development of canonical and isothermal–isobaric
partition functions for large systems of atoms that have nonreactive ground
electronic states and may only access translational states. Expressions are obtained
for the internal energy, entropy, pressure, constant-volume heat capacity, Gibbs
energy, and the chemical potential associated with translational motion. None
of the final thermodynamic expressions for these quantities obtained using
the two partition functions differ in the thermodynamic limit. Expressions are
also developed for nonzero nuclear spins and excited electronic atomic state
contributions to the thermodynamic functions. Vibrational contributions to the
thermodynamic internal energy, heat capacity, entropy, and Helmholtz energy
are obtained for an ensemble of simple harmonic oscillators. Simple harmonic
oscillator concepts are then utilized to develop the Einstein and Debye models for
the constant-volume heat capacity of monatomic crystalline solids.
5.1 Ground Electronic Term Atoms
We have at this point covered the most difficult aspects of equilibrium statistical
thermodynamics, namely the concepts of ‘ensemble’, ‘partition function’, and their
connections to thermodynamics itself. What remains to be achieved in order to use
the subject matter is to be able to perform an actual evaluation of a partition function
and of the quantities that we have derived from it in a formal fashion. In order to
accomplish this goal, we have to be able to reduce the problem of evaluating the
statistical thermodynamic expressions to their simplest terms. We shall accomplish
this goal by reducing the problem ultimately to the evaluation of these quantities in
terms of the properties of individual atoms and molecules.
© 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_5
213
Atomic Systems
This chapter begins with the development of canonical and isothermal–isobaric
partition functions for large systems of atoms that have nonreactive ground
electronic states and may only access translational states. Expressions are obtained
for the internal energy, entropy, pressure, constant-volume heat capacity, Gibbs
energy, and the chemical potential associated with translational motion. None
of the final thermodynamic expressions for these quantities obtained using
the two partition functions differ in the thermodynamic limit. Expressions are
also developed for nonzero nuclear spins and excited electronic atomic state
contributions to the thermodynamic functions. Vibrational contributions to the
thermodynamic internal energy, heat capacity, entropy, and Helmholtz energy
are obtained for an ensemble of simple harmonic oscillators. Simple harmonic
oscillator concepts are then utilized to develop the Einstein and Debye models for
the constant-volume heat capacity of monatomic crystalline solids.
5.1 Ground Electronic Term Atoms
We have at this point covered the most difficult aspects of equilibrium statistical
thermodynamics, namely the concepts of ‘ensemble’, ‘partition function’, and their
connections to thermodynamics itself. What remains to be achieved in order to use
the subject matter is to be able to perform an actual evaluation of a partition function
and of the quantities that we have derived from it in a formal fashion. In order to
accomplish this goal, we have to be able to reduce the problem of evaluating the
statistical thermodynamic expressions to their simplest terms. We shall accomplish
this goal by reducing the problem ultimately to the evaluation of these quantities in
terms of the properties of individual atoms and molecules.
© 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_5
213