Statistical Thermodynamics

5Canonical Ensemble of Gas Molecules

Having learned the partition function Z of a canonical ensemble and how to obtain thermodynamic functions from Z, we apply these tools to an ideal gas in this chapter. In Section 5.1, we apply the canonical distribution function to find a distribution of the velocity of molecules. In Section 5.2, we consider the thermodynamics of an ideal gas using classical representation of the energy. In Section 5.3, we consider its quantum‐mechanical version.

The thermodynamic functions we consider in this chapter are the internal energy and heat capacity only. We need to learn the indistinguishability in Chapter 7 before being able to correctly express the chemical potential.

In Section 5.4, we consider the most probable state for the rotation of a diatomic molecule. Section 5.5 considers conformations of a molecule.

5.1 Velocity of Gas Molecules

We apply the canonical distribution to find the velocity distribution of molecules in vapor phase. Molecules in a gas collide with each other to change their velocities. At equilibrium, a steady distribution of the velocity is established. It was James Clark Maxwell who first derived the correct distribution function in 1857, long before statistical mechanics was introduced or the concept of Boltzmann distribution was established.

Considering monatomic gas makes the derivation easy, since the molecules do not have rotational or vibrational motion. Since only the center‐of‐mass movement is involved in the velocity ...