The Perfect Classical gas

We have introduced the perfect gas in Chapter 1, and wrote down its equation of state in Section . This is a relation between pressure, volume, temperature and number of particles, which is valid whenever the gas is in a state of equilibrium, that is when these macroscopic variables are well defined, meaning that they are uniform throughout the gas. The perfect gas is an idealisation, but a good one for real gases at low enough density, when the potential energy due to the interactions between the particles is much lower than their kinetic energy.

Under the assumption of no interaction, each particle of gas is independent from every other particle and so we can think of the gas as a collection of particles, each having its own private energy from the list of allowed values $\epsilon_1 \le \epsilon_2 \le \dots$ ^4.1. This list of energies is the same for every particle and it depends on the volume of the enclosure containing the gas. There will be $n_1$ particles in single particle microstate 1 with energy $\epsilon_1$, $n_2$ particles in single particle microstate 2 with energy $\epsilon_2$ and so on, and each of the $n_r$'s can vary between 0 and the total number of particles $N$, so we can write the energy of a particular microstate of the gas:

$$E(n_1,n_2,\dots) = \sum_r n_r \epsilon_r,$$ (4.1)

with the auxiliary condition

$$N = \sum_r n_r.$$ (4.2)

The sum runs over all possible single particle microstates and so it may be infinite. An equivalent way of writing the energy of the gas microstate would be:

$$E(r_1,r_2,\dots,r_N) = \sum_{i=1}^N \epsilon_{r_i},$$ (4.3)

where $\epsilon_{r_i}$ is the energy of particle $i$ in single particle microstate $r_i$.