The particles are assumed to be non-interacting, so the total energy of the gas in state , where there are particles in quantum state , et cetera, is simply

(8.16) |

where the sum extends over all possible quantum states, . Furthermore, because the total number of particles in the gas is known to be , we must have

(8.17) |

In order to calculate the thermodynamic properties of the gas (i.e., its internal energy or its entropy), it is necessary to calculate its partition function,

Here, the sum is over all possible states, , of the whole gas. That is, over all the various possible values of the numbers .

Now, is the relative probability of finding the gas in a particular state in which there are particles in state 1, particles in state 2, et cetera. Thus, the mean number of particles in quantum state can be written

A comparison of Equations (8.18) and (8.19) yields the result

Here, .