where is the number of atoms, and depends only on the energy of the gas (and is independent of the volume). We obtained this result by integrating over the volume of accessible phase-space. Since the energy of an ideal gas is independent of the particle coordinates (because there are no interatomic forces), the integrals over the coordinates just reduced to simultaneous volume integrals, giving the factor in the above expression. The integrals over the particle momenta were more complicated, but were clearly completely independent of , giving the factor in the above expression. Now, we have a statistical rule which tells us that

(266) |

(267) |

(268) |

(269) |

The above derivation of the ideal gas equation of state is rather elegant. It is certainly far easier to obtain the equation of state in this manner than to treat the atoms which make up the gas as little billiard balls which continually bounce of the walls of a container. The latter derivation is difficult to perform correctly because it is necessary to average over all possible directions of atomic motion. It is clear, from the above derivation, that the crucial element needed to obtain the ideal gas equation of state is the absence of interatomic forces. This automatically gives rise to a variation of the number of accessible states with and of the form (6.6), which, in turn, implies the ideal gas law. So, the ideal gas law should also apply to polyatomic gases with no interatomic forces. Polyatomic gases are more complicated that monatomic gases because the molecules can rotate and vibrate, giving rise to extra degrees of freedom, in addition to the translational degrees of freedom of a monatomic gas. In other words, , in Eq. (265), becomes a lot more complicated in polyatomic gases. However, as long as there are no interatomic forces, the volume dependence of is still , and the ideal gas law should still hold true. In fact, we shall discover that the extra degrees of freedom of polyatomic gases manifest themselves by increasing the specific heat capacity.

There is one other conclusion we can draw from Eq. (265). The statistical
definition of temperature is [Eq. (187)]

(270) |

(271) |

(272) |

The volume independence of the internal energy can also
be obtained directly from the ideal gas equation of state.
The internal energy of a gas can be considered as a general function of the
temperature and volume, so

(273) |

(274) |

(275) |

(276) |

However, is the exact differential of a well-defined state function, . This means that we can consider the entropy to be a function of temperature and volume. Thus, , and mathematics immediately tells us that

(278) |

One well-known property of partial differentials is the equality of second derivatives, irrespective of the order of differentiation, so

(281) |

(282) |

(283) |

(284) |