Statistical Mechanics of a Perfect Gas in Boltzmann Statistics

Here XjyjZj are the coordinates of the center of gravity, Px Vv V i the components of total momentum, of the molecule. 

The integration over z, y, z is to be carried over the volume of the container and gives simply a factor V. The integrations over px, py, ps arc carried from — oo to oo, and can be found by Eqs. (2.3) of Chap. IV. The integral depending on the internal coordinates and momenta will not be further discussed at present we shall abbreviate it 

is the internal partition function of a single molecule. The second way of writing it, in terms of a summation, by analogy with Eq. (5.17) of Chap. Ill, refers to a summation over all cells in a 2s-dimen-sional phase space in which qi p8 are the dimensions. We note, for future reference, that the quantity Zi depends on the temperature, but not on the volume of the gas. 

From the partition function (3.5), we can now find the Helmholtz free energy, entropy, and Gibbs free energy of our gas. Using the equation 

From A we can find the pressure by the equation P = — (dA/dV)T. We have at once 

---