# Partition Function for a Ideal Gas - Statistical Mechanics

While I was studying statistical mechanics, I saw this in the book that I'm following: We can divide the partition function into a product,

$$\zeta = \zeta_\text{trans}\zeta_\text{int}$$

where $$\zeta_\text{trans}$$ and $$\zeta_\text{int}$$ correspond to translational and internal energy (rotational, vibrational, etc). The book said that, for monoatomic gases, we can just set $$\zeta_\text{int} = 1$$. It didn't explain why. The explicit expression for $$\zeta_\text{int}$$ is : $$\zeta_\text{int} = \frac{1}{h^{l}}\int e^{-\beta H}$$ , $$l = n-3$$

where $$n$$ is the degrees of freedom. Why we can do this for the monoatomic gas?

• Where would internal energy come from for monatomic gases? – BioPhysicist Apr 6 '20 at 23:27
• From nowhere, my point is : what happens to the $$\zeta_{int}$$ integration? – Matheus Barreto Alves Apr 6 '20 at 23:30
• What are you integrating over in your expression for $\zeta_\mathrm{int}$? – By Symmetry Apr 11 '20 at 13:12
• Hi, came upon the same question myself. Is the book you were using Schroeder? If so, can you point out where it is said that Z_int =1 ? – Lopey Tall Jul 13 '20 at 19:20

In gas interatomic interactions are negligible. Therefore $$H$$ is always zero, and $$e^{-\beta H} = 1$$ for any temperature and any point. As the gas is monoatomic, all degrees of freedom are translational. Apparently the book authors choose integration range of each degree of freedom to be equal to Plank's constant $$h$$. Hence the integral equals $$h^l$$ and the interaction partition function $$\zeta_{int} = 1$$.
• You mean $e^{-\beta H} = 1$, right? – Clara Diaz Sanchez Apr 11 '20 at 12:17