4. Equation of State and Related Descriptions

From Hansen and Kawaler.

4.4B. Ideal monatomic gas

Elementary statistical mechanics leads to the following expression for the number of particles of species i in phase-space volume d 3r d 3p :

ni(p) d 3r d 3p = (d 3r d 3p / h3) ∑ j gj {e [−μi + εj + ε(p)] / kT ± 1}−1 .

For a gas of distinguishable particles, we may neglect the ±1, or in the event that we can't, we may expand the expression above in a power series. For now, neglect. Then in the isotropic, non-relativistic case,

n(p) dp = (4πp2dp / h3) g e [μεp2/2m] / kT /cm3.

Here ε allows one to refer to a single excitation state of some species.
Integrating over p must result in the total number density n cm−3. Note again: the only notational distinction between n(p) and n is the (p); we probably should use a subscripted np for the former...

The integration is easy and gives the following result for the chemical potential μ:

e μ / kT = n (h3/g) (2πmkT )−3/2 eε / kT .

If we now rewrite the last equation as n = f (μ, T ), and divide the momentum distribution n(p) by n, then g, μ, and ε cancel out and we end up with the Maxwell distribution:

(1/n) n(p) dp = (4πp2dp) (2πmkT )−3/2 ep2/2mkT .

Inserting n(p) and the expression for e μ / kT into the pressure and energy integrals give the familiar

P = nkT and E = (3/2) nkT .