Outline

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 .