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*^{ 3}**r** *d*^{ 3}**p** :

**n**_{i}(*p*) *d*^{ 3}**r** *d*^{ 3}**p** =
(*d*^{ 3}**r** *d*^{ 3}**p** / *h*^{3})
∑_{ j} *g*_{j} {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* =
(4π*p*^{2}*dp* / *h*^{3})
*g* e^{ [μ −
ε −
p2/2m] / kT} /cm^{3}.

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 *n*_{p} for the former...

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

e^{ μ / kT} = *n* (*h*^{3}/*g*)
(2π*mkT* )^{−3/2} e^{ε / 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* = (4π*p*^{2}*dp*)
(2π*mkT* )^{−3/2} e^{−p2/2mkT} .

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* .