Classical ideal-gas partition function

For

N

indistinguishable noninteracting classical particles,

Z_{N} = z_{1}^{N} / N!

, where

z_{1}

is the normalized one-particle phase-space integral.

Thermal wavelength

The three-dimensional translational momentum integral is

1/ λ^{3}

per unit volume, where

λ = \frac{2 π ℏ ^{2}}{m k _{B} T} .

p c + U (x)

Z_{1} = \frac{( k _{B} T ) ^{3}}{π ^{2} ( ℏ c ) ^{3}} \int e^{- U (x) / (k_{B} T)} d^{3} x .

U (r) = r^{2 n} / V^{2 n /3}

, then

Z_{1} = \frac{4}{π ( ℏ c ) ^{3}} V (k_{B} T)^{3 + 3/ (2 n)} I_{n}, I_{n} = \int_{0}^{\infty} u^{2} e^{- u^{2 n}} d u .

Consequently

p V = N k_{B} T

, while the mean energy and variance are

N A k_{B} T

and

N A (k_{B} T)^{2}

with

A = 3 + 3/ (2 n)

The radial density is proportional to

r^{2} e^{- r^{2 n} / (V^{2 n /3} k_{B} T)}

, whose mode is

r = V^{1/3} (\frac{k _{B} T}{n})^{1/ (2 n)} .

In the classical dilute regime, a monatomic ideal gas has entropy

S = N k_{B} [lo g (\frac{V}{N λ ^{3}}) + \frac{5}{2}] .

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