Ideal Gas Partition Function

Derive the partition function for an ideal gas and extract thermodynamic quantities from it.

Single Particle in a Box

Energy levels for a particle in a 3D box of side $L$ :

E_{n_{x}, n_{y}, n_{z}} = \frac{ℏ^{2} π^{2}}{2 m L^{2}} (n_{x}^{2} + n_{y}^{2} + n_{z}^{2})

Single-particle partition function — convert sum to integral (valid when $k_{B} T ≫$ level spacing):

z = \int_{0}^{\infty} d n_{x} d n_{y} d n_{z} e^{- β \frac{ℏ^{2} π^{2}}{2 m L^{2}} (n_{x}^{2} + n_{y}^{2} + n_{z}^{2})} = {(\frac{L}{λ})}^{3} = \frac{V}{λ^{3}}

where $λ = \sqrt{\frac{2 π ℏ^{2}}{m k_{B} T}}$ is the thermal de Broglie wavelength.

Particles are indistinguishable, so divide by $N!$ :

Z = \frac{z^{N}}{N!} = \frac{1}{N!} {(\frac{V}{λ^{3}})}^{N}

\ln Z = N \ln V - 3 N \ln λ - \ln N!

Using Stirling's approximation ( $\ln N! \approx N \ln N - N$ ):

⟨ E ⟩ = - \frac{\partial \ln Z}{\partial β} = \frac{3}{2} N k_{B} T

P = k_{B} T \frac{\partial \ln Z}{\partial V} = \frac{N k_{B} T}{V} \Rightarrow P V = N k_{B} T

S = k_{B} (\ln Z + β ⟨ E ⟩) = N k_{B} [\ln \frac{V}{N λ^{3}} + \frac{5}{2}] e x t (S a c k u r - T e t r o d e)

The classical partition function arrives at the same result via phase space integration:

Z = \frac{1}{N! h^{3 N}} \int d^{3 N} p d^{3 N} q e^{- β H}

The $h^{3 N}$ factor sets the fundamental phase space cell size — needed to make $Z$ dimensionless and $S$ extensive.