Partition Function and Ideal Gas

Partition function encodes all thermodynamic properties of a system from its energy spectrum.

Partition Function

For a system with energy states $E_{i}$ :

Z = \sum_{i} e^{- E_{i} / k_{B} T}

Key quantities derived from $Z$ :

Quantity	Formula
Free energy	$F = - k_{B} T \ln Z$
Average energy	$⟨ E ⟩ = - \frac{\partial \ln Z}{\partial β}$ , where $β = \frac{1}{k_{B} T}$
Heat capacity	$C_{V} = \frac{\partial ⟨ E ⟩}{\partial T}$
Entropy	$S = k_{B} (\ln Z + β ⟨ E ⟩)$
If a system is composed of independent subsystems: $Z = Z_{1} \cdot Z_{2} \cdot \dots$

Energy levels: $E_{n} = ℏ ω (n + \frac{1}{2})$ , $n = 0, 1, 2, \dots$

Z = \sum_{n = 0}^{\infty} e^{- β ℏ ω (n + 1 / 2)} = \frac{e^{- β ℏ ω / 2}}{1 - e^{- β ℏ ω}}

⟨ E ⟩ = \frac{ℏ ω}{2} + \frac{ℏ ω}{e^{β ℏ ω} - 1}

For $N$ non-interacting particles in volume $V$ , the partition function factorizes: $Z = z^{N}$ where $z$ is the single-particle partition function.

z = \frac{V}{λ^{3}}, λ = \sqrt{\frac{2 π ℏ^{2}}{m k_{B} T}}

( $λ$ is the thermal de Broglie wavelength)

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

The $N!$ accounts for indistinguishability (fixes Gibbs paradox, see Gibbs Paradox).
Key results:

⟨ E ⟩ = \frac{3}{2} N k_{B} T

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

S = N k_{B} [\ln \frac{V}{N λ^{3}} + \frac{5}{2}] (Sackur-Tetrode equation)