(Solved): (e) The single-particle partition function for the ideal gas is given by, \[ Z_{1}=\int_{0}^{\infty ...

(e) The single-particle partition function for the ideal gas is given by, \[ Z_{1}=\int_{0}^{\infty} e^{-\beta \epsilon} g(\epsilon) \mathrm{d} \epsilon, \] Use the expression for \( g(\epsilon) \) in part (c), to show that the single-particle partition function for a ultra-relativistic gas can be written as, \[ Z_{1}=\frac{8 \pi V}{(\beta h c)^{3}} . \] (f) We can also write the single-particle partition function as \( Z_{1}=V / \Lambda_{\mathrm{th}}^{3} \), where \( \Lambda_{\mathrm{th}} \) is the thermal wavelength. Show that the thermal wavelength is given by, \[ \Lambda_{\mathrm{th}}=\frac{\hbar c \pi^{2 / 3}}{k_{B} T} . \] (g) The partition function for \( N \) molecules in a gas is given by, \[ Z_{N}=\frac{1}{N !}\left(\frac{V}{\Lambda_{\mathrm{th}}^{3}}\right)^{N} \text {. } \] Find the internal energy and show that the heat capacity is given by, \( C_{V}=3 \mathrm{Nk}_{B} \). Comment on its value with respect to the non-relativistic ideal gas. (h) The Helmholtz free energy can be determined by, \( F=-k_{B} T \ln Z_{N} \). Use this in a relation from the practice question sheets to find an expression for the pressure, \( p \), and therefore the ideal gas equation. (i) Show that the average energy density for a relativistic gas is given by \( \bar{u}=3 p \), where \( p \) is the pressure. (j) In the early stages of the universe we can consider the relativistic gas to be expanding adiabatically whilst remaining in equilibrium. If this gas is monatomic, find \( \gamma \equiv C_{p} / C_{V} \) and hence show that, if the volume of the universe expands as \( \left(V \rightarrow a^{3} V\right) \), where \( a(t) \) is the scale factor that describes the expansion, then the energy density of the gas decreases as, \[ \bar{u} \propto a^{-4} \]