Evaluating quantities from given Z(T,V,N)
(a) After some calculations, the partition function of a 3D classical non-relativistic ideal gas is found to be
Z(T,V,N)=N!1zᴺ
where z is the single-particle partition function given by z(T,V)=∫0[infinity]g(ϵ)e−ϵ/kTdϵ=4π2V(ℏ22m)3/2∫0[infinity]ϵ1/2e−ϵ/kTdϵ Evaluate the integral (hint: Gamma Functions may be useful) and show that the result is the familiar Z(T,V,N)=N!1(λth3V)N where λth(T) is the thermal de Broglie wavelength.
(b) After some calculations, the partition function of a 3 D classical ultra-relativistic ideal gas is found to be Z(T,V,N)=N!1zN
where z is the single-particle partition function given by
z(T,V)=∫0[infinity]g(ϵ)e−ϵ/kTdϵ=c3h34πV∫0[infinity]ϵ2e−ϵ/kTdϵ
(i) Evaluate the integral and hence obtain Z(T,V,N)
(ii) Evaluate the pressure P and obtain an equation of state for the gas.
(iii) Evaluate the entropy S and check that your answer is indeed extensive.
(iv) Evaluate the energy ⟨E⟩ (which is U in thermodynamics) and hence find the heat capacity CV. [Remarks: An ultra-relativistic gas refers to systems in which the particles' energies are much bigger than the rest energy, and so ϵ=c2p2+m2c4≈cp=cℏk. The dispersion relation ϵ=cℏk is different from ϵ=ℏ2k2/2m for non-relativistic particles. Therefore, the density of (s. p) states is different for ultra-relativistic particles. ]