Past exam of the thermal and statistical physics course of the University of Cambridge / 2018 / Section A / A1

First solve the equation of state for the volume:The Gibbs free energy has differential $dG=-SdT+Vdp+\mu dN$. At fixed temperature and particle number, integrate $dG=Vdp$ between the two pressures:ConsequentlyUse $S=-(\partial G/\partial T{)}_{p,N}$, keeping both endpoint pressures fixed while differentiating the difference:Equivalently, the Maxwell relation $(\partial S/\partial p{)}_{T,N}=-(\partial V/\partial T{)}_{p,N}$ gives $-N{k}_B/p+N{A}^{\prime }(T)$, whose pressure integral is the same result. These are the Gibbs free-energy change for a pressure-linear gas correction identities. The physical path requires $p_0,p_1>0$ and positive volume throughout. The correction $A(T)$ has units of volume per particle, so each term in $\Delta G$ is an energy and each term in $\Delta S$ has units of energy per kelvin.

For a phonon mode of angular frequency $\omega$, the equilibrium occupation isWell above the Debye temperature, the relevant vibrational energies are small compared with $k_{B}T$. Expanding the exponential givesThe spectrum and its cutoff are fixed in the model, so the remaining sum is temperature independent. ThusFor example, the three-dimensional Debye model with $D(\omega )=9N{\omega }^2/{\omega }_D^3$ gives $N_{\mathrm{ph}}\simeq (9/2)NT/{\Theta }_D$. This is the high-temperature phonon number in the Debye model. The number of phonon quanta can exceed $3N$; $3N$ counts modes, not their total occupation. Under the stipulated proportionality to phonon number, only the phonon contribution to resistivity has been calculated. A temperature-independent residual resistivity would add an intercept in a real sample.

Particles within each species are indistinguishable, while the two species are physically distinguishable and their numbers are fixed. Independence makes the canonical partition function factor by species:Since $Z_1\propto T^{3/2}$ at fixed volume, $U=-{\partial }_{\beta }\ln Z=3N{k}_{B}T/2$. The canonical entropy is thereforeFor large species populations, Stirling's approximation gives the equivalent Sackur-Tetrode equation formFor structureless particles with the same translational partition function, $Z_1=V/{\lambda }^3$, where $\lambda$ is their common thermal de Broglie wavelength. More generally the supplied common constant factor in $Z_1$ should be retained in the entropy.

Writing $x_i=N_i/N$ separates the total-number contribution from the composition term:The last term is the entropy of mixing relative to separated ideal components at the same temperature and pressure. This is the entropy of a two-species classical ideal gas. Artificially labeling otherwise identical chemical atoms would not justify dividing by separate factorials or adding a mixing entropy. The pure-species limit is obtained by $x\ln x\to 0$.