Paper 1, Section II, D

Statistical Physics
Part II, 2019

(a) Explain, from a macroscopic and microscopic point of view, what is meant by an adiabatic change. A system has access to heat baths at temperatures T1T_{1} and T2T_{2}, with T2>T1T_{2}>T_{1}. Show that the most effective method for repeatedly converting heat to work, using this system, is by combining isothermal and adiabatic changes. Define the efficiency and calculate it in terms of T1T_{1} and T2T_{2}.

(b) A thermal system (of constant volume) undergoes a phase transition at temperature TcT_{\mathrm{c}}. The heat capacity of the system is measured to be

C={αT for T<Tcβ for T>TcC= \begin{cases}\alpha T & \text { for } T<T_{\mathrm{c}} \\ \beta & \text { for } T>T_{\mathrm{c}}\end{cases}

where α,β\alpha, \beta are constants. A theoretical calculation of the entropy SS for T>TcT>T_{\mathrm{c}} leads to

S=βlogT+γS=\beta \log T+\gamma

How can the value of the theoretically-obtained constant γ\gamma be verified using macroscopically measurable quantities?