(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 $T_{1}$ and $T_{2}$, with $T_{2}>T_{1}$. 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 $T_{1}$ and $T_{2}$.

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

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

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

$$S=\beta \log T+\gamma$$

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