Throughout this question you should consider a classical gas and assume that the number of particles is fixed.

(a) Write down the equation of state for an ideal gas. Write down an expression for the internal energy of an ideal gas in terms of the heat capacity at constant volume, $C_{V}$.

(b) Starting from the first law of thermodynamics, find a relation between $C_{V}$ and the heat capacity at constant pressure, $C_{p}$, for an ideal gas. Hence give an expression for $\gamma=C_{p} / C_{V}$.

(c) Describe the meaning of an adiabatic process. Using the first law of thermodynamics, derive the equation for an adiabatic process in the $(p, V)$-plane for an ideal gas.

(d) Consider a simplified Otto cycle (an idealised petrol engine) involving an ideal gas and consisting of the following four reversible steps:

$A \rightarrow B:$ Adiabatic compression from volume $V_{1}$ to volume $V_{2}<V_{1}$;

$B \rightarrow C$ : Heat $Q_{1}$ injected at constant volume;

$C \rightarrow D:$ Adiabatic expansion from volume $V_{2}$ to volume $V_{1}$;

$D \rightarrow A:$ Heat $Q_{2}$ extracted at constant volume.

Sketch the cycle in the $(p, V)$-plane and in the $(T, S)$-plane.

Derive an expression for the efficiency, $\eta=W / Q_{1}$, where $W$ is the work out, in terms of the compression ratio $r=V_{1} / V_{2}$. How can the efficiency be maximized?