(a) Describe the Carnot cycle using plots in the $(p, V)$-plane and the $(T, S)$-plane. In which steps of the cycle is heat absorbed or emitted by the gas? In which steps is work done on, or by, the gas?

(b) An ideal monatomic gas undergoes a reversible cycle described by a triangle in the $(p, V)$-plane with vertices at the points $A, B, C$ with coordinates $\left(p_{0}, V_{0}\right),\left(2 p_{0}, V_{0}\right)$ and $\left(p_{0}, 2 V_{0}\right)$ respectively. The cycle is traversed in the order $A B C A$.

(i) Write down the equation of state and an expression for the internal energy of the gas.

(ii) Derive an expression relating $T d S$ to $d p$ and $d V$. Use your expression to calculate the heat supplied to, or emitted by, the gas along $A B$ and $C A$.

(iii) Show that heat is supplied to the gas along part of the line $B C$, and is emitted by the gas along the other part of the line.

(iv) Calculate the efficiency $\eta=W / Q$ where $W$ is the total work done by the cycle and $Q$ is the total heat supplied.