2.8. Air approximates an ideal gas obeying Boyle's law $p \propto n$ at constant temperature, and its kmolar heat capacity at constant volume is $C_V = \frac{5}{2}R_0$, independent of temperature (see Section 9.5). 1 kmole of air, initially at a pressure $p_0$ and a temperature $\theta_0$ on the ideal gas scale ($\theta = \frac{p}{k_B n}$), is subjected to the following cycle. First, it is heated at constant volume until its pressure has increased by a factor $r$. It is then heated at constant pressure until its volume has increased by a factor of $r$. It is then cooled at constant volume to its initial pressure, and finally cooled at constant pressure to its initial volume. Assume that each process is carried out reversibly.
(a)Draw the $p$, $V$ diagram analogous to Figure 2.3 for the cycle.
(b) Calculate the work done by the gas in one cycle (Caution: watch signs!).
(c) Calculate the heat input to the gas in the first two stages, and the heat output in the second two. Show that the first law of thermodynamics is obeyed, and show that the efficiency $\eta$, defined as the ratio of work out to heat in, is $\frac{2(r-1)}{7r+5}$.
