Past exam of the physics B course of the University of Cambridge / 2017 / Paper 2 / Section C / C10 / Heat exchanges and efficiency / Solution

Use heat into the gas as positive. The two adiabatic stages have $Q_{12}=Q_{34}=0$. During constant-pressure heating and constant-volume cooling,The first law over a complete cycle gives $W_{\mathrm{out}}=Q_{23}+Q_{41}$, since the internal energy returns to its initial value. ThereforeSubstitute the temperatures:HenceEquivalently, define compression ratio $r=V_1/V_2$ and cutoff ratio $\rho =V_3/V_2$:These formulas use the ideal constant-heat-capacity model and reversible adiabatic strokes. The heat-addition and heat-rejection strokes occur over changing gas temperatures, unlike the isothermal heat exchanges of a Carnot engine.