= Solution
A clockwise engine <Carnot cycle> has four reversible steps. At hot <temperature> $T_h$, <isothermal expansion> absorbs $Q_h=T_h\Delta S>0$ and does positive <work> by the gas. <Adiabatic expansion> then cools it to $T_c$, with zero <heat> and positive <work> by the gas. <Isothermal compression> at $T_c$ emits $Q_c=-T_c\Delta S$ and requires <work> on the gas. Finally <adiabatic compression> returns it to $T_h$, with zero <heat> and <work> on the gas.
For an <ideal gas>, the $p,V$ isotherms obey $pV=\text{constant}$ and the adiabats $pV^\gamma=\text{constant}$, the latter being steeper. In the $T,S$ plane, the isotherms are horizontal and the adiabats vertical, forming a clockwise rectangle. <Heat> occurs only on the horizontal legs, while <work> occurs on all four $p,V$ legs.
$$
\boxed{W=(T_h-T_c)\Delta S,\qquad\eta_{\rm Carnot}=1-T_c/T_h.}
$$
Reversing the arrows gives a refrigerator, reversing the <heat> and <work> signs.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2017/ii/paper-3-carnot.png]
{title=The <Carnot cycle> in thermodynamic state space}
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