 Solution

ID: past-exam-of-the-physics-b-course-of-the-university-of-cambridge/2022/paper-2/section-c/c8/a/solution

Use an ideal gas to illustrate the Carnot cycle, with hot and cold heat baths at

T_{h} = T_{2}

and

T_{c} = T_{1}

. The four processes are all reversible thermodynamic processes. Lettered states below belong only to this illustration; the engine later in the question uses numbered states.

Starting at

A

, perform the following four stages:

$A \to B$ : isothermal expansion at $T_{h}$ . The gas absorbs heat $Q_{h}$ from the hot heat bath and does work. Its internal energy is unchanged, so $Q_{h} = W_{A B} = n R T_{h} ln (V_{B} / V_{A})$ .
$B \to C$ : reversible ideal-gas adiabat during expansion. Insulation prevents heat transfer; work done by the gas reduces its internal energy and lowers its temperature from $T_{h}$ to $T_{c}$ .
$C \to D$ : isothermal compression at $T_{c}$ . Work done on the gas is rejected as heat $Q_{c}$ to the cold heat bath, with $Q_{c} = n R T_{c} ln (V_{C} / V_{D}) > 0$ .
$D \to A$ : insulated adiabatic compression, performed reversibly, raises the temperature back to $T_{h}$ and closes the cycle.

The clockwise enclosed area in the pressure-volume diagram is the positive net work per cycle,

W = \oint p d V = Q_{h} - Q_{c}

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