Carnot Engine and its Efficiency

The Carnot engine is an idealised heat engine, conceived by the French engineer Sadi Carnot, built entirely out of reversible steps and operating between only two fixed temperatures: it absorbs heat only while at the higher temperature $T_1$, and rejects heat only while at the lower temperature $T_2$, with no heat exchanged at any other temperature in between. Its working substance is carried through a closed, repeating sequence of exactly four reversible steps, together called the Carnot cycle:

1. Isothermal expansion at $T_1$ (the working substance is in contact with the hot reservoir throughout): the gas expands slowly, absorbing heat $Q_1$ from the source, doing work on its surroundings, with its temperature held fixed at $T_1$ throughout (Section 11.7).
2. Adiabatic expansion: the working substance is thermally isolated and allowed to expand further, doing more work at the expense of its own internal energy, with no heat exchanged, so its temperature falls smoothly from $T_1$ down to $T_2$ (Section 11.8).
3. Isothermal compression at $T_2$ (now in contact with the cold reservoir): the gas is compressed slowly, rejecting heat $Q_2$ to the sink, with work now being done on the gas, at the fixed lower temperature $T_2$.
4. Adiabatic compression: the working substance is again thermally isolated and compressed further, with work done on it raising its temperature back up smoothly from $T_2$ to exactly $T_1$ -- returning it precisely to its original starting state, and closing the cycle.

On a $P$-$V$ diagram, these four steps trace out a closed, curved, roughly leaf-shaped loop (Fig. 1); because the working substance returns to its original state after step 4, $\Delta U = 0$ over the complete cycle, so by the first law the net work done by the engine over one full cycle, $W = Q_1 - Q_2$, is exactly equal to the area enclosed by this closed loop.

Efficiency of the Carnot engine. Working out the heat exchanged during the two isothermal steps (using the isothermal work/heat formula of Section 11.7) and the temperature relations for the two adiabatic steps (using the adiabatic equations of Section 11.8) together, the volume ratios in the two isothermal legs turn out to be related by the two adiabatic legs in such a way that everything but the temperatures cancels out, leaving the strikingly simple result

so that the efficiency of a Carnot engine, from $\eta = 1 - Q_2/Q_1$, works out to

where $T_1$ and $T_2$ must be expressed on an absolute (kelvin) temperature scale for this formula to be valid. Note that this efficiency depends only on the two operating temperatures $T_1$ and $T_2$, and not at all on the particular working substance used, nor on any detail of how the engine is built -- a remarkable and very general result. …