Though we have shown the net flow of energy and entropy, we haven't
proposed a more specific mechanism for the heat engine. The most basic
cycle is known as the Carnot cycle, and is simple if not completely
accurate for a real engine. Still, it is beneficial to see a simplified
picture to understand the basic concepts.

The Carnot cycle consists of four phases. Refer to
as we trace the steps of the cycle. At point A, the gas (it needn't be
a gas necessarily) is at temperature τ_{h} with entropy σ_{L}
where the latter represents the lowest entropy attained by the system
during the cycle and is distinct from σ_{l}. The gas is then
expanded at constant temperature and the entropy is increased to
σ_{H} at point B. The expansion is isothermal, that is, performed
at a constant temperature.

Now, the gas is expanded further, but at constant entropy. The
temperature falls to τ_{l} during this isentropic process and
arrives at point C. The gas is then compressed isothermally to point D, and
is compressed isentropically back to point A, thus completing one cycle.

The total work accomplished by the system can be written from our
previous results as W = Δτ×σ_{h}. Looking at the figure
again, we see that this is merely the area enclosed by the rectangle.
This yields a nice graphical method of understanding a simple version of
a heat engine.

Energies Revisited

We have stressed throughout that knowing well the energy identities
makes problem solving much easier, and we have seen this in many of the
problems we have tackled. It appears again here, as we discuss
processes performed on a gas.

For an isothermal expansion or compression, we wish to deal with an
energy where τ appears as a differential. Conventionally, the
Helmholtz free energy is used. Barring
any diffusive exchange, we can see that dF gives us dU - dQ, which is
exactly the work done on the system.