The Carnot Cycle
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 τh with entropy σL
where the latter represents the lowest entropy attained by the system
during the cycle and is distinct from σl. The gas is then
expanded at constant temperature and the entropy is increased to
σH 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 τl 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×σh. 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.
Figure 2.1: A Carnot Cycle
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.
For an isobaric process, we wish to use the
Enthalpy, for the pressure appears in the
differential there. This choice allows us to carefully account for the
work done on the system and that done on the environment in a process.
For a process that is both isobaric and isothermal, it makes sense to
look at the Gibbs Free Energy.
Therefore, while solving problems, look for what is being held constant
so that you can make an appropriate choice of energy.
