A heat engine is an engine that uses heat to produce mechanical work by carrying a working substance through a cyclic process. The Carnot heat engine uses a particular thermodynamic cycle studied by Nicolas Léonard Sadi Carnot in the 1820s.

The Carnot cycle consists of the following steps:

1. Reversible isothermal expansion of the gas at the "hot" temperature, T_H. During this step, the expanding gas causes the piston to do work on the surroundings. The gas expansion is driven by absorption of heat from the high temperature reservoir.
2. Reversible adiabatic expansion of the gas. For this step we assume the piston and cylinder are thermally insulated, so that no heat is gained or lost. The gas continues to expand, doing work on the surroundings. The gas expansion causes it to cool to the "cold" temperature, T_C.
3. Reversible isothermal compression of the gas at the "cold" temperature, T_C. Now the surroundings do work on the gas, causing heat to flow out of the gas to the low temperature reservoir.
4. Reversible adiabatic compression of the gas. Once again we assume the piston and cylinder are thermally insulated. During this step, the surroundings do work on the gas, compressing it and causing the temperature to rise to T_H. At this point the gas is in the same state as at the start of step 1.

The amount of work produced by the Carnot cycle, w_cy, is the difference between the heat absorbed in step 1, q_H and the heat rejected in step 3, q_C. Or in equation form:

The efficiency η of a heat engine is defined as the ratio of the work done on the surroundings to the heat input at the higher temperature. Thus for the Carnot cycle:

It can also be shown that for the Carnot cycle q_C/q_H = T_C/T_H, so in terms of temperature, the efficiency is:

From Equation 3 it is clear that in order to maximize efficiency one should maximize T_H and minimize T_C.

Carnot's theorem states that No engine operating between two heat reservoirs can be more efficient than a Carnot engine operating between the same reservoirs. Thus, Equation 3 gives the maximum efficiency possible for any engine using the corresponding temperatures. A corollary to Carnot's theorem states that: All reversible engines operating between the same heat reservoirs are equally efficient. So Equation 3 gives the efficiency of any reversible engine.

In reality it is not practical to build a thermodynamically reversible engine, so real heat engines are less efficient than indicated by Equation 3. Nevertheless, Equation 3 is extremely useful for determining the maximum efficiency that could ever be expected for a given set of thermal reservoirs.

Although Carnot's cycle is not practical to build, the expression of Carnot efficiency can find usefulness if TH and TC are replaced by "TH average " and "TC average" in the expression. By "TH average", we mean the average temperture at which heat is input; likewise, "TC average" stands for the average temperature at which heat is rejected. Needless to say, for Carnot cycle or its equivalent, the TH average is the hightest temperature available and TC average is the lowest temperature available. For other less efficient cycles, TH average will be lower than TH , and TC average will be higher than TC. This can help illustrating, for example, why a reheater or a regenerator can improve thermal efficiency.