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 Sadi Carnot in the 1820s.

The Carnot cycle consists of the following steps:

**Reversible isothermal expansion of the gas at the "hot" temperature, T**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._{H}.**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}.**Reversible isothermal compression of the gas at the "cold" temperautre, T**Now the surroundings do work on the gas, causing heat to flow out of the gas to the low temperature reservoir._{C}.**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:

w_{cy}= q_{H}- q_{C}(1)

The efficiency 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:

efficiency = w_{cy}/q_{H}= (q_{H}- q_{C})/q_{H}= 1 - q_{C}/q_{H}(2)

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:

efficiency = 1 - T_{C}/T_{H}(3)

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 theory 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.

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