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A cyclic process is one that returns the system, but not the environment, to its original state. A closed cycle consisting of two isothermal and two adiabatic transformations is called a Carnot cycle after the French physicist Sadi Carnot, who first discussed the implications of such cycles. During the Carnot cycle occurring in the operation of a heat engine, a definite quantity of heat is absorbed from a reservoir at high temperature; part of this heat is converted into useful work, but the balance is expelled into a low-temperature reservoir and thus "wasted." The greater the temperature difference between the two reservoirs, which in a steam engine are represented by the boiler and the condenser, the greater the fraction of absorbed heat that is converted into useful work. It is, however, theoretically impossible to convert all the heat extracted from the reservoir into useful work.
In general it is impossible to perform a transformation whose only final result is to convert into useful work heat extracted from a source that is at the same temperature throughout. This statement is Lord Kelvin's version of the second law of thermodynamics. Another version of this law, formulated by R. J. E. Clausius, states that a transformation is impossible whose only final result is to transfer heat from a body at a given temperature to a body at higher temperature; in other words, the spontaneous flow of heat from hot to cold bodies is reversible only with the expenditure of mechanical or other nonthermal energy. These two versions of the second law of thermodynamics can be shown to be entirely equivalent.
The second law is expressed mathematically in terms of the concept of entropy. When a body absorbs an amount of heat Q from a reservoir at temperature T, the body gains and the reservoir loses an amount of entropy S=Q/T. Thus, in a reversible adiabatic process (no heat change) there is no change in the total entropy. If an amount of heat Q flows from a hot to a cold body, the total entropy increases; because S=Q/T is larger for smaller values of T, the cold body gains more entropy than the hot body loses. The statement that heat never flows from a cold to a hot body can be generalized by saying that in no spontaneous process does the total entropy decrease.
In all real physical processes entropy increases; in ideal reversible processes entropy remains constant. Thus, in the Carnot cycle, which is reversible, there is no change in the total entropy. The engine itself experiences no net change in entropy because it is returned to its original state at the end of the cycle. The entropy gained by the low temperature reservoir is equal to the entropy lost by the high temperature reservoir. However, according to the formula S=Q/T, less heat need be expelled into the low temperature reservoir than is extracted from the high temperature reservoir for equal and opposite changes in entropy. In the Carnot cycle this difference in heat appears as useful mechanical work.
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