In 1824, a French scientist N.L.Sadi Carnot, suggested an idealized engine which is called the Carnot engine, which has an intimate relation with the second law of thermodynamics. To understand the principle, let us consider an ideal gas taken in a cylinder. The bottom of the cylinder is diathermic, whereas the rest of it is adiabatic. An adiabatic piston is fitted into the cylinder. Also, suppose we have two large bodies, one at a constant high temperature T
1 and the other at a lower temperature T
2.
The figure (a) shows the basic process of a Carnot engine on a PV diagram. The other parts of the figure represent the process schematically. Consider a cylinder being kept in contact with the high-temperature body, at temperature T
1 in a compressed state. This state is represented by the point a in the PV diagram. The gas is isothermally expanded to a state b (figure b). Work is done by the gas and an amount of heat Q
1 is supplied to it by the body at temperature T
1. The cylinder is now kept on an adiabatic platform and the gas is allowed to expand further to the state c (figure c). This is an adiabatic expansion and the temperature falls from T
1 to T
2. Work is done by the gas. At this stage, the cylinder is put in contact with the lower-temperature body at temperature T
2. It is isothermally compressed to a state d (figure d). Work is done on the gas and the gas rejects an amount of heat Q
2 to the body at the lower temperature T
2. Finally, it is kept on the adiabatic platform and is further compressed to reach the state a, where the temperature is T
1 (figure e).
The process represented by abcda in figure(a) is a cyclic process. If the piston is frictionless and is moved very slowly, the process is a reversible cyclic process.
Efficiency of a carnot engine
The basic process of a Carnot engine, described above, is again shown in the above figure in a T-S (temperature-entropy) diagram. The points a, b, c and d represent the same states as in figure (a). Let the entropy in state a be S1. An amount of heat Q1 is supplied to the system in the isothermal process ab at the temperature T1. The entropy increases in this part, as heat is supplied to the system. Also, by definition,
The entropy remains constant in the part bc as it describes an adiabatic process. Therefore, the entropy in state c is S2. In the part cd, the system gives a heat Q2 at the lower temperature T2 and its entropy decreases. The part da represents an adiabatic process and the entropy remains constant. As the entropy in state a is S1, the entropy in state d is also S1. Using the definition of change in entropy for the process cd,
From (i) and (ii),
The efficiency of the engine is
Thus, the efficiency of the engine depends only on the temperatures of the hot and cold bodies between which the engine works.
Carnot's theorem
Carnot engine is a reversible engine. It can be proved from the second law of thermodynamics that:
All reversible engines operating between the same two temperatures have equal efficiency and no engine operating between the same two temperatures can have efficiency greater than this.
This theorem is called Carnot's theorem. It is a consequence of the second law and puts a theoretical limit
to the maximum efficiency of heat engines.
