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Carnot Engine

In the beginning of the nineteenth century, different types of heat engines were in use. The efficiency of these engines was found to vary between 5% and 55%, which means that at the most only 55% of the total heat energy supplied is converted into work. The question, therefore, naturally arises, whether the low efficiency was due to bad designing of the engine or whether there is something in the very nature (i.e. whether there is a law of nature) of heat engines that forbids the complete conversion of heat into work.

It was the brilliant work of a French engineer Sadi Carnot (1824) that clearly explained how and to what extent work could be obtained from heat and proved that it was not possible to convert the whole of heat into useful work and that this is a consequence of a certain law of nature.

An ideal heat engine is free from all the imperfections of an actual engine and hence cannot be realized in practice. It was conceived by Carnot. It consists of
  • A cylinder, with perfectly non-conducting walls but with a perfectly conducting bottom, containing one gram-molecule of a perfect gas as the working substance.
  • A hot body, of high thermal capacity, maintained at a constant high temperature T1K to serve as the source of heat.
  • A cold body at a constant lower temperature T2K to serve as the sink.
  • A perfectly non-conducting platform to serve as the stand for the cylinder.

Carnot's engine has to be operated in a special way in order that it may supply continuous work. The special way is called the Carnot's cycle.

Carnot's Cycle
The gas in the cylinder is subjected to a series of operations as follows.

Operation One: System is placed on source


The cylinder is first placed on the source, so that the gas acquires the temperature T1 of the source. It is then allowed to undergo a quasi-static expansion. As the gas expands, it does external work against the pressure on the piston and its temperature tends to fall. The fall in temperature is compensated by the heat passing into the cylinder through its perfectly conducting base which is in contact with the source.

The gas therefore undergoes a slow isothermal expansion at a constant temperature T1. This operation is represented by the isothermal AB on the indicator diagram.

In accordance with the first law of thermodynamics the quantity of heat absorbed Q1, must be equal to the external work done W1 by the gas during isothermal expansion along AB from (V1, P1) to (V2, P2). Hence


Operation Two: System is placed on insulating stand


The cylinder is now removed from the source to the insulating stand, so that the gas is thermally isolated from the surroundings. The system is allowed to undergo a slow adiabatic expansion, performing external work at the expense of its internal energy, until its temperature falls to T2, the same as that of the sink.

The gas therefore undergoes a slow adiabatic expansion; this operation is represented by the isothermal BC on the indicator diagram.


Operation Three: System is placed on sink


The cylinder is now transferred from the insulating stand to the sink at temperature T2 and the piston is gradually moved downwards i.e. work is done on the system. So the gas is gradually compressed. The temperature of the gas naturally rises due to the heat produced by the compression.

The gas therefore undergoes a slow isothermal compression; this operation is represented by the isothermal CD on the indicator diagram.


Operation Four: System is placed on insulating stand


The cylinder is again removed to the insulating stand, to thermally isolate the gas and the slow downward movement of the piston continues. Work thus continues to be done on the gas which continues to be further compressed adiabatically and the temperature rises.

The gas therefore undergoes a slow adiabatic compression; this operation is represented by the adiabatic DA on the indicator diagram.


Total work done by the gas in one complete cycle,


The efficiency of the Carnot engine is

.... (1)


We can write Since W2 3 is an adiabatic process,

..... (2)

...... (3)

..... (4)

Similarly Since W4 1 is an adiabatic process,

.... (5)

.... (6)

…… (7)

From (4) and (7),

The efficiency becomes,
This is the maximum efficiency and it is independent of the nature of the system performing the Carnot cycle of operations.
This is a universal relation independent of the nature of the system. The efficiency of heat engine depends upon the temperatures T1 and T2 of the source and the sink respectively and the greater the value of (T1-T2), the higher the efficiency. Since (T1-T2) must always be less than T1, it is clear that the efficiency of any heat engine is always less than 1 or 100%.

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