Carnot’s theorem proof and Carnot’s refrigerator

  • In today’s article, we are going to study about ‘Carnot’s theorem proof and Carnot’s refrigerator’. We will also proof of carnot theorem, carnot’s refrigerator, coefficient of performance of refrigerator, definition of carnot cycle etc in detail. So, take your notebooks in you hand and get ready to study physics in an easy and sorted way.

Carnot’s theorem

The Carnot’s theorem states that all reversible engines working between the same two temperature of the source and sink are equally efficient, whatever may be the working substance.

To prove this theorem let us assume that there are two reversible engines A and B both of which operate between the same source and the sink. Both engine are arranged in such a way that A works in forward direction. And B works in opposite direction as shown in fig. Suppose the efficiency of engine A is ηA and that of B is ηB and engine A is more efficient than B i.e. ηA> ηB.

Engine A absorbs a certain quantity of heat (q2) from the source at T2, converts a certain part of it into work w, and gives up the remaining part -(q2 – w) to the sink at T1. Therefore, engine B absorbs the same amount of heat from the source but being less efficient converts a smaller part into work (w’). And thus, transfers a large quantity of heat (q2 – w’) to the sink.


When these reversible engines are coupled the composite system of engine is obtained in which A operates in the direct manner while B in the opposite direction i.e. it withdraws heat from the sink. Same work is done on it and then given up that to source, thus acts as a refrigerator. Now using the convention that heat absorbed by system is taken positive and given by systems is taken as negative. And work done by the system is negative. Work done on the system is positive the work heat transfer for both engines are as follows :

Since w> w’, hence composite engine is able to convert heat absorbed (q1′ – q1)  at the lower temperature completely into work. This is, however against the second law of thermodynamics. Thus, ηA can not be more than ηB. Similarly we can prove that ηB can not be more than ηA. Thus ηA = ηB.

Carnot’s refrigerator

Carnot cycle is completely reversible, in which working substance absorbs heat q2 from a hot reservoir at temperature T2, does some work w and rest of the heat q1 is given up to sink at lower temperature T1. If carnot engine works in reverse direction all processes will be reversed. Thus, engine will absorb heat q1 from sink at T1, some work w will be done on the engine and q2 heat will be transferred to source at T2. Now the engine will work as refrigerator i.e. heat is taken from a cooler sink and given to hot source. Working of engine and refrigerator are shown in fig.

Carnot's theorem proof and Carnot's refrigerator

Coefficient of performance of Refrigerator

Ratio of work done on the engine to the heat absorbed at lower temperature is the coefficient of performance (∈ )

Thus   ε = work done the engine/ heat absorbed from sink  … (1)

ε = w/ q1

ε = q2 – q1 / q1    …(2)

In terms of temperature ε = T2 – T1 / T1  …(3)

Carnot cycle

The extent to which heat can be changed in to work carnot explained a cyclic process, known as carnot cycle.

The carnot heat engine consists of a cylinder fitted with an ideal piston and ideal gas as the system. The engine works reversibly in a cycle between two large heat reservoirs one at higher temperature (T2) which acts as the source of heat and other at lower temperature (T1) which acts as sink. Complete cycle of carnot engine is shown in fig.

Carnot's theorem proof and Carnot's refrigerator


So our article is finished and after completely reading this article, one can easily tell what is carnot’s theorem and what is carnot’s refrigerator.

So one can say that they got a detailed information about carnot’s refrigerator and Carnot’s theorem.

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