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V V
A  A  A  A  A  RT 1 ln 2  C V T  T 2  RT 2 ln 4  C V (T  T 1 ) 
41
34
2
23
12
1
V V
1 3
V V V V
 RT 1 2  RT 2 ln 4  RT 1 ln 2  RT 2 ln 3 ) 1 . 9 . 2 (
V 1 V 3 V 1 V 4

So effective work by Carnot cycle is done due to difference of work
expansion and compression . If we take into account that

V V
A  RT 1 ln 2  Q and A  RT 2 ln 3  Q , effective work of Carnot
34
2
12
1
V 1 V 3
cycle is equal to

V V
A  Q  Q  RT 1 ln 2  RT 2 ln 3 (2.9.2)
1
2
V 1 V 4

Consumed energy is a quantity of heat Q , and, therefore, efficiency 
1
of Carnot cycle is equal to
V V
RT 1 ln 2  RT 2 ln 3
A Q  Q V V
  1 2  1 4 (2.9.2)
Q 1 Q 1 RT ln V 2
1
V 1
According to the Poisson's equation for adiabatic expansion
  1
 1   

T 1  V 3  T 1   V 4 

 and compression   we obtain

T 2  V 2  T 2  V 

V V V V
3  4  2  3 (2.9.3)
V 2 V 1 V 1 V 4

Therefore,

A Q  Q T  T
  1 2  1 2 (2.9.4)
Q Q T
1 1 1

Carnot engine efficiency does not depend on the nature of a working

substance, but depends on the temperatures of hot and cold reservoirs. No
real engine can have efficiency greater than that of a Carnot engine working

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