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Derive the formula for efficiency of the Carnot engine.

Answer
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Hint: Carnot engine was proposed by Nicolas Leonard sadi Carnot in 1824. Carnot engine which gives an estimate of the maximum possible efficiency of a heat engine during the conversion process of heat into work. A carnot engine is used in heat pumps and working fluid has a fixed mass throughout a cycle.

Complete step-by-step solution:
Carnot cycle consist of following four processes and carnot engine is an ideal reversible closed thermodynamic cycle
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1) Isothermal expansion
Heat Q1 is absorbed from the reservoir at temperature T1 and gas is taken from P1,V1,T1 to P2,V2,T2. The total change in internal energy is zero and work done by the gas on the environment is equal to the heat absorbed by the gas.
Work done by the gas is given by:
W12=Q1=μRT1×lnv2v1
2) Adiabatic expansion
From P2,V2,T1 to P3,V3,T2 gas expands adiabatically
Work done by the gas is given by:
W23=μRγ1(T1T2)
3) Isothermal compression:
From P3,V3,T2 to P4,V4,T2 gas is compressed isothermally
Work done by the gas is given by:
W34=μRT2lnv3v4
4) Adiabatic compression
From P4,V4,T2 to P1,V1,T1 gas is compressed adiabatically
Work done by the gas is given by:
W41=μRγ1(T1T2)
Total work done by the gas in one complete cycle is given by:
W=W12+W23+W34+W41W=μRT1lnv2v1μRT2lnv3v4
Net efficiency (Ne) of engine is equal to ratio of net work done by the gas to the heat absorbed by the gas
Ne=WQ1=Q1Q2Q1Ne=1T2T1lnv3v4lnv2v1
(1) and (2) process is an adiabatic process
T1V2γ1=T2V3γ1
(4) And (1) process we can write
v1v2=(T2T1)1γ1
Net efficiency of Carnot engine =1T2T1

 Note: The efficiency of a carnot engine is independent of the nature of the working substance and dependent on the temperature of the hot and cold reservoirs. Working fluid in each cycle passes through equilibrium to non-equilibrium states (same physical state) and it consists of one and the same phase throughout the cycle .There are no entropy changes in the engine.
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