CBSE Class 11science Answered
Show four steps of carnot engine in P – V graph write the equation of each step and obtain the work
done by the system. Also obtain the efficiency of a carnot engine.
Asked by juzarsiddhapurwala  07 Mar, 2021, 10:08: PM
Expert Answer
Carnot cycle consists of 4 process as explained below
(1) Isothermal expansion. Heat energy is absorbed from hot reservoir of high temperature T_{H} ans work is done by system.
In figure, this process is shown from state A to state B
(2) Adiabatic expansion . System is cooled from High temperature T_{H} to lower temperature T_{L} adiabatically.
In figure, this process is shown from state B to state C
(3) Isothermal compression . Thermodynamic fuel is compressed to intermediate pressure isothermally at low temperature T_{L}.
In figure, this process is shown from state C to state D
(4) adiabatic compression . Thermodynamic fuel is compressed adiabatically to initial pressure to close the cycle.
In figure , this process is shown from state D to state A
Process  Heat Transfer Q  Workdone W  Remarks 
state A to state B
Isothermal expansion

Q_{AB }= n R T_{H} ln( V_{B} / V_{A} )  W_{AB }= n R T_{H} ln( V_{B} / V_{A} ) 
From First law of thermodynamics , Q = ΔU+W
Where Q is heat trasferred , ΔU is change in internal energy
and W is workdone . For isothermal process ΔU = 0
Hence we have Q = W
n is number of moles , R is universal gas constant,
T_{H} is hot reservoir temperature,
V_{A} and V_{B }are volume at stateA and stateB respectively

State B to State C
Adiabatic expansion

Q_{BC} = 0  W_{BC} = [ n R ( T_{L}  T_{H}) ] / (1  γ)_{ } 
Adiabatic process heat transfer is zero
T_{L} is cold reservoir temperature and
γ is ratio of specific heat of thermodynamic fuel

State C to state D
Isothermal Compression

Q_{CD }= n R T_{L} ln( V_{D} / V_{C} )  W_{CD }= n R T_{L} ln( V_{D} / V_{C} ) 
V_{C} and V_{D }are volume at stateC and stateD respectively

State D to State A
Adiabatic compression

Q_{DA }= 0  W_{DA} = [ n R ( T_{H}  T_{L}) ] / (1  γ) 
Net Workdone = W_{AB} + W_{CD} = n R ( T_{H}  T_{L} ) ln( V_{B} / V_{A} )
( In Carnot cycle , V_{B }/ V_{A} = ( V_{C} / V_{D }) , i.e. volume ratio for both isothermal process is same ,
We have used this in above expression to get net heat transfer )
Heat transfer from Hot reservoir = Q_{AB} = n R ( T_{H}  T_{L} ) ln( V_{B} / V_{A} )
Efficiency = Net workdone / Heat transfer from Hot reservoir
Efficiency = [ n R ( T_{H}  T_{L} ) ln( V_{B} / V_{A} ) ] / [ n R T_{H} ln( V_{B} / V_{A} )]
Efficiency = ( T_{H}  T_{L} ) / T_{H} = [ 1  ( T_{L} / T_{H} ) ]
Answered by Thiyagarajan K  08 Mar, 2021, 12:33: AM
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