Explain tne working of carnot ’s neat engine and find its efficiency?

The working substance is taken through a cycle of four operations known as Carnot’s cycle in a Carnot engine.

Four operations of Carnot cycle are:
(i) isothermal expansion,
(ii) adiabatic expansion,
(iii) isothermal compression, and
(iv) adiabatic compression.
Initially the working substance is heated to a temperature of source T₁. 
The initial thermodynamic state (P₁,V₁,T₁) of working substance. 
(i) Isothermal expansion: The cylinder is placed on source and working substance is allowed to expand slowly so that it expands isothermally.
Let the thermodynamic states of working substance change from (P₁,V₁,T₁) to (P₂,V₂,T₁).
Temperature remains constant as it expands isothermally. 
This process is depicted on PV diagram by curve AB.
Let Q₁ be the heat absorbed by the gas and W₁ be the work done by it. 

(ii) Adiabatic expansion: Now, the cylinder is placed on an insulated pad and allowed to expand quickly so that the temperature of working substance falls to the temperature of sink T₂.
The thermodynamic states of working substance in this process changes from (P₂,V₂,T₁) to (P₃,V₃,T₂).
This process on PV diagram is represented by curve BC.
In this process, the heat absorbed by gas is zero.
Work done on the gas is W₂. 

(iii) Isothermal compression: When an isothermal compression is carried out, the cylinder is placed on the sink and working substance is compressed slowly so that it compresses isothermally.
The thermodynamic states of working substance change from (P₃,V₃,T₂) to (P₄,V₄,T₂).
This process is shown on PV diagram by curve CD.
Let Q₂ be the heat evolved and W₃ be the work done on the working substance. 

(iv) Adiabatic compression: Finally, the cylinder is placed on an insulated pad and compressed quickly so that the working substance returns to initial state (P₁,V₁,T₁).
This process on PV diagram is illustrated by curve DA.
In this process, the heat evolved by gas is zero.
Work done on the gas is W₄. 
 
Net amount of work done by the working substance in 1 complete cycle. 
In isothermal and adiabatic expansion, the work W₁ and W₂ is done by the gas.
Using sign convention W₁ and W₂ are positive.
In isothermal and adiabatic compression, the work W₃ and W₄ is done on the gas.
Therefore W₃ and W₄ are negative.
Thus net work done by working substance in one complete cycle is, 
W = W₁ + W₂ + (-W₃) + (-W₄)
From equation (2) and (4), 
W₂ = W₄
Therefore, 
W = W₁ - W₃ = Q₁ - Q₂            ... (5) 

That is in a Carnot's heat engine, the mechanical work obtained during each cycle is equal to the area of the loop on PV diagram.

Efficiency of Carnot’s cycle.
The efficiency of Carnot heat engine is given by, 
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