Question

Define the Carnot's cycle. Find the expression for its efficiency.

Answer 1

Carnot Cycle:- Operation 1. Isothermal expansion- The cyclinder containing an ideal gas as working substance having pressure p₁ and volume V₁ is placed over the source at temperature T₁. The working substance also attains temperature T₁. Let this state of the working substance is represented by point A. The pressure on the piston of the cylinder is allowed to decrease and hence the volume of the working substance increases. As a result, work is done by the working substance. As the base of the cylinder is perfectly conducting, so the working substance absorbs heat. The temperature raised by the absorption of heat is lowered during the expansion, so the temperature of the working substance remains constant at T₁. Thus, the process is isothermal. In this process, work done is equal to amount of heat absorbed. Let Q₁ be the heat absorbed and the volume of the working substance changes from V₁ to V₂ and pressure changes from p₁ to p₂. The final stage of the working substance is represented by point B on the p-V diagram.

Now work done during this isothermal process is

Operation 2. Adiabatic expansion- Now the cylinder is placed on the insulated stand. The pressure on the piston is further decreased and hence the volume of the working substance increases. Since there is no exchange of heat between system and its surroundings, so the temperature of the working substance decreases during the expansion. The process is completely adiabatic. The process continues till the pressure, volume and temperature of the working substance becomes p₃, V₃ and T₂ respectively.

This state is represented by point C on the p-V diagram.

Work done by the system during adiabatic expansion is given by

= Area BCSR   ...(ii)

Operation 3. Isothermal compression- The cylinder is placed over the sink and the pressure on the piston is increased. As a result of this, the working substance is compressed and its volume decreases and temperature increases. Since the base of the cylinder is conducting, so there is transfer of heat energy from the working substance to the sink. Let Q₂ be heat energy rejected to sink. Thus, the temperature of the working substance remains constant at T₂ and the process is completely isothermal. The process continues till point D is reached.

Work done on the system during this isothermal process is

Operation 4. Adiabatic compression- Finally the cylinder is placed on the insulated stand. The pressure on the piston is further increased so that the working substance is compressed adiabatically attains the initial state A (p₁.V₁).

Work done on the system during this adiabatic process is

Now the total work done during the complete cycle is

W = W₁ + W₂ + (-W₃) + (-W₄)

Since W₂ = W₄

∴ W = W₁ - W₃ = Area ABCD

Efficiency of the Carnot's engine is given by

η = {Work done(output)}/{Heat absorbed from source (input)}

or, η = W/Q₁ = {W₁ - W₃}/{W₁} = {Q₁ - Q₂}/{Q₁} = 1 - {W₃}/{W₁} = 1 - {Q₂}/{Q₁}

[Using eqn (i) and (iii)]

Since points B and C lie on same adiabatic curve,

We have,

Since points D and A lies on the same adiabatic curve,

again,

From eqn. (vi) and (vii), we have