# Carnot Cycle

## Carnot Cycle

Carnot engine is a theoretical thermodynamic cycle by Leonard Carnot. It gives the estimate of the maximum possible efficiency that a heat engine during the conversion process of heat into work and conversely, working between two reservoirs, can possess.  As the engine works, the working substance of the engine undergoes a cycle known as Carnot Cycle. The Carnot Cycle consists of the following four strokes.

First Stroke (Isothermal Expansion, Curve AB): The cylinder containing ideal gas as working substance allowed to expand slowly at this constant temperature T₁.

Work done = Heat absorbed by the system

∴ $${{W}_{1}}={{Q}_{1}}=\int\limits_{{{V}_{1}}}^{{{V}_{2}}}{PdV}=R{{T}_{1}}\,{{\log }_{e}}\left( \frac{{{V}_{2}}}{{{V}_{1}}} \right)=Area\,\,ABGE$$.

Second Stroke (Adiabatic Expansion, Curve BC): The cylinder is then placed on the non – conducting stand and the gas is allowed to expand adiabatically till the temperature falls from T₁ to T₂.

∴ $${{W}_{2}}=\int\limits_{{{V}_{2}}}^{{{V}_{3}}}{PdV}=\frac{R}{(\gamma -1)}\left[ {{T}_{1}}-{{T}_{2}} \right]=Area\,BCHG$$.

Third Stroke (Isothermal Compression, Curve CD): The cylinder is placed on the sink and the gas is compressed at constant temperature T₂.

Work done = Heat released by the system

∴ $${{W}_{3}}={{Q}_{2}}=-\int\limits_{{{V}_{3}}}^{{{V}_{4}}}{PdV=-R{{T}_{2}}\,{{\log }_{e}}\frac{{{V}_{4}}}{{{V}_{3}}}}=R{{T}_{2}}\,{{\log }_{e}}\frac{{{V}_{3}}}{{{V}_{4}}}=Area\,CDFH$$.

Fourth Stroke (Adiabatic Compression, Curve DA): Finally, the cylinder is again on non – conducting stand and the compression is continued so that gas returns to its initial stage.

∴ $${{W}_{4}}=-\int\limits_{{{V}_{4}}}^{{{V}_{1}}}{PdV}=-\frac{R}{\gamma -1}\left( {{T}_{2}}-{{T}_{1}} \right)=\frac{R}{\gamma -1}\left( {{T}_{1}}-{{T}_{2}} \right)=Area\,ADFE$$.