Carnot engine is a theoretical thermodynamic cycle proposed 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.

According to Carnot Theorem:

Any system working between two given temperatures T_{1} (hot reservoir) and T_{2} (cold reservoir), can never have an efficiency more than the Carnot engine working between the same reservoirs respectively.

Also, the efficiency of this type of engine is independent of the nature of the working substance and is only dependent on the temperature of the hot and cold reservoirs.

For an ideal gas operating inside a Carnot cycle, the following are the steps involved:

Isothermal expansion: The gas is taken from P_{1}, V_{1}, T_{1} to P_{2}, V_{2}, T_{2}. Heat Q_{1} is absorbed from the reservoir at temperature T_{1}. Since the expansion is isothermal, the total change in internal energy is zero and the heat absorbed by the gas is equal to the work done by the gas on the environment, which is given as:

\(W_{1\rightarrow 2}=Q1=\mu \times R\times T_{1}\times ln\frac{v_{2}}{v_{1}}\)

Adiabatic expansion: The gas expands adiabatically from P_{2}, V_{2}, T_{1} to P_{3}, V_{3}, T_{2}.

**Here work done by the gas is given by:**

\(W_{2\rightarrow 3}=\frac{\mu R}{\gamma -1}(T_{1}-T_{2})\)

Isothermal compression: The gas is compressed isothermally from the state (P_{3}, V_{3}, T_{2}) to (P_{4}, V_{4}, T_{2}).

**Here, the work done on the gas by the environment is given by:**

\(W_{3\rightarrow 4}=\mu RT_{2}ln\frac{v_{3}}{v_{4}}\)

Adiabatic compression: The gas is compressed adiabatically from the state (P_{4}, V_{4}, T_{2}) to (P_{1}, V_{1}, T_{1}).

**Here, the work done on the gas by the environment is given by:**

\(W_{4\rightarrow 1}=\frac{\mu R}{\gamma -1}(T_{1}-T_{2})\)

**Hence, the total work done by the gas on the environment in one complete cycle is given by:**

\(W= W_{1\rightarrow 2}+W_{2\rightarrow 3}+W_{3\rightarrow 4}+W_{4\rightarrow 1}\\ \\ W=\mu \: RT_{1}\: ln\frac{v_{2}}{v_{1}}-\mu \: RT_{2}\: ln\frac{v_{3}}{v_{4}}\)

\(Net\; efficiency =\frac{Net\; workdone\; by\; the\;gas}{Heat\; absorbed\; by\; the\;gas}\)

\(Net\; efficiency =\frac{W}{Q_{1}}=\frac{Q_{1}-Q_{2}}{Q_{1}}=1-\frac{Q_{2}}{Q_{1}}=1-\frac{T_{2}}{T_{1}}\frac{ln\frac{v_{3}}{v_{4}}}{ln\frac{v_{2}}{v_{1}}}\)

Since the step 2–>3 is an adiabatic process, we can write T_{1}V_{2}^{Ƴ-1 }= T_{2}V_{3}^{Ƴ-1}

Or,

\(\frac{v_{2}}{v_{3}}=(\frac{T_{2}}{T_{1}})^{\frac{1}{\gamma -1}}\)

Similarly, for the process 4–>1, we can write

\(\frac{v_{1}}{v_{2}}=(\frac{T_{2}}{T_{1}})^{\frac{1}{\gamma -1}}\)

This implies,

\(\frac{v_{2}}{v_{3}}=\frac{v_{1}}{v_{2}}\)

**So, the expression for net efficiency of carnot engine reduces to:**

\(Net\; efficiency=1-\frac{T_{2}}{T_{1}}\)