Question

Derive the equation to find efficiency of Carnot engine cycle.

Answer 1

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

Step 1:

Isothermal expansion: The gas is taken from P₁, V₁, T₁ to P₂, V₂, T₂. Heat Q₁ is absorbed from the reservoir at temperature T₁. 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:

Step 2:

Adiabatic expansion: The gas expands adiabatically from P₂, V₂, T₁ to P₃, V₃, T₂.

Here work done by the gas is given by:

Step 3:

Isothermal compression: The gas is compressed isothermally from the state (P₃, V₃, T₂) to (P₄, V₄, T₂).

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

Step 4:

Adiabatic compression: The gas is compressed adiabatically from the state (P₄, V₄, T₂) to (P₁, V₁, T₁).

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

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

Since the step 2–>3 is an adiabatic process, we can write T₁V₂^Ƴ-1 = T₂V₃^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

 \(\frac{v_1}{v_2} = (\frac{T_2}{T_1})^{\frac{1}{\gamma - 1}}\)

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 - \(\frac{T_2}{T_1}\)