A diatomic ideal gas is used in a car engine as the working substance. If during the adiabatic expansion part of the cycle, volume of the gas increases from $$V$$ to $$32\,V,$$  the efficiency of the engine is

Solution :
The efficiency of cycle is $$\eta = 1 - \frac{{{T_2}}}{{{T_1}}}$$
For adiabatic process $$T{V^{\gamma - 1}} = {\text{constant}}$$
For diatomic gas $$\gamma = \frac{7}{5}$$
$$\eqalign{ & {T_1}V_1^{\gamma - 1} = {T_2}V_2^{\gamma - 1} \cr & {T_1} = {T_2}{\left( {\frac{{{V_2}}}{{{V_1}}}} \right)^{\gamma - 1}} \cr & {T_1} = {T_2}{\left( {32} \right)^{\frac{7}{5} - 1}} = {T_2}{\left( {{2^5}} \right)^{\frac{2}{5}}} = {T_2} \times 4 \cr & {T_1} = 4{T_2} \cr & \therefore \eta = \left( {1 - \frac{1}{4}} \right) = \frac{3}{4} = 0.75 \cr} $$