Two cylinders $$A$$ and $$B$$ fitted with pistons contain equal amounts of an ideal diatomic gas at $$300\,K.$$  The piston of $$A$$ is free to move, while that $$B$$ is held fixed. The same amount of heat is given to the gas in each cylinder. If the rise in temperature of the gas in $$A$$ is $$30\,K,$$  then the rise in temperature of the gas in $$B$$ is

Solution :
$$A$$ is free to move, therefore heat will be supplied at constant pressure
$$\therefore \Delta {Q_A} = n{C_p}\Delta {T_A}\,......\left( {\text{i}} \right)$$
$$B$$ is held fixed, therefore heat will be supplied at constant volume.
$$\eqalign{ & \therefore \Delta {Q_B} = n{C_v}\Delta {T_B}\,......\left( {{\text{ii}}} \right) \cr & {\text{But}}\,\Delta {Q_A} = \Delta {Q_B}\,\left( {{\text{given}}} \right) \cr & \therefore n{C_p}\Delta {T_A} = n{C_v}\Delta {T_B} \cr & \therefore \Delta {T_B} = \left( {\frac{{{C_p}}}{{{C_v}}}} \right)\Delta {T_A} \cr & = \gamma \left( {\Delta {T_A}} \right)\,\,\left[ {\gamma = 1.4\left( {{\text{diatomic}}} \right)} \right] \cr & = \left( {1.4} \right)\left( {30\,K} \right) \cr & \therefore \Delta {T_B} = 42\;K \cr} $$