During an adiabatic process, the pressure of a gas is found to be proportional to the cube of its absolute temperature. The ratio $$\frac{{{C_P}}}{{{C_V}}}$$  for the gas is

Solution :
$$\eqalign{ & P\, \propto \,{T^3} \cr & \Rightarrow \,\,P{T^{ - 3}} = {\text{constant }}.....\left( {\text{i}} \right) \cr} $$
But for an adiabatic process, the pressure temperature relationship is given by
$$\eqalign{ & {P^{1 - \gamma }}{T^\gamma } = {\text{constant}} \cr & \Rightarrow {\text{ }}P{T^{\frac{\gamma }{{1 - \gamma }}}} = {\text{constant }}.....\left( {{\text{ii}}} \right) \cr} $$
From (i) and (ii)
$$\eqalign{ & \frac{\gamma }{{1 - \gamma }} = - 3 \cr & \Rightarrow \,\,\gamma = - 3 + 3\gamma \cr & \Rightarrow \,\,\gamma = \frac{3}{2} \cr} $$