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

For an adiabatic process,

$$PV^{\gamma}=\text{constant}$$

Also ideal gas law gives

PV=nRT

or

$$V=\frac{nRT}{P}$$

Substitute into adiabatic equation:

$$P\left(\frac{nRT}{P}\right)^{\gamma}=\text{constant}$$

$$P\cdot\frac{(nR)^{\gamma}T^{\gamma}}{P^{\gamma}}=\text{constant}$$

$$P^{1-\gamma}T^{\gamma}=\text{constant}$$

Given pressure is proportional to cube of absolute temperature,

$$P\propto T^3$$

So

$$P=kT^3$$

Comparing with

$$P^{1-\gamma}T^{\gamma}=\text{constant}$$

write

$$P\propto T^{\frac{\gamma}{\gamma-1}}$$

Thus,

$$=3\frac{\gamma}{\gamma-1}$$

Solving,

$$γ=3γ−3$$

$$2γ=3$$

$$γ=\frac{3}{2}$$

But for an adiabatic process so,

$$\gamma=\frac{C_P}{C_V}$$