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

For an adiabatic process: $$PV^\gamma = \text{constant}$$ and $$TV^{\gamma-1} = \text{constant}$$.

From the ideal gas law: $$PV = nRT$$, so $$V = \frac{nRT}{P}$$.

Substituting: $$P\left(\frac{nRT}{P}\right)^\gamma = \text{const}$$

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

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

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

$$P \propto T^{\gamma/(\gamma-1)}$$

Given $$P \propto T^3$$:

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

$$\gamma = 3\gamma - 3$$

$$2\gamma = 3$$

$$\gamma = \frac{3}{2}$$

The answer is $$\frac{C_p}{C_v} = \frac{3}{2}$$, which corresponds to Option (2).