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

$P\propto T^4$

$P{T}^4$= constant       ---(i)

For an adiabatic process,

$P{T}^{\gamma /1-\gamma }$=constant        ---(ii)

Comparing (i) and (ii), we get

$$\begin{gathered} \frac{\gamma }{1-\gamma }=-4 \\ \gamma =-4+4\gamma \\ -3\gamma =-4or\gamma =\frac{4}{3} \\ As \gamma =\frac{C_p}{C_v}\therefore \frac{C_p}{C_v}=\frac{4}{3} \end{gathered}$$