During an adiabatic process, the pressure of a gas is found to be...

It is given that

$P ∝ {T}^{3} ⇒ P{T}^{- 3} = \text{constant} . . . \left(1\right)$

For an adiabatic process,

$P{V}^{\gamma} = \text{constant} . . . \left(2\right)$

Assuming the gas to be a perfect gas, it can be written from equation (2) that

$P{\left(\dfrac{nRT}{P}\right)}^{\gamma} = \text{constant} ⇒ {P}^{1 - \gamma}{T}^{\gamma} = \text{constant} ⇒ PT\dfrac{\gamma}{1 - \gamma} = \text{constant}$

By comparing with equation(1), we get

$⇒ \dfrac{\gamma}{1 - \gamma} = - 3 ⇒ \gamma = - 3 + 3\gamma ⇒ 3 = 2\gamma ⇒ \gamma = \dfrac{3}{2}$