Physics - Heat & Thermo Question with Solution | TestHub

It is given that

$$\begin{gathered} P\propto T^3 \\ \Rightarrow P{T}^{-3}=\text{constant} ...\left(1\right) \end{gathered}$$

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

$$\begin{gathered} P{\left(\frac{nRT}{P}\right)}^{\gamma }=\text{constant} \\ \Rightarrow P^{1-\gamma }{T}^{\gamma }=\text{constant} \\ \Rightarrow P{T}^{\frac{\gamma }{1-\gamma }}=\text{constant} \end{gathered}$$

By comparing with equation(1), we get

$$\begin{gathered} \Rightarrow \frac{\gamma }{1-\gamma }=-3 \\ \Rightarrow \gamma =-3+3\gamma \\ \Rightarrow 3=2\gamma \\ \Rightarrow \gamma =\frac{3}{2} \end{gathered}$$