A hypothetical gas expands adiabatically such that its volume...

Let $\gamma$ be the ratio of $\dfrac{{C}_{p}}{{C}_{v}}$.

Then for adiabatic process $P{V}^{\gamma} =$Constant

$⇒ \dfrac{{P}_{i}}{{P}_{f}} = {\left(\dfrac{{V}_{f}}{{V}_{i}}\right)}^{\gamma}$

$⇒ \left(\dfrac{81}{16}\right) = {\left(\dfrac{27}{8}\right)}^{\gamma} ⇒ \left[{\left(\dfrac{3}{2}\right)}^{3}\right]\dfrac{4}{3} = {\left(\dfrac{27}{8}\right)}^{\gamma}$

$⇒ \gamma = \dfrac{4}{3}$