At ${27}^{°}C$ a gas is compressed suddenly such that its pressure becomes $\left(\frac{1}{8}\right)$ of original pressure. Final temperature will be $(\gamma =\frac{5}{3})$

$$\begin{gathered} \mathrm{Adiabatic} \mathrm{relation} \mathrm{between} \mathrm{temperature} \mathrm{and} \mathrm{pressure} \mathrm{is} \\ {\mathrm{P}}^{1-\mathrm{\gamma }}{\mathrm{T}}^{\mathrm{\gamma }} = \mathrm{constant} \end{gathered}$$

${\mathrm{P}}_1^{1-\mathrm{\gamma }}{\mathrm{T}}_1^{\mathrm{\gamma }}={\mathrm{P}}_2^{1-\mathrm{\gamma }}{\mathrm{T}}_2^{\mathrm{\gamma }}$

$\mathrm{Here} {\mathrm{P}}_1=\mathrm{P}, {\mathrm{P}}_2=\frac{\mathrm{P}}{8} {\mathrm{T}}_1=300\mathrm{K}, {\mathrm{T}}_2=?$

$$\begin{gathered} \mathrm{Substituting} \mathrm{we} \mathrm{get} \\ {8}^{1-\mathrm{\gamma }}{300}^{\mathrm{\gamma }}={\left({\mathrm{T}}_2\right)}^{\mathrm{\gamma }} \\ {\mathrm{T}}_2=\frac{300}{4^{{\displaystyle \raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$5$}\right.}}}\approx 131 \mathrm{K}=-{142}^0\mathrm{C} \end{gathered}$$