The volume of a gas is reduced to $\dfrac{1}{4}$ of its initial volume adiabatically at $27^o$. The final temperature of the gas (if $\gamma= 1.4$) will be

The equation of state for an adiabatic process:

$TV^{\gamma - 1} = constant$

i.e. $T_{1}V_{1}^{\gamma - 1} = T_{2}V_{2}^{\gamma - 1}$

Given:

$T_{1} = 300K$, $V_{2} = \dfrac{V_{1}}{4}$

$T_{1}V_{1}^{\gamma - 1} = T_{2}V_{2}^{\gamma - 1}$

$300\times V_{1}^{0.4} = T_{2}\dfrac{V_{1}^{0.4}}{4}$

$T_{2} = 300 \times (4)^{0.4}$