The volume of a gas is reduced adiabatically to $\text{(}\frac{1}{4})$ of its volume at ${27}^0\mathrm{C}. \mathrm{If} \mathrm{\gamma }$ = 1.4, the new temperature will be

For adiabatic change the relation between temperature and volume is

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

where $\mathrm{\gamma }$ is ratio of specific heats of the gas,

Given, ${\mathrm{T}}_1 = 27 + 273 = 300 \mathrm{K}, {\mathrm{V}}_1 = \mathrm{V}, {\mathrm{V}}_2 = \frac{\mathrm{V}}{4}$

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

$\Rightarrow {\mathrm{T}}_2 = {\left(\frac{{\mathrm{V}}_1}{{\mathrm{V}}_2}\right)}^{\mathrm{\gamma }-1}\times {\mathrm{T}}_1$

             = ${\left(\frac{V}{{\displaystyle \frac{V}{4}}}\right)}^{1.4-1}\times 300$

            = ${\left(4\right)}^{0.4}\times 300 = 300 \times {\left(4\right)}^{0.4}\mathrm{K}$