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Q.
An ideal monoatomic gas at 27$^\circ$C is compressed adiabatically to $\frac{8}{27}$ times the present volume. The increase in temperature of the gas is
Let the initial volume of the gas be $V$ i.e $V_{1}=V$
Thus final volume of the gas $V_{2}=\frac{8}{27} V$
Given: $ T _{1}=27^{\circ} C =300 K$
For monoatomic gas, $\gamma=\frac{5}{3}$
Using $TV ^{\gamma}=$ constant
$
\Rightarrow \frac{ T _{2}}{ T _{1}}=\left(\frac{ V _{1}}{ V -2}\right)^{\gamma-1}
$
$
\therefore \frac{ T _{2}}{300}=\left(\frac{27}{8}\right)^{\frac{5}{3}-1}
$
$
\begin{array}{l}
\text { OR } \frac{ T _{2}}{300}=\left(\frac{27}{8}\right)^{\frac{2}{3}}=\left(\frac{3}{2}\right)^{2} \Rightarrow T _{2}=300 \times 2.25=675 K \\
\Longrightarrow T _{2}=675-273=402^{\circ} C
\end{array}
$
Thus increase in temperature $\Delta T = T _{2}- T _{1}=402-27=375^{\circ} C$