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Q.
A certain mass of gas at $273\, K$ is expanded to $81$ times its volume under adiabatic conditions. If $\gamma$ = $1 \times 25$ for the gas then its final temperature is
Given: Initial temperature $\left(T_{1}\right)=273 \,K ;$ Initial volume $\left(V_{1}\right)=V ;$ Final volume $\left(V_{2}\right)=81 \,V$ and $\gamma=1.25$
We know that under adiabatic condition $T_{1} V_{1}^{\gamma-1}$
$=T_{2} V_{2}^{\gamma-1}$ or $\frac{273}{T_{2}}=\left(\frac{V_{2}}{V_{1}}\right)^{1-1}=\left(\frac{81 V}{V}\right)^{125-1}=(81)^{0 \cdot 25}=3$ or
$T_{2}=\frac{273}{3}=91 K =-182^{\circ} C$
(where $T_{2}$ is the final temperature of the gas).