An ideal gas at ${27^o}C$ is compressed adiabatically to $\frac{8}{{27}}$ of its original volume. If $\gamma = \frac{5}{3}$, then the rise in temperature is........ $K$
AIPMT 1999, Medium
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(b) $\frac{{{T_2}}}{{{T_1}}} = {\left( {\frac{{{V_1}}}{{{V_2}}}} \right)^{\gamma - 1}} \Rightarrow {T_2} = 300\,{\left( {\frac{{27}}{8}} \right)^{\frac{5}{3} - 1}} = 300\,{\left( {\frac{{27}}{8}} \right)^{\frac{2}{3}}}$
$ = 300\,{\left\{ {{{\left( {\frac{{27}}{8}} \right)}^{1/3}}} \right\}^2} = 800\,{\left( {\frac{3}{2}} \right)^2} = 675K$
$ \Rightarrow \;\Delta T = 675 - 300 = 375\;K$
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