The temperature of a hypothetical gas increases to $\sqrt 2 $ times when compressed adiabatically to half the volume. Its equation can be written as
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(a) $T{V^{\gamma - 1}}$= constant
$\therefore \frac{{{T_1}}}{{{T_2}}} = {\left( {\frac{{{V_2}}}{{{V_1}}}} \right)^{\gamma - 1}}$or ${\left( {\frac{1}{2}} \right)^{\gamma - 1}} = \sqrt {\frac{1}{2}} $
$\therefore \gamma - 1 = \frac{1}{2}$or $\gamma = \frac{3}{2}$
$\therefore \frac{{{T_1}}}{{{T_2}}} = {\left( {\frac{{{V_2}}}{{{V_1}}}} \right)^{\gamma - 1}}$or ${\left( {\frac{1}{2}} \right)^{\gamma - 1}} = \sqrt {\frac{1}{2}} $
$\therefore \gamma - 1 = \frac{1}{2}$or $\gamma = \frac{3}{2}$
$P{V^{3/2}}$= constant
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