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A perfect gas is found to obey the relation, ${PV}^{3 / 2} =$ constant, during an adiabatic process. If such a gas initially at a temperature $T$ , is compressed to half of its initial volume, then its final temperature will be

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${(2)}^{1 / 2} T$

${(2)}^{- 1 / 2} T$

$2 T$

$4 T$

answer is C.

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Detailed Solution

${PV}^{3 / 2} = constant$

$\Rightarrow (\frac{nRT}{V}) V^{3 / 2} = constant$

$\Rightarrow {TV}^{1 / 2} = const.$
$\Rightarrow T (V)^{1 / 2} = T_{2} (\frac{V}{2})^{1 / 2}$

$\Rightarrow T_{2} = (2^{1 / 2}) T$

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