A monatomic ideal gas follows the process $T V^{3 / 2} =$ constant. If the molar heat capacity for this process is $(\frac{5 R}{y})$ , then the value of $y$ is

3

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5

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6

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8

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Solution

The correct option is C $6$
Given,
$T V^{3 / 2} =$ constant
$\Rightarrow (\frac{P V}{n R}) V^{3 / 2} =$ constant
$[P V = n R T]$
$\Rightarrow P V^{5 / 2} =$ constant
[ $n$ is fixed for a given ideal gas]
Comparing with $P V^{x} =$ constant, we get,
$x = 5 / 2$
Molar specific heat capacity for polytropic process is given by,
$\Rightarrow C = C_{v} + \frac{R}{1 - x}$
$\Rightarrow C = \frac{f R}{2} + \frac{R}{1 - x}$
$\Rightarrow C = \frac{3 R}{2} + \frac{R}{1 - \frac{5}{2}}$
[ $f = 3$ for monatomic gas]
$\Rightarrow C = \frac{3 R}{2} - \frac{2 R}{3} = \frac{9 R - 4 R}{6}$
$\Rightarrow C = \frac{5 R}{6}$
$\Rightarrow C = \frac{5 R}{y}$ [given]
$\Rightarrow y = 6$

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