An ideal gas undergoes a process in which $T = T_{0} + a V^{3}$ , where $T_{0}$ and $^{'} a^{'}$ are positive constants and $V$ is molar volume. The molar volume for which pressure will be minimum is :

{(\frac{T_{0}}{2 a})}^{1 / 3}

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{(\frac{T_{0}}{3 a})}^{1 / 3}

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{(\frac{a}{2 T_{0}})}^{1 / 3}

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None of these

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Solution

The correct option is B ${(\frac{T_{0}}{2 a})}^{1 / 3}$
Given, $T = T_{0} + a V^{3}$
We know that $P V = n R T$
$\Rightarrow T = \frac{P V}{n R}$
$\Rightarrow \frac{P V}{n R} = T_{0} + a V^{3}$
$\Rightarrow \frac{P}{n R} = \frac{T_{0}}{V} + a V^{2}$
Now, for pressure to be minimum $\frac{d P}{d V} = 0$
$- \frac{n R T_{0}}{a V^{2}} + n R a 2 V = 0$
$V = (\frac{T_{0}}{2 a})^{\frac{1}{3}}$

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