The volume $\mathrm{V}$ of a given mass of monatomic gas changes with temperature $\mathrm{T}$ according to the relation $\mathrm{V}=\mathrm{KT}^{\frac{2}{3}}$. The work done when temperature changes by $90 \mathrm{~K}$ will be $\mathrm{xR}$. The value of $\mathrm{X}$ is_______ [ $R$ =universal gas constant]
(60)
Given: $V=k T^{2 / 3}$
$V^{3 / 2}=(k)^{3 / 2} T$
$\mathrm{TV}^{-3 / 2}=$ const. $\ldots(1)$
and $\mathrm{TV}^{\gamma-1}=$ const $\ldots(2)$
From (1)\&(2)
$-\frac{3}{2}=\gamma-1$
$\gamma=-\frac{1}{2}$
Work done $(\mathrm{w})=\frac{n R_{\Delta} T}{\gamma-1}$
$W=\frac{1 \times R \times 90}{-\frac{1}{2}-1} \quad|W|=60 R \quad x=60$
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