# The volume V of a given mass of monatomic gas changes

Question:

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]

Solution:

(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 $(\mathrm{w})=\frac{n R_{\Delta} T}{\gamma-1}$

$W=\frac{1 \times R \times 90}{-\frac{1}{2}-1} \quad|W|=60 R \quad x=60$