Question

Find the amount of work done (in joule) to increase the temperature of one mole of an ideal gas by $30^{\circ}C$ if it is expanding under the condition Foe $V\propto T^{2/3}.\left(R =1.99 \,cal/ mol - K\right)$

Answer 1

Given, $V=k T^{2 / 3}$
$\therefore d V=k \times \frac{2}{3} T^{-1 / 3} d T=\frac{2}{3} k T^{-1 / 3} d T$
Work done $d W=P d V$
$ =\frac{R T}{V} d V $
$=\frac{R T}{k T^{2 / 3}} \times \frac{2}{3} k T^{-1 / 3} d T=\frac{2}{3} R(d T) $
Total work done $W=\frac{2}{3} R \int_{T_{1}}^{T_{2}} d T$
$ =\frac{2}{3} R\left[T_{2}-T_{1}\right] $
$ -2 \times 1.99 \times 3 $
$ =2 \times 1.99 \times 30=39.8\, cal =167\, J $