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 $V\propto T^{2/3}.\left(R=1.99cal/mol-K\right)$

Answer 1

Answer: 167

Given, $V=k{T}^{2/3}$
$\therefore dV=k\times \frac{2}{3}{T}^{-1/3}dT=\frac{2}{3}k{T}^{-1/3}dT$
Work done $dW=PdV$
$<br/>\begin{array}{l}<br/>=\frac{RT}{V}dV\\ <br/>=\frac{RT}{k{T}^{2/3}}\times \frac{2}{3}k{T}^{-1/3}dT=\frac{2}{3}R(dT)<br/>\end{array}<br/>$
Total work done $W=\frac{2}{3}R{\int }_{T_1}^{T_2}dT$
$<br/>=\frac{2}{3}R\left[T_2-T_1\right]<br/>$
$<br/>-2\times 1.99\times 3<br/>$
$<br/>=2\times 1.99\times 30=39.8cal=167J<br/>$