Question

Find the amount of work done to increase the temperature of one mole of an ideal gas by ${30}^{o}C$, if it is expanding under the condition $V\propto T^{2/3}$. (R $=$ $8.31J/mole/K$)

• A. $16.62J$
• B. $166.2J$
• C. $1662J$
• D. $1.662J$

Answer 1

The correct option is B $166.2J$

Given $V\alpha T^{2/3}$ .....(1)
Using ideal gas equation $PV=nRT=RT$ since $n=1$
Substituting $T=\frac{PV}{R}$ in eqn(1) we get
$V=constant(\frac{PV}{R}{)}^{\frac{2}{3}}$
$\therefore P^{2/3}{V}^{-1/3}=constant$
$\therefore P{V}^{-1/2}=constant$
$\therefore$ polytropic exponent $\eta$ is $(-\frac{1}{2})$
Work done in a polytropic process : $W=\frac{-R\Delta T}{\eta -1}$ ....(2)
Substituting the values in (2)
$W=\frac{-8.31\times 30}{\frac{-1}{2}-1}=166.2$