First slide

Work done by gas

Question

A gas expands with temperature according to the relation $V = {KT}^{1 / 3} .$ What is the work done when the temperature changes by $30^{o} C$

Moderate

A

10 R
B

20 R
C

30 R
D

40 R

Solution

$V = {KT}^{1 / 3}$
$\Rightarrow dV = \frac{1}{3} {KT}^{- \frac{2}{3}} dT$
$\Rightarrow = \frac{1}{3} \frac{V}{T^{1 / 3}} T^{- \frac{2}{3}} dT$
$\Rightarrow dV = \frac{1}{3} V . T^{- 1} dT$
$\Rightarrow \frac{dV}{V} = \frac{1}{3} \frac{dT}{T} \Rightarrow W = \int \begin{matrix} T_{2} \\ T_{1} \end{matrix} \frac{RT}{T} dT$
$= \frac{1}{3} R (T_{2} - T_{1})$
$\Rightarrow W = \frac{1}{3} \times R \times 30 = 10 R$

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