Question

One mole of an ideal gas is kept enclosed under a light piston (area $={10}^{-2}{\text{m}}^2)$ connected by a compressed spring (spring constant $=200\text{N/m}$). The volume of gas is $0.83{\text{m}}^3$ and its temperature is $100\text{K}$. The gas is heated so that it compresses the spring by further $0.1\text{m}$. Find the workdone by the gas during the process (in joule). (Take $R=8.3\text{J/k-mole}$ and suppose that there is no atmosphere)

Answer 1

Before just heating the gas let the pressure of gas be $P_1$.
Applying equilibrium condition for the piston:
$P_{1}A=k{x}_1$
$\Rightarrow x_1=\frac{P_{1}A}{k}$
$\Rightarrow x_1=\left(\frac{nR{T}_1}{V_1}\right).\frac{A}{k}$
or, $x_1=\frac{(1\times 8.3\times 100)}{0.83\times 200}\times {10}^{-2}$
$\therefore x_1=0.05\text{m}$

After the gas is heated, the compression produced in gas is 0.1 m.
Thus final compression in spring will be $x_2=0.1+0.05$
$\Rightarrow x_2=0.15\text{m}$

From energy conservation:
Workdone by gas = Increase in elatic P.E. of spring
$\Rightarrow W_{gas}=\frac{1}{2}k(x_2^2-x_1^2)$
or, $W_{gas}=\frac{1}{2}\times 200\times \left[\right(0.15{)}^2-\left(0.05{)}^2\right]$
or, $W_{gas}=100\times [0.0225-0.0025]$
$\therefore W_{gas}=100\times 0.02=2\text{J}$

$\begin{array}{c}\text{Why this question}?\\ \text{Tip: on heating the gas, expansion of gas will}\\ \text{occur. This will lead to further compression}\\ \text{in spring hence the workdone by gas during whole}\\ \text{process will be stored as increase in elastic}\\ \text{potential energy of the spring.}\end{array}$