Question

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

• A. 3 J
• B. 6 J
• C. 9 J
• D. 1.5 J

Answer 1

The correct option is D 1.5 J

Work done by the gas = change in spring potential energy = $\frac{1}{2}k(x_2^2-x_1^2)$
Before heating, let the pressure of gas be $P$.
At equilibrium piston,
$PA=k{x}_1$
$\therefore x_1=\frac{PA}{k}=\left(\frac{nRT}{V}\right)\frac{A}{k}=\frac{1\times 8.3\times 100\times {10}^{-2}}{0.83\times 100}={10}^{-1}=0.1m$
Since during heating process, the spring is compressed further by $0.1m$
$\therefore x_2=0.2m$
$\therefore$ work done by the gas $=\frac{1}{2}.100({0.2}^2-{0.1}^2)=\frac{1}{2}.100.\left(0.1\right)\left(0.3\right)=1.50=1.5J$
Since the gas is expanding, the work done by the gas is positive.