Question

Two moles of a monoatomic ideal gas $[U=\frac{3}{2}nRT]$ is enclosed in an adiabatic vertical cylinder fitted with a smooth light adiabatic piston. The piston is connected to a vertical spring of spring constant 200 N/m as shown in figure. The area of cross-section of the cylinder is $20.0c{m}^2$. Initially, the spring is at its natural length and temperature of the gas is 300K. The atmospheric pressure is 100 kPa. The gas is heated slowly for some time by means of an electric heater so as to move the piston up through 10 cm.


The work done by the gas in the whole process is

• A. 21 J
• B. 12 J
• C. 2.1 J
• D. 1.2 J

Answer 1

The correct option is A 21 J

The force exerted by the gas on the piston at the moment when compression of the spring is x, is given by

$F=P_{0}A+kx$

where $P_0=$ 100 k Pa = atomospheric pressure

$A=20c{m}^2$ = Area of cross-section of the piston and k = 200 N/m = spring constant. Hence, work done by the gas as the piston moves through

$\ell$ = 10cm, is given by :

$W={\int }_0^{l}Fdx={\int }_0^l(P_{0}A+kx)dx=P_{0}A\ell +\frac{1}{2}k{\ell }^2$

$=(100\times 103N/m^2)\times (20-{10}^{-4}{m}^2)\times (10\times {10}^{-2}m)+\frac{1}{2}\times \left(200N/m\right)\times (100\times {10}^{-4}{m}^2)$

$=20J+1J=21J$