Two mole of an ideal monatomic gas are confined within a cylinder by a mass less spring loaded with a frictionless piston of negligible mass and crossectional area `4xx10^(-3)m^(2)`. The gas is heated by a heater for some time. During this time the gas expands and does `50J` of work in moving the piston through a distance of `0.01`m. The temperature of gas increases by `50`k.

The force constant of spring is

Options

A. `189.6 N m ^(-1)`
B. `18.96 N m ^(-1)`
C. `1896 N m ^(-1)`
D. `2896 N m ^(-1)`

Answer

C

Explanation

When the gas is heated it expands `&` pushes the piston by x. If k is force constanat of spring and `A` is area of crossection of the piston. It `P_(0)` is atmospheric pressure then at equilibrium of piston the pressure of the gas on the piston
`P=P_(0)+(kx)/(A)`
The increas in the volume of the gas by samll movement of x of piston is
`dv=Adx`
`W=int_(0)^(X)Pdv=P_(0)Ax+(1)/(2)kx^(2)`
Putting `A=4xx10^(-3)m^(2),x=0.1m, W=50 J`.
`P_(0)=1.013xx10^(5) Nm^(-2)` in above equation
`:. k=1896 Nm^(-1)`