Two moles of an ideal monoatomic gas are confined within a cylinder by a massless and frictionless spring loaded piston of cross-sectional area `4 xx 10^(-3)m^(2)`. The spring is, initially in its relaxed state. Now the gas is heated by an electric heater, placed inside the cylinder, for some time. During this time, the gas expands and does `50J` of work in moving the piston through a distance `0.10m`. The temperature of the gas increases by `50K`. Calculate the spring constant and the heat supplied by the heater. `P_(atm) = 1 xx 10^(5)N//m^(2)R = 8.314 J//mol-K`

Explanation

`n=2`,`A=4xx10^-3m^2`, monoatomic gas,
Let as some instant, spring is compressed by `x` force in spring: `kx`
pressure exerted by spring `P=(kx)/(A)`
Pressure exerted by gas `=P_0+(kx)/(A)`,`P_0`: atm pressure
`dW=PdW=(P_0+(kx)/(A))Adx`
`W=int_(0)^(x0)(P_0+(kx)/(A))Adx`
`=P_(A)x_0+(1)/(2)kx_0^2`
`=10^5xx4xx10^-3xx0.1+(1)/(2)k(0.1)^2`
`50=40+(0.01)/(2)k`
`k=2000(N)/(m)`
`DeltaU=nC_VDeltaT=2xx(3R)/(2)xx50`
`=150xx8.3=1245J`
`DeltaQ=DeltaU+DeltaW=1245+50`
`=1295J`