An ideal gas has a volume `0.3 m^(3)` at `150 k Pa`. It is confined by a spring-loaded piston in a vertical cylinder. Initially, the spring is in relaxed state. If the gas is heated to a final state of `0.5 m^(3)` and pressure `600 k Pa` find the work done on the spring (atmospheric pressure, `P_(0) = 1 xx 10^(5) N//m^(2))`.

Explanation

`P_(at) = 10^(5)` Pa
Initial pressure of gas,
`P_(1) P_(1 at) + P_(1 spri ng) + P_(1 p i s t o n)`
`P_(s p ri n g)` : pressure due to spring
`P_(p is t o n)` : pressure due to piston
Initially, the pressure is in relaxed position,
`P_(1 s p ri n g) = 0`
`P_(1) = 150 k Pa = 1.5 xx 10^(5) Pa`
`1.5 xx 10^(5) = 10^(5) + 0 + P_(p i s t o n)`
Pressure due to piston `= mg//A = 0.5 xx 10^(5) N// m^(2)`
In the final condition,
`P' = P_(0) + P'_(s p ri n g) + P'_(p i s t o n) (P_(p i s t o n) = P'_(p i s t o n))`
`P'_(s p r i n g) = 6 xx 10^(5) - 1 xx 10^(5) - 0.5 xx 10^(5)`
`4.5 xx 10^(5) N//m^(2)`
Initial load in the spring , `F_(1) = 0`
Final load in the spring,
`F_(2) = P'_(s p r i n g) A = (4.5 xx 10^(5) A) N`
Work done on the spring
= (Average force) (distance moved)
`= ((F_(1) + F_(2)))/(2) ((V_(2) - V_(1)))/(A)`
`= [((0 + 4.5 xx 10^(5))/(2)) A xx ((0.5 - 0.3)/(A))]`
`= 0.45 xx 10^(5) J = 45 kJ`