A non conducting piston of mass m and area of cross section A is placed on a non conducting cylinder as shown in figure. Temperature, spring constant, height of the piston are given by T,K,h respectively. Initially spring is relaxed and piston is at rest. Find
(a) Number of moles
(b) Work done by gas to displace the piston by distance d when the gas is heated slowly.
(c ) Find the final temperature

Explanation

(i) `PV = nRT rArr (P_(atm) +(mg)/(A)) Ah = nRT`
`rArr n = ((P_(atm)+(mg)/(A))Ah)/(RT)`
(ii) `1^(st)` method
Applying Newton's law on the piston
`mg + P_(atm) A + Kx = P_(gas)A`
`W_(gas) = in_(0)^(d)t P_(gas) A dx`
`= int_(0)^(d) (mg +P_(atm)A +Kx) dx`.
`rArr W_(gas) = mgd +P_(atm)dA +(1)/(2) Kd^(2)`
`2^(nd)` method
Applying work energy theorem on the piston
`W_(all) = DeltaKE`
Since piston moves slowely therefore `DeltaKE = 0`
`W_("gravity") +W_(gas) +W_("atm") +W_("spring") = 0`
`-mgd +W_(gas) +(-P_(atm)Ad) +[-((1)/(2)Kd^(2)-0)] = 0`
`rArr W_(gas) = mgd +P_(atm) dA +(1)/(2) Kd^(2)`