Consider one mole of a perfect gas in a cyclinder of unit cross section with a piston attached, (figure) A spring (spring constant K) is attached (unstretched length `L`) to the piston and the bottom of the cylinder. Initially, the spring is unstretched and the gas is in equilibrium. A certain amount of heat `Q` is supplied to the gas causing an increase of volume from `V_(0) "to" V`.
(a) What is the initial pressure of the system?
(b) What is the final pressure of the system?
(c) Using the first law of thermodynamics, write down a relation between `Q, P_(a), V, V_(0)` and `K`.

Explanation

(a) Initially, the spring is unstretched and the gas is in equilibrium, therefore, initial pressure of the system `P_(i)`= atmospheric pressure= `P_(a)`
(b) On supplying heat, the gas expands from `V_(0) "to" V`.
`:.` Increase in volume of gas `= V-V_(0)`
As cross sectional are of piston, `A=1`,
`:.` extension in spring , `x= ((V-V_(0)))/A= (V-V_(0))`
Force exerted by stretched spring on the piston,
`F= kx=k(V-V_(0)) ( :. A=1)`
`:.` Final pressure of the system, `P_(f)= P_(i)+P= P_(a)+K(V-V_(0))`
(c ) If `T` is final temp. of the gas, then increase in internal energy, `dU= C_(v)(T-T_(0))`
Where `T_(0)= =(P_(0)V_(0))/R=(P_(a)V_(0))/R`, and `T=(P_(f)V)/R= [P_(a)+k(V-V_(0))]V/R`
Also, `dW= P_(a)(V-V_(0))+1/2kx^(2)`
From first law of thermodynamics, `dQ=dU+dW`
`:. Q=C_(v)(T-T_(0))+P_(a)(V-V_(0))+1/2k(V-V_(0))^(2)`
This is the required. relation.