Consider one mole of perfect gas on a cylinder of units cross-section with a piston attached (figure). A spring (spring constant K ) is attached ( unstretched length L ) to the piston and to 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 value from `V_(0)` to `V_(1)`.
(a) What is the initial pressure of the system ?
(b) What is the final pressure of the system?
(c) Using the first law of thermodynamic, write down a relation between `Q, p_(a),V,V_(0)` and K.

Explanation

(a) Initially the piston is in equilibrium hence,`p_9i)= p_(a)`

(b) On supplying heat, the gas expands form `V_(0)` to `V_(1)`
`therefore ` Increase in volume of the gas = `V_(1) -V_(0)`
`x = (V_(1) - V_(0))/("Area") = V_(1)-V_(0)`
`therefore ` Force exerted by the spring on the piston
`= F = kx = (V_(1) - V_(0))`
Hence, Final pressure `= p_(f) = p_(a) + kx `
`= p_(a) + k xx (V_(1) - V_(0))`

(c) From first law of therodynamics dQ = dU + dW
If T is final temperature of hte gas, then increase in the internal energy
`dU = C_(V) (T, T_(0)) = C_(v) (,T_(0))`
We can write , `T= (p_(f) V_(1))/(R) = [(p_(a) + K (V_(1) - V_(0)))/(R)](V_(1))/(R)`
Work done by the gas = pdV + increase in PE of the spring
`p_(a) (v_(1) -V_(0)) +(1)/(2)kx^(2)`
Now, we can write dQ = dU + dW
`C_(v) (T- T_(0)) + p_(a) (V - V_(0)) + (1)/(2)kx^(2)`
`=C_(v) (T- T_(0)) + P_(a) (V- V_(0)) +(1)/(2)(V_(1) - V_(0))^(2)`
This is the required relation .