Consider one mole of perfect gas in a cylinder of unit cross section with a piston attached (Fig. 12.12). 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 volume from Vo to V1.
(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, Pa, V, Vo and 
k.
a. As initially the spring was unstretched and there was only external pressure (Atmospheric pressure). So, internal pressure should be equal to external pressure.
=> Pa=Pin
b. In this part we will use simple mechanics and not thermodynamics.
We can find the relax length of spring by A x l= Vo
Where A=cross sectional area of plate
Therefore,
Similarly, in the stretched position
So, pressure due to stretched spring 
Pin(final pressure)=Pa +
c. Ideally, the mechanical energy cannot be generated by heat, but let’s suppose it is happening in this case.
Using 1st law of thermodynamics which states that the change in internal energy of a system is equal to the heat added to the system minus work done by the system.
But here work is done on the system. So,
Now, the work done on the system is by the atmospheric pressure and the spring.
Work done by atmospheric pressure, (WPa)= F.d
WPa = F.dx
Since, Pa= 
F=Pa A
And 
WPa
Since volume has expanded and Force is in opposite direction so there will be negative sign in work done.
)
Work done by the spring=
=
Here, negative sign implies that displacement is against the force.
So, work done by the system,
And 
n=1
We know that ,
(here n=1)
