How much heat energy must be added to the gas to expand the cylinder length to 16.0 cm ?

Missing figure (and details) here:
http://i.imgur.com/l6n4Sry.jpg

Solution:
1) Let's use the ideal gas state law to find the number of moles of the gas:

where  is the initial pressure,  the initial volume of the gas, n the number of moles,  and  the initial temperature of the gas.
We need to find the pressure and the volume first. The pressure is given by the spring acting on the piston, with a force , where  is the spring compression. The area of the piston is , therefore the pressure is

The volume is instead the length of the cylinder (L=13 cm=0.13m) multiplied by the area of the piston:

Using these values and , we can find the number of moles:


2) After the heat is added, the cylinder expands to from L=13 cm to L=16 cm. Therefore, all the conditions of the gas (p,V and T) have changed, but the number of moles is still the same. However, we can calculate the new pressure and volume of the gas.
The spring has compressed by 3 cm (because the cylinder is now 3 cm longer), so . Therefore the new pressure is

The length of the cylinder is now L=16 cm=0.16m, therefore the new volume is

And so, from the ideal gas law we can find the new temperature of the gas:


3) Now we can calculate the variation of internal energy of the gas. This is given by:

where  for a monoatomic gas, and . Therefore we find


4) However, not all the heat added is used to increase the internal energy of the gas: part of it is used to compress the spring, doing a work of


5) And finally, using the first thermodynamics law, we can find the total heat added to the system: