# Monatomic gas work

Recently I was trying to solve the following problem: monatomic gas expanded from 0.2 to 0.5 m^3 and pressure increased from 404 to 808 kPa. Find work done by gas, heat gained and change in internal energy. As I undersand both volume and pressure changes happen simultaniously. All I can think of is that here the temperature increases five times, but I still need help to derive the answers. Or should I consider that pressure increased after the expansion and use iso-law formulas? Any help is very much appreciated!

• A few questions: 1) Is it an ideal gas? 2) Does it expand adiabatically or isothermally? – Ian Jun 10 '15 at 20:19
• The point is that I said everything about the problem. It is supposed to be a simple problem, so my interpretation is that pressure increases after volume.. – FringeEvent Jun 10 '15 at 20:33

## 1 Answer

Assuming this can be treated as a classical ideal gas, by the equipartition theorem

$$U=\frac{3}{2}N k_B T$$

and the ideal gas law

$$PV=Nk_B T$$

we find that the internal energy is

$$U=\frac{3}{2}PV$$

Therefore, the internal energy is multiplied by a factor of 5 in this process. Energy is a state function; this is why we can determine the change in internal energy by only considering the initial and final states.

With the given information you cannot tell anything about the work or the heat. This is because work and heat are not state functions. To determine the work and heat you need to know the path taken to get from the initial to the final state.

One way to see that there are many possible values of heat and work for any given change in internal energy is to look at the first law of thermodynamics:

$$\Delta U=Q+W$$

Notice that many different combinations of heat and work can add to give the same change in internal energy.