thermodynamics - Energy of room. Ideal gas law

I have been following Blundel's "Concepts of thermal Physics" and I got to the derivation of the ideal gas law. And it all made sense, we made a couple of assumptions and approximations, but then I got to an exercise that doesn't make a lot of sense to me:

"If a room is initially at 18ºC and is then heated up to 25ºC what happens to the total energy of the air in the room?"

That seems a straight forward question and an attempt of solution is: $$\langle E\rangle =uV=\frac{1}{2}Nm\langle v^2 \rangle$$ as $\langle v^2 \rangle=3\frac{k_B T}{m}$ we get: $$\langle E\rangle =\frac {3}{2}NK_BT$$ where $u$ is the energy density, $V$ is the volume of the room, and $\langle E\rangle$ is the mean kinetic energy of the gas in the room. After this we could just compare for both temperatures.

But doing this we are only considering kinetic energy. Throughout the derivation we ignored all kinds of motion a particle and considered only translational energy, but even doing this a particle still has 2 ways of having energy: kinetic and potential. We obviously didn't ignore the mass of a particle otherwise it wouldn't have kinetic energy either. so the particles still have mass, therefore it must have potential energy! I get that in the comparison of the energy at both temperatures, as the room stays still, the potential energy cancels out, but then saying that the mean energy of the room at a temperature $T$ is $\langle E\rangle =\frac {3}{2}NK_BT$ is inclomplete!

How do I describe the total energy of a room at temperature $T$?

Answer

Let's look at the ratios between the average kinetic energy (for one particle) and the change in potential energy between the bottom and top of our container. $$\frac{K}{\Delta U}=\frac{\frac 32 k_BT}{mgh}$$

Air is made of many types of particles, but nitrogen is the most common, so let's work with that. The mass of an $\rm N_2$ molecule is $4.65\times10^{-26}\ \rm{kg}$, and let's say the height of the room is about $3\ \rm m$. Then for $T=295 \ \rm K$ (around the middle of your temperature range) we have $$\frac{\frac 32 k_BT}{mgh}\approx 4.5\times 10^3$$

So the kinetic energy is much larger than the potential difference between the top and bottom of the room, which is why we tend to ignore it. You do have do consider potential energy when looking at larger heights, like when analyzing the atmosphere for example.