Can we use electrostatics in place of the Pauli Exclusion Principle to crudely model "touching"?

I know similar questions have been asked before, but I was thinking about this and curious if it’s a valid way to approach the question — especially when we are introducing high school students to the concept of electrostatic repulsion. So think of this as an exercise I might take students through as a lesson-set to illustrate why certain theories or laws are not complete, and what their limits are — I would love to her what people think. Be aware that this would be geared to people who have NOT had calculus, and many might struggle with math, though they might have had chemistry.

We know that the Pauli Exclusion Principle is why two objects can’t be in the same space, because particles like electrons (fermions) won’t occupy the same quantum state. This is why for example you can’t have more than 2 electrons in the lowest-energy orbitals of an atom, and more than 8 in the next one, et cetera (and please, tell me if I am mis-stating this).

That’s well and good. But let’s think about the usual electrostatic repulsion, which is more common to see at the macro scale (like when we see things affected by static electricity and can do tricks like making styrofoam peanuts stick to a cat).

Let’s imagine a piece of some material, let’s say it’s carbon. There are four electrons in the outer shell of carbon and room for four more (in theory, I realize this never really happens to free carbon atoms in nature). In 1 mole of carbon there are $(6 times 10^{23})(4) = 2.4 times 10^{25}$ electrons. They aren’t all on the surface, though. Amorphous carbon is ~2 g/ml and a mole of carbon 12 grams, so the volume is 24 cubic centimeters and that would be a cube 2.88 cm on a side. That’s about $8.44 times 10^{7}$ layers of carbon atoms, and each layer has about $7.12 times 10^{15}$ atoms in it, so that’s about how many carbon atoms will be in contact with our hypothetical surface. Assuming only one in four electrons is in contact with said surface at any given time (I know this is not quite right but we aren’t assuming quantum effects here) that’s $7.12 times 10^{15}$ electrons.

If we assume our surface is also carbon then we get the same number of electrons in contact with our cube of carbon.

We can’t assume zero separation between the surfaces, but we can figure out what the minimum distance would be for the repulsive force to overcome gravity and be in stasis, if we are doing classical calculations. Gravity pulls on our 12-gram carbon cube with a force of

$(0.012 kg)(9.81 m/s^2) = 0.1172 N sim 0.12 N$

And for a crude model we can see how many Newtons per electron that is, and we get $0.12/7.12 times 10^{15} = 1.69 times 10^{-17} N$ per atom

$$1.69 times 10^{-17} N = frac{kq_1q_2}{r^2} = frac{(8.99 times 10^9)(1.6 times 10^{-19})^2}{r^2} = frac{2.3014 times 10^{-28}}{r^2} rightarrow r^2 = frac{2.3014 times 10^{-28}}{1.69 times 10^{-17}} = 1.36 times 10^{-11} m
r = 3.69 times 10^{-6}m$$

Which makes a lot of sense as that is so close that "touching" seems as good an approximation as any.

Now, I know this is wrong. But it is interesting that you would get these kinds of results using classical methods, and what struck me was that this method shows why the classical explanation is in fact, inadequate! Aside from the phenomenon of vacuum welding, which by itself would show that electrostatic repulsion isn’t the whole story. The fact that electrons lose energy (giving off radiation as they move) would say that atoms should collapse, and they don’t. (I realize I am to an extent duplicating the thinking of past scientists here; that is kind of the point).

But this exercise does show that even electrostatic repulsion can have macroscopic effects, and again, my thought is it might illustrate some of the outer bounds, as it were.

What I was thinking of is if this is a good way to at least get some ideas of what the limits are; if we know that electrons repel each other classically we know that there has to be at least that kind of distance if we’re talking QM. This is something I was thinking of as a way of getting physics students into the history a bit too; if we can show the kinds of calculations people could do before quantum mechanics was developed, and then get into why it works — but only to a point — and also why it doesn’t. I’m still sort of turning this over in my head, so I am not at all sure if it’s even a good plan. But I thought of it in the context of the way we often teach physics and chemistry.

ETA: square-rooting the r, which gives an answer that looks more "normal." I’m still thinking this gives something workable, as it demonstrates Coulomb repulsion, but I am playing around a bit to see if we can’t sort-of-duplicate the historical thinking.