Re: Why is sodium stable?

I will extend my previous answer, because I think you are focusing on the wrong aspect of the problem.

First, a comment: you wrote, I am not about to tell them just take my word on faith. My question to you is, Why not? This is the basic pattern of learning.

While epistemology is not something we will handle here in depth, I think Michael Polanyi's Personal Knowledge pretty thoroughly demonstrates that we learn as part of a community, and that we must always accept at least something known by that community on faith before we can know it to be true by experience.¹ Sometimes we never do have the opportunity to experience the truth of some particular bit of knowledge... so we must always take it on faith. One must have some confidence in one's predecessors and contemporaries, after all... the alternative is a barren, radical skepticism.

Question Authority is an incomplete slogan. Before one has a right to question, one must first accept and learn. But I'm afraid that Accept authority. Learn from it. Then tear it to bits doesn't fit so well on a bumper sticker.

End of editorial; back to our previously-scheduled answer.

Chemistry is not something that happens to isolated atoms. Physics is what happens to isolated atoms. Therefore, it is illegitimate to demand a "complete explanation" of CHEMISTRY in terms of isolated atoms or ions. Now, we can use the Lewis picture to provide some hints... but in the last analysis, reactions happen because THE SYSTEM after the reaction has a lower free energy than THE SYSTEM before the reaction. THE SYSTEM is typically made as large as it needs to be to include the components from which free energy is removed as well as those which may gain free energy.

I will continue answering you point-by-point: Sodium has one electron in its valence shell. Now if the atom loses this electron its atomic radii decreases. This brings the electrons closer to the protons which seems to make a stable ion. In fact it would seem that this would be more stable than the neutral sodium ion which is much bigger. However energy is required to ionize sodium to this form! This suggests that the neutral sodium atom is more stable than the ion... What makes the neutral sodium stable?

In interstellar chemistry there are lots of gas-phase cations, but producing those ions involves the input of energy -- the impact of a photon or of another molecule is required to knock an electron loose. Even in sodium, it costs 118.5 kcal/mol to remove an electron. For other elements in the same row, it costs rather more. The trend in ionization energies as one moves across (and also down) the Periodic Table can be explained on the basis of the shell model of the atom; but the point you are making, and to which I am responding, is that the ionization energy is always positive.

The reason for this is that, no matter how loosely-held it is, an electron in an atom has a lower free energy than a free electron. Neutral sodium (even gas-phase, atomized sodium) is more stable than isolated sodium cations plus isolated electrons because even an outer-shell electron is always closer to the positively-charged nucleus than it would be as a free electron. The wonder is, not that neutral sodium is stable, but that it takes so little energy to knock an electron loose. And that can be explained using shell theory!

By-the-bye, the decrease in atomic radius upon the ionization of sodium is primarily due to the fact that the lone outer electron occupies a rather large orbital. Lose that electron (so that you are left only with electrons in the smaller, inner shells) and you automatically lose a good deal of the atomic radius! A 3s electron has comparatively little "penetration" and so the inner electrons really do not see much more nuclear charge than they did when the outer electron was present. Of course, the inner shells shrink somewhat after ionization; but this effect on atomic radius is decidedly second-order.

Electron affinities follow the same pattern. An electron in an atom (even if the atom becomes negatively-charged thereby) is usually² more stable-in-itself than a free electron, because now it has at least some interaction with a positive charge (the atomic nucleus).

Actually, chlorine is not quite stable as a neutral atom. It costs 29 kcal/mol to break a chlorine-chlorine bond, and rather more to break bonds to other elements. Now, the electron affinity³ of chlorine is 83.4 kcal/mol, so that it is energetically favorable to produce Cl^- from gaseous Cl₂ (the element's standard state). This is not necessarily the case for other elements; except for the halogens, bond energies are usually too great for electron affinity to compensate for breaking covalent bonds.

If you examine trends in electron affinities,³ the shell model provides an excellent rationalization of what is observed, just as it does for ionization energies. Here it is:

What I am missing? ... Particularily, am I looking too much at atoms as particles? Yes, you are. What you are missing is the real and essential interactions between atoms which make up chemistry. To summarize: