How was it decided how many electrons were assigned to each “row” in each element? Is the progession mathematically determined?

You’re referring to filling electron shells in elements, right— 2, 8, 18, etc.? Yes, they’re mathematically (well, physically) determined.

Short answer: Electrons in an atom have four quantum numbers, which are usually denoted n, l, m[SUB]l[/SUB], and m_[SUB]s[/SUB]. (The second item there is a lowercase l. It’s usually represented as a lowercase script letter to avoid confusion with capital I.) No two electrons can have exactly the same set of numbers because of the Pauli exclusion principle, and electrons fill up those quantum numbers in order of increasing energy*. For various quantum mechanical reasons, the four quantum numbers can only take on very specific values:

- n has to be a positive integer: 1, 2, etc. (It describes which shell the electron is in and is essentially the energy of the electron.)
- l ranges from 0 to n - 1. (If you’re familiar with the idea of orbitals, this is the number corresponding to them: l = 0 is the s-orbital, l = 1 is the p-orbital, etc.)
- m[SUB]l[/SUB] is an integer from -l to l. (It represents the angular momentum eigenvalue, in some direction.)
- m[SUB]s[/SUB] is either -1/2 or 1/2. (It represents the spin eigenvalue, in some direction.)

If you work out the combinatorics of the allowed quadruples (n, l, m[SUB]l[/SUB], m[SUB]s[/SUB]), you get the progression you describe.

- The order in which the orbitals are filled is a bit complicated. There’s a point at which you really need to start involving chemistry or atomic physics, rather than just straightforward classical quantum mechanics, to get an accurate model of larger atoms.

Longer answer: The four quantum numbers parametrize solutions of the Schrodinger equation for a hydrogen atom, and they’re eigenvalues (modulo some bookkeeping) of various operators: n corresponds to total energy, l corresponds to total angular momentum L^2, m[SUB]l[/SUB] corresponds to some angular momentum L[SUB]z[/SUB] along the z-axis (or your favorite axis), and m[SUB]s[/SUB] corresponds to spin S[SUB]z[/SUB] along the z-axis (or, again, your favorite axis). The last is easy: electrons are spin-1/2 particles, and so m[SUB]s[/SUB] is either -1/2 or 1/2. The others involve a bit more math than I can really write in this format. It’s straightforward and illustrative— in fact, it’s an exemplary problem in quantum mechanics— but fairly mathy; you’re basically doing some representation theory of sl(2), which is well-known but can be tricky if you’re not familiar with the general theory. This problem was a triumph of quantum mechanics and reasonably easy to derive, so it should be in any first course in the subject; Griffiths’ *Introduction to Quantum Mechanics* (the one with the cat on the cover) is as good a resource as any. Wikipedia is also surprisingly good for this sort of thing.

It’s often said that n determines the energy, but that’s an oversimplification. All four quantum numbers have an effect on the energy. The effect from n is the largest, but out in the higher-numbered shells, it’s possible to have the energy states from different n values overlapping.

I think the OP is not asking why they are that way, but how physicists first arrived at the theory that they were that way. The simple answer was that those numbers were the difference between the atomic numbers of pairs of elements with similar properties, such as the halogens, the highly reactive metals, and inert gases. Once that difference was more or less consistent and recognized, it formed a theoretical platform that could be pursued.

I was referring to part of that in the footnote above. Unfortunately, the exact energy in different shells gets a bit complicated (I think Cu is the first clear exception?), and I’m not familiar with the details. The quantum number n refers to the radial part of the wave equation after separation of variables, which I think dominates the total energy for low atomic number, but I’m not sure where exactly the model breaks down.

The Periodic Table came first. In the mid to late 1800s, chemists discovered a large number of elements, and it was noticed that there were periodic trends in their properties. The idea introduced by Mendeleev was to list the elements by weight per mole and to restart the rows every so often, so that similar elements ended up in vertical columns. This is the Periodic Table. The reasons for its structure – how long the rows had to be to make the similar elements end up under each other – was unknown.

As soon as it became known that the atom was made of a positive core and negative electrons, roughly 1900, it was proposed that the force that bonded atoms together into compounds and molecules was the mutual attraction of the positive cores for the negative electrons, which works the same way a pretty woman can keep two men who don’t like each other in the same vicinity at a party. 19th century chemists had already worked out the chemical bonding tendencies of each element, e.g. how many chemical bonds Group 7A elements formed with carbon, versus how many their neighbors in Group 6A would. This was called the elements “valence,” from the Latin for “strength.” As soon as the idea arose that chemical bonds were made of electrons, the conclusion that an element’s valence had something to do with the number of electrons “available” for bonding was natural. So then we found that the number of “valence electrons” corresponded very well to the arrangement of elements in the Periodic Table (with some minor shuffling, since the original PT was arranged by atomic weight, not number of electrons – the two orderings are very similar but not identical).

Once you know the number of valence electrons for each atom, you can deduce the number of electrons that must be added in each row to give elements in the same column the same number of valence electrons. But why is that number what it is? At that point, chemistry can tell you no more, and it was up to the physicists to figure it out from first principles. As noted above, the first triumph was establishing the fact that electrons refuse to be in the exact same state (the Pauli Principle), which gives us a reason why there is structure to the way electrons are arranged in atoms in the first place – why we have shells of electrons that get filled up, and force the construction of a new shell further out. The second triumph was using basic principles of physics – the attraction between the nucleus and the electrons – plus the Pauli Principle, plus the general requirement that the energy be minimized, to work out exactly how the electrons can be arranged in an atom.

As it turns out, the higher the energy of an electron, the more ways it can orbit the nucleus, which is not hugely surprising – it’s a general principle of thermodynamics that the entropy of a state increases with its energy. So that explains why the number of electrons you can fit around an atom at roughly the same energy increases as the energy goes up, which tells you why the length of the rows in the Periodic Table gets longer and longer. But to get the exact number, you have to solve the equations, and I can think of no direct physical principle that tells you to expect 2 for the first row, 8 for the second and third, 18 for the fourth and fifth, and so on.

Already in principle for the H atom the energy depends on n and l. It does not depend on m_l or m_s in the absence of a magnetic field. That’s because the radial part of the Schroedinger equation contains a “centrifugal barrier” term that depends on l. In the absence of any electron-electron repulsion, it just turns out that the energy does not depend on l – there is an accidental degeneracy (although I vaguely remember arguments that it isn’t very accidental). Obviously you can’t solve the SE exactly for more than one electron, but when you solve it in a mean-field approximation (and then do perturbation theory to recover some of the effects of electron correlation) you find that the importance of l to the energy grows with the number of electrons. Using the hydrogenic model to calculate electronic energy is already noticeably wrong for He, and gets worse after that.

But you can still use the relative ordering of hypothetical one-electron orbitals (which do not, strictly speaking,exist, but which are convenient basis functions for the many-electron wavefunction) to explain the Periodic Table, mostly. The first weirdness is the failure of the 3d subshell to be filled after the 3p, i.e. the fact that after Ar we get [Ar]4s1 instead of [Ar]3d1, or, in other words, the odd fact that Period 3 is the same length as Period 2. The hydrogenic model cannot explain this. But we can just take it as an empirical rule and keep going, and we mostly do OK. There are about 10-15 exceptions in the main table (not counting the inner transition elements) to the result of using the hydrogenic model plus a few empirical rules about the effect of l. Some are amenable to simple explanations, such as Cr ([Ar]3d5,4s1 instead of [Ar]3d4,4s2) and the one you mentioned, Cu which is [Ar]3d10,4s1 instead of [Ar]3d9,4s2. We invoke the extra stability of a half-filled or filled subshell.

Others, however, have no easy explanation of which I am aware, e.g. Nb, which is [Kr]4d4,5s1 instead of [Kr]4d3,5s2. V above it is [Ar]3d3,4s2 and Ta below it is [Xe]4f14,5d3,6s2. Why is niobium the odd man out? I don’t know.

Wow, that’s really interesting. Thanks!

Strictly speaking, the energy of the hydrogen atom does depend on m_l and m_s, even in the absence of an external field, because the proton itself has a magnetic field. The difference in energy is tiny, though. Changing from one to the other is what’s called the hyperfine transition, and is associated with 22 cm radio waves.

I went to Wikipedia, and equation I was looking for was 2nsq. IOW-2,8,18, etc.