S,P,D,F subshells allow for 1,3,5,7 orbitals respectively. Is there some obvious reason why successive subshells always allow for 2 more orbitals (besides obviously that it will allow for the total electrons to be 2N^2’ which I’m assuming is tautological)? Thanks in advance!
This relates to the angular momentum of the orbital, which is governed by two quantum numbers, l and m.
Long story short, l does go up by 1 each “level”, so the S orbitals are ones where l=0, the P orbitals are the ones where l = 1, &c. For each level, though, m is constrained to be between l and -l. So S orbitals have m=0, P orbitals have m ∈ {-1, 0, 1}, &c, which accounts for the “increasing by 2”, since every single increase in l increases the number of possible values of m by 2.
(In the classical analogue, where angular momentum is a 3-vector, l determines the length of the vector, and m determines the value of one component of it, and the constraint between l and m is the constraint that a vector has to be at least as long as its longest component.)
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If the state of an electron is determined by the angular momentum, the problem boils down to enumerating the possible irreducible finite-dimensional representations of SU(2) (or its Lie algebra). The number l will essentially be the highest weight of the representation, and the weights that occur will be integers between -l and l, as stated above. The angular momentum is thus quantized.
My chemistry teacher in high school claimed that the very simple formula for the number of electrons in each shell was clear proof that the universe was designed. Well, ok, maybe. There’s no simple reason, but its provable in quantum mechanics. The reasons are very complex and took physicists a long time to figure out, even once they knew about atoms and protons and electrons. As mentioned, it has to do with the various states of angular momentum that are possible for an electron in an atom, but why would take two college courses at least to explain.
If you’re interested in learning, I suggest trying out MIT’s Open Course Ware. You probably can start with 804 (Quantum Physics; the 8 means Physics, by the way. This is the 4th course in Physics.), but learning 803 (Waves mechanics) helps a lot even if it’s just classical waves, because quantum theory is based on waves that have many similar principles to classical waves. 804 made a lot more sense after I went back and did 803. Knowing basic Electricity/Magnetism and Maxwell’s equations’ theoretical results helps a lot too even if you can’t work out any problems (I certainly can’t, but I know the physical meanings of all of it). You’ll learn a good deal about the energy levels in Hydrogen in 804, and a whole lot more in 805 if you’re willing to put up with learning a whole ton of math (I already knew the linear algebra required as a grad student in math).
Is this discussion of possible angular momenta at all related to Pauli’s Exclusion Principle?
As a hand-wavy heuristic explanation, you can think of the subshell angular momentum like this:
[ul][li]Angular momentum tells you about the shape of the orbital. You could describe shape via a factor like[/li]x[sup]i[/sup] y[sup]j[/sup] z[sup]k[/sup]
where x, y, and z are meant to be the three directions. Since we’re only interested in shape and not in spatial extent, we can assume
x[sup]2[/sup] + y[sup]2[/sup] + z[sup]2[/sup] = 1.
In terms of the kind of angular factor I’ve described above, the angular momentum is
l~i+j+k.
[li]But this isn’t quite right. We’re actually only interested in the unique shape factors, and a lot of the ones generated by the procedure I’ve outlined above are redundant. The unique new shapes at angular momentum l wind up being 2l+1 in number, for reasons discussed earlier.[/li][/ul]
So let’s make this concrete:
[ul][li]For l=0 I just get one possibility: x[sup]0[/sup] y[sup]0[/sup] z[sup]0[/sup] = 1. This is the shape of an s orbital, which has no angular structure at all.[/li]
[li]For l=1 I have three possibilities: x, y, and z. These are the shapes of the p orbitals, and there are 3.[/li]
[li]But then we come to l=2. If you work through it, you have 6 possibilities: x[sup]2[/sup], y[sup]2[/sup], z[sup]2[/sup], xy, xz, yz. But one combination of these things is[/li]x[sup]2[/sup] + y[sup]2[/sup] + z[sup]2[/sup] = 1
and 1 is the shape factor for an s orbital. In other words, there is one combination of these 6 potential “d” orbital shape factors which we already had. And that means we have to exclude it from our list of d orbitals because the shape isn’t new. From the original list of 6 possibilities, we only have 5 new shapes.
[li]Similarly, if you tried to do this with f orbitals which have l=3, you should come up with 10 possibilities, but among these would bex[sup]3[/sup] + x y[sup]2[/sup] + x z[sup]2[/sup] = x[/li]x[sup]2[/sup] y + y[sup]3[/sup] + y z[sup]2[/sup] = y
x[sup]2[/sup] z + y[sup]2[/sup] z + z[sup]3[/sup] = z
and these have the same shape as the p orbitals. You’re left with 7 new things from the original 10.
[/ul]
And so on.
Well, electrons with different orbital angular momenta, or different electron spin (also a type of angular momentum) will not be excluded from being jammed into the same atomic orbital. Pauli originally stated his principle for electrons; maybe someone here is familiar with his original proof/derivation?
Erratum: where I typed SU(2) above, read SO(3) (group of rotations in three dimensions). This is in fact a crucial point, since this rotational symmetry when analyzing angular momentum is precisely what (after some algebra) leads to spherical harmonics and the fact that the irreducible unitary representations have dimensions 1, 3, 5, …
One can also realize these representations as homogeneous harmonic polynomials as explained by g8rguy: so the 3-dimensional representation will be spanned by x, y, and z; the 5-dimensional one will be spanned by x[sup]2[/sup]-y[sup]2[/sup], x[sup]2[/sup]-z[sup]2[/sup], xy, xz, and yz; etc.
This mathematics may seem a bit ad hoc but spherical harmonics will in fact show up when solving the spherically symmetric Schrödinger’s equation.