Part 5 of our article (one more to go!), in which we hope to develop an understanding of atomic structure and the periodic table, from the beginning of secondary school/junior-high level, to first/second year undergraduate physics/chemistry, via one fictional conversation. Find the others at our blog page sciencebydegrees.com/blog. Sigrid has just asked Sally what spin has to do with electron configuration…
SALLY: Well, one orbital holds up to two electrons, as you said, but two electrons in the same orbital have to have opposite spins. So to summarise all that new information, go and draw an energy level diagram for n=1 to n=4 with all the orbitals in it.
SIGRID: OK. Here it is…
SALLY: That’s a great effort. It’s not quite right – but there’s no reason you should have realised why yet. Look at my version and compare the two…
SIGRID: Hmm. The different types of orbital within a shell have different energies. Why is that? And does it matter?
SALLY: It really matters, because it explains why the potassium and calcium start to fill the n=4 shell before the n=3 shell is full, and why the transition metals, actinides and lanthanides have blocks on the periodic table of their own. As for why it happens, there’s a three stage explanation. Firstly, remember that ‘more attraction from the nucleus’ is the same as ‘more tightly bound’ is the same as ‘lower energy’ is the same as ‘more chemically stable’! Next, p-orbitals are on average further from the nucleus than s-orbitals of the same shell, because of their dumbbell, rather than spherical shape. In turn, d-orbitals are further than p, and f-orbitals are further than d. Finally, electrons that are far from the nucleus are ‘shielded’ from its attraction by the effect of the electrons closer in.
SIGRID: That must mean that f-orbitals have more shielding than d, which have more shielding than p, which have more shielding than s.
SALLY: Exactly – so what will be the effect on the energy levels?
SIGRID: s electrons are the least shielded and the closest to the nucleus, so they will have lower energy than p electrons. And so on. The order will be s<p<d<f.
SALLY: And that’s exactly what is shown the diagram I just showed you.
SIGRID: So because the energy levels for each shell are spread out, and because the energies for each n get closer together as n increases, that means that the energy of the n=3 p-orbitals (called 3p for short) are actually higher than the n=4 s-orbital (called 4s for short). And that’s why the n=4 shell starts to fill before the n=3 shell is full. It’s the bit labelled in red on the diagram…
SALLY: If we are going to build some atoms, in what order should we fill the orbitals?
SIGRID: Like this I guess: 1s, 2s, 2p, 3s, 3p, 4s, 3d, etc, by working up the new energy level diagram.
SALLY: You know how you built some atoms before, using electronic configurations like 2.8.1 for sodium? Well, why don’t you repeat that with your new notation? You’ll find you can get further than the first 20 elements.
SIGRID: Ok, well for hydrogen, we obviously put the single electron in the 1s orbital. The ‘1’ signifies the first shell (energy level), the s specifies the orbital within it.
SALLY: Yes, and we label that 1s1, where the superscript ‘1’ means ‘1 electron in this orbital’.
SIGRID: Then hydrogen is 1s1 and helium is 1s2. We can draw them like this:
SALLY: Good – keep going…
SIGRID: OK, well I’ve done up to boron (element number 5). Here it is.
But for carbon I need to put another electron in an n=2 orbital, and I don’t know whether it makes any difference where it goes. I mean, wherever it goes, I write 1s22s22p2, but it affects how I draw the diagram.
SALLY: It does matter! There is something called Hund’s rules to help you. Basically, you add an electron to each orbital with spins in the same directions (we say ‘parallel spins’), before adding the second electron to any orbital.
SIGRID: Why does that work?
SALLY: Basically, electrons in the same orbital are on average closer than electrons in different orbitals. That means they have greater mutual repulsion and thus higher energy.
SIGRID: So the lower energy state is the one utilising more orbitals without doubling up.
SALLY: Exactly.
SIGRID: OK, now I know that I can carry on. I’ll go up to 20 just to show I can match the previous method. Here you are…
SALLY: Well done. And now look at your list of order of filling, and see what happens to the 21st element.
SIGRID: OK, so the order is 1s, 2s, 2p, 3p, 4s, 3d, and the first 20 elements get us up to the 4s orbital. Element 21 (scandium) must use the 3d orbital. So when I wrote it before as 2.8.9.2 instead of 2.8.8.3, that 9th electron in the n=3 shell corresponds to the 3d electron in our new notation: 1s22s22p63s23p64s23d1.
SALLY: And whichever way you write it, when filling shells, that 21st electron is the first we have seen that doesn’t go into the outer shell.
SIGRID: So the first row of the transition metals look like this…
I thought Cr would be 1s22s22p63s23p63d44s2 and Cu would be 1s22s22p63s23p63d94s2, but when I cheated (I mean ‘checked my working out against a number of textbooks’) I found this table is definitely correct. It seems like Cr and Cu are preferentially half-filling and filling the 3d subshell.
SALLY: That’s right. An electron shifts from 4s into 3d to gain extra stability. And remember – by extra stability, we mean ‘it’s a lower energy state that way’… Can you see why the transition metals are often called the d-block elements?
SIGRID: Yes! Because that whole rectangle of the periodic table are the elements for which the ‘most recently’ filled orbital is a d-orbital. And the rectangle is 10 elements wide because each shell from n=3 onwards has 5 d-orbitals, each of which can hold 2 electrons.
SALLY: Do you realise you have just explained that ‘gap’ between beryllium and boron, and between magnesium and aluminium?! It’s because lower down the periodic table, you need to fit in the d-block elements. And can you explain the octet rule now?
SIGRID: OK! For the noble gas electronic configurations, the outer shells have fully occupied s and p orbitals. And that accounts for 8 electrons in the outer shell, even if there are unoccupied d and f orbitals in inner shells. And that’s the octet rule…
SALLY: And what about the 2n2 rule?
SIGRID: That was about how many electrons in total you can fit in each level, or value of n, if you will. Well that hinges on the odd number of orbitals of each type in each shell. n=1 has 1 orbital, n=2 has 1+3 orbitals, n=3 has 1+3+5 orbitals, and so on, and this rule generates the square numbers, n2. Each orbital holds 2 electrons. So 2n2!
SALLY: And what about the lanthanides and actinides?
SIGRID: Well, we said before that 2n2 causes us to need to fit more in further down. So just like the rectangle of transition elements that we squeeze in is because of the d-orbitals, are the actinides and lanthanides the elements with occupied f-orbitals.
SALLY: That’s exactly it. Like this…
SIGRID: Hmmm. That’s pretty weird.
SALLY: Try being a nuclear physicist! A similar energy level structure exists for protons and neutrons in nuclei too, but it’s much worse there! The energy levels overlap almost straight away, so you don’t even get the first 20 behaving themselves, like we did here! At least those first 20 provided you with a step on the road to a better understanding.
SIGRID: Yeah, OK. Maybe I’ll stick to atomic physics and chemistry…
SALLY: Just one thing. That energy level diagram that shows the order of filling orbitals? That applies to the ground states of atoms. And basically it’s the interactions between electrons with the nucleus and also with the other electrons (including the shielding that we mentioned) that lead s, p, d and f orbitals to have different energies within the same energy level. But hydrogen with 1 electron also has an n=3 state – it’s just an excited state, rather than the ground state. And for hydrogen there’s no energy difference between, say, the 3s and 3p excited states, because there are no other electrons to affect them. The equivalence of energy in different states is called degeneracy, so we could say that the 3s and 3p states are degenerate in hydrogen, but not, say in chlorine. We’ll look at the idea of degeneracy a bit more later on.
SIGRID: OK, so now I’ve got a more sophisticated idea of what an atom ‘looks like’ in terms of orbitals, rather than electron orbits. But you’ve only shifted the point of attack of my question. Before, I asked why the shells can only hold 2, 8 etc electrons, and we explained that in terms of orbitals. But we’ve just transformed the same question into ‘why are there as many orbitals as there are?’
SALLY: Hmm. I was rather hoping you’d be satisfied with what we’ve done so far, but I can tell you want to know more. So we’ll carry on. Just one thing – be wary of ‘why?’ questions. Quite often, we do end up explaining one thing after another until we get to an idea that just seems to work and we don’t know why. Richard Feynman used to talk about that.
SIGRID: What’s the next step? I promise not to be angry if I still have questions after that.
SALLY: Do you remember how the number n came out of Bohr’s theory of the hydrogen atom? We called it the ‘principal quantum number’? Well, the next step involves a different model of the atom, which uses the ‘wavefunction’ of the system. That’s, if you like, a map in space and time, of the probabilities of finding the electrons in particular places.
SIGRID: Like a mathematical version of the orbitals we have drawn.
SALLY: Exactly, and in fact we know what shape to draw orbitals because people have determined mathematical functions for the wavefunction.
SIGRID: Where does the wavefunction come from?
SALLY: From a thing called the Schrödinger equation, which you can use to tell you the amount of energy an atom has. When you solve this equation, the solution involves n, just like in Bohr’s model. But it also involves a second quantum number, which we call l.
SIGRID: And that corresponds to some property of an electron, other than energy?
SALLY: Yes – the orbital angular momentum we talked about earlier.
SIGRID: OK, but how does that help?
SALLY: Well, when you solve the Schrödinger equation, which we won’t do here, it turns out that l cannot have just whatever value it likes…
SIGRID: …It is quantised, like n was…
SALLY: Exactly, and in particular units, it too can only have whole number values. But it is also constrained by the value of n. It turns out that the value of l for an electron in an atom can have values 0, 1, 2, etc up to n-1.
SIGRID: One less than n…
SALLY: Exactly.
SIGRID: But how can l be 0? That would mean it has 0 angular momentum and isn’t orbiting!
SALLY: I thought we had got away from thinking about electrons in terms of their orbits?
SIGRID: Oh, yeah. Sorry. Anyway, by those rules, for an n=1 electron, l would have to be 0.
SALLY: Keep going.
SIGRID: An n=2 electron could have l values of 0 or 1. An n=3 electron could have l values of 0, 1 or 2. And so on.
SALLY: Do you notice any similarities here with our discussion on orbitals?
SIGRID: I can see that n=1 has only one value of l available, and we know the n=1 shell has only a single s-orbital. So does l=0 correspond to the s-orbital?
To be continued…