Hydrogen orbitals: where the electron is found

Imagine a hydrogen atom, one proton with one electron, prepared in its lowest-energy state, the state it settles into when left alone. We look for the electron once and mark where it was found with a dot. Then we take another hydrogen atom prepared exactly the same way, look again, and add another dot; and so on. Press “Look once” a few times: each press adds one dot. A single dot tells us almost nothing, and the dots do not follow one another along any path.

Now press “Look 1000 times” and keep adding. A pattern appears: the dots crowd thickly near the nucleus and thin out further away, equally in every direction, so the pattern is a ball. Where the dots are dense, the electron is often found; where they are sparse, it is rarely found. This is not a picture of many electrons: each dot represents a separate position measurement on a hydrogen atom prepared in the same state. These are simulated samples from the model, not a recording of one electron moving through these dots in order. This pattern of where the electron is found is called an orbital. This one, the smallest, is named 1s: “1” for the first shell, and “s” for its round shape.

Notice the word: an orbital is not an orbit. An orbit is a path, like a planet’s; an orbital is a cloud of places, and the electron has no definite path between one look and the next.

How far from the nucleus is the electron usually found? Cut the cloud into thin layers, like the skins of an onion, each 0.1 Bohr radius thick, and count the dots in each layer. The chart beside the picture does exactly that with the dots you have collected. The counts rise, reach a peak, and fall away.

Why is there a peak, when the dots are densest right at the nucleus? Because a layer close to the nucleus is a tiny ball, with very little room in it, while a layer further out is a large sphere with much more room. The number of dots in a layer is how crowded the dots are there times how much room the layer has. Close in, the crowding is high but the room is small; far out, the room is large but the crowding has fallen. The product is largest in between, at 52.9 pm from the nucleus, 52.9 trillionths of a metre.

That distance is not new. It is exactly the radius of the first circle on the Bohr page, called the Bohr radius. Switch on “Bohr’s circle” to compare. So Bohr’s circle survives, not as the path of the electron but as the distance at which it is most likely to be found.

Does the cloud have an edge? No: the further out you go, the rarer the dots become, but they never stop completely. To draw an orbital as a shape, we choose a surface that holds most of the dots, usually 90 %. For 1s that surface is a sphere of radius 141 pm (switch on “90 % boundary”). It is a convenient outline, not a wall: over many identically prepared trials, about 10% of detections are expected outside it. A particular batch of ten need not contain exactly one outside result.

A hydrogen atom given more energy can be found in bigger patterns. These belong to the second shell, the third shell and so on, the shells of the Bohr page. Choose them in the picture and collect dots for each.

The second shell has two kinds of orbital. 2s is round like 1s but bigger, and has an empty layer inside it, at 2 Bohr radii, where the electron is never found; look at the dots from the side to see the gap. The three 2p orbitals have a new shape, two lobes on opposite sides of the nucleus, like a dumbbell, with no dots at all in the flat plane between the lobes. They are the same shape pointing three ways: along x, along y and along z.

The third shell has three kinds: one 3s, three 3p, and five 3d orbitals, most of which have four lobes. The letters s, p, d (and f, which first appears in the fourth shell) name the shape families. They come from old descriptions of the lines in the light given out by atoms, and today they are just names.

Count them shell by shell. The theory of the atom, confirmed by its light, says that shell $n$ has $n$ kinds of orbital, and that the kinds come in groups of 1 ($s$), 3 ($p$), 5 ($d$) and 7 ($f$). So shell 1 has 1 orbital, shell 2 has $1 + 3 = 4$, shell 3 has $1 + 3 + 5 = 9$, and shell 4 has $1 + 3 + 5 + 7 = 16$. These are $1^2$, $2^2$, $3^2$, $4^2$: adding up the first $n$ odd numbers always gives $n^2$. The picture shows why: a square of $n$ by $n$ dots grows into a square of $(n + 1)$ by $(n + 1)$ by adding an L-shaped border of $n + n + 1 = 2n + 1$ dots, which is the next odd number.

Atoms other than hydrogen have more than one electron, and each orbital can hold at most two of them. This is a rule found from experiment, and the two electrons sharing an orbital must differ in a property called spin, which comes in exactly two kinds, usually drawn as an up arrow and a down arrow. So the most electrons a shell can hold is two for every orbital: $2 \times 1 = 2$ in shell 1, $2 \times 4 = 8$ in shell 2, $2 \times 9 = 18$ in shell 3, $2 \times 16 = 32$ in shell 4. These are the numbers from the Bohr page, now explained.

Electrons fill the places of lowest energy first. In an atom with many electrons, the orbitals of one shell no longer all have the same energy: $s$ is lowest, then $p$, then $d$. That is why chemists write where the electrons are as a list such as $1s^2 2s^2 2p^6 3s^1$ for sodium. Read it as: the number is the shell, the letter is the kind of orbital, and the small raised number is how many electrons are in that kind. Adding the raised numbers, $2 + 2 + 6 + 1 = 11$, gives sodium’s 11 electrons, and grouping them by shell gives 2, 8, 1, as on the Bohr page.

One surprise explains the 2, 8, 8 of potassium and calcium. After argon, $1s^2 2s^2 2p^6 3s^2 3p^6$ (2, 8, 8), the third shell still has room for ten more electrons in its 3d orbitals. But in these atoms the 4s orbital has slightly lower energy than 3d, so the 19th electron, potassium’s, goes into 4s instead: 2, 8, 8, 1. Only after 4s is full do the 3d orbitals start to fill, in the transition metals. Why 4s comes below 3d needs more than this page can show, and the order changes again in some ions, so treat 1s, 2s, 2p, 3s, 3p, 4s, 3d as a guide for neutral atoms, not a law.