Why Eight? Unpacking the Octet

As we approached the end of the school year, our chemistry department gave us the chance to research and present on a topic of our choice. I wanted something I was genuinely curious about and could blend with some physics, so I picked a question that had bothered me since GCSE: why do atoms obey the octet rule?

Every chemistry student learns that atoms react to end up with eight electrons in their outer shell. It is one of the most useful rules in the subject, and almost nobody explains it. Why eight? Why not six, or twelve, or five?

The answer turned out to be more involved than I expected, running through thermodynamics and quantum mechanics before it gets back to chemistry. Rather than narrowing the topic, I tried to build it up from simple analogies so my classmates could follow it. This is the write-up of that presentation.

What does “stable” actually mean?#

In general, something is stable if it tends to stay in the state it is in. Picture a ball in a valley: nudge it and it rolls back to the bottom. A ball balanced on top of a hill is unstable, because the smallest push sends it rolling down and converts its energy from one store to another. Stability, in other words, is about sitting at low potential energy.

At A-level, a stable species is defined as one that tends not to react, one that stays at the bottom of its valley. But at the fundamental level, a chemical reaction is a rearrangement of electrons: they are transferred or shared. That gives a more useful definition. A stable species is one that holds on to its own electrons and doesn’t take on anyone else’s.

To describe why some atoms seem “generous” with their electrons and others “greedy”, we need two properties:

• Ionisation energy: the energy needed to remove an electron from an atom.
• Electron affinity: the energy released when an atom gains an electron.

Reactive metals like sodium have low ionisation energies and give electrons away easily. Reactive non-metals like chlorine have high electron affinities and take them readily. A species that has neither tendency, that is hard to strip and gains nothing from an extra electron, is stable.

Why does anything favour stability?#

The energy balance#

A reaction is favourable when it releases more energy than it absorbs, leaving the products lower in potential energy than the reactants. That difference is the enthalpy change, $\Delta H$.

It is tempting to say this just means electron affinity has to beat ionisation energy, but the numbers show it is more subtle. Taking an electron from sodium costs $496\text{ kJ/mol}$, and giving it to chlorine only releases $349\text{ kJ/mol}$, so on its own the transfer loses energy. What pays for it is the attraction between the ions once they have formed: packing $\text{Na}^+$ and $\text{Cl}^-$ into a lattice releases about $787\text{ kJ/mol}$, many times more than the shortfall. The same logic applies to covalent bonds, where the energy comes from both nuclei attracting a shared pair.

Entropy: the real driving force#

But why should the universe favour losing energy at all? The answer is the Second Law of Thermodynamics: in any real process, the total entropy of the universe increases.

Entropy is best understood as a measure of how spread out energy is, which comes down to counting arrangements. There are thousands of ways for a room to be messy and only one way for it to be perfectly tidy, so a room left alone drifts towards mess. Energy behaves the same way. (I’ve written more about this in Why Must Time Flow Forward.)

When a reaction moves to a lower energy state, the energy it releases escapes as heat, and heat is the most disordered form energy can take. That heat raises the entropy of the surroundings by

so an exothermic reaction (negative $\Delta H$) increases the entropy of everything around it. The “drive for stability”, including the drive to reach an octet, is really the universe maximising its entropy by releasing and spreading out energy.

That explains why atoms settle into low energy arrangements. It doesn’t yet explain why the lowest energy arrangement so often involves exactly eight electrons. For that we need to look at orbitals.

Rethinking orbitals#

An orbital isn’t a little track the electron runs around. It is a wavefunction: a description of the electron as a wave, whose intensity at each point gives the probability of finding the electron there. A node is a place where that wave has zero amplitude, so the electron is never found there.

The best way I found to picture this is a standing wave on a string. The simplest standing wave has no nodes between the ends; the next one up has one node in the middle; the next has two. Each extra node squeezes the wave into a shorter wavelength, and a shorter wavelength means more energy. Electron waves follow the same pattern: more nodes means more energy.

This gives a simple rule. An orbital in energy level $n$ has exactly $n - 1$ nodes. Those nodes come in two kinds:

• A radial node is a sphere around the nucleus where the electron can’t be.
• An angular node is a flat plane through the nucleus where the electron can’t be.

Now count how many different orbitals each level allows.

Level 1 has zero nodes. There is only one way to have no nodes: a plain sphere, the 1s orbital.

Level 2 must have exactly one node, and there are only two ways to add it:

1. Make it a sphere. A sphere around the nucleus has only one possible orientation, which gives the single 2s orbital.
2. Make it a plane. A plane through the nucleus can face along the $x$, $y$ or $z$ axis, giving three separate orbitals: the three 2p orbitals.

That makes four orbitals in the second level, and nothing else is allowed.

Bringing it all together#

The last ingredient is the Pauli Exclusion Principle: no two electrons in an atom can have exactly the same set of quantum numbers. Electrons have only two spin states, “up” and “down”, so each orbital can hold at most two electrons, one of each spin.

Four orbitals, two electrons each:

That is where the eight comes from. It isn’t a coincidence or a rule of thumb; it is the capacity of the second energy level, set by the geometry of standing waves and the Pauli principle.

An atom with its 2s and 2p orbitals completely full, like neon, is exceptionally stable:

• Removing an electron takes a huge amount of energy. Neon’s first ionisation energy is $2081\text{ kJ/mol}$, compared with $496\text{ kJ/mol}$ for sodium, the very next element.
• Adding an electron gains nothing. The level is full, so an extra electron would have to go into level 3, which is higher in energy.

Other atoms react because gaining, losing or sharing electrons moves them towards that same filled arrangement, releasing energy and increasing the entropy of the universe as they do.

The same counting works for every level:

Where eight stops working#

Seeing where the number comes from also explains its exceptions, which is how you can tell an explanation is doing real work.

• Hydrogen and helium only have level 1, which holds two electrons. That is why hydrogen is happy with two (a “duet”), not eight.
• Period 3 can in principle hold 18, but the 3d orbitals sit higher in energy than 4s. After 3s and 3p fill there is still a big energy gap, so eight remains the magic number for elements like chlorine and argon.
• Transition metals do use their d orbitals: one s, three p and five d make nine orbitals, or 18 electrons. Chemists use exactly this as the 18-electron rule for transition metal complexes.
• “Expanded octets” like $\text{SF}_6$ and $\text{PCl}_5$ are often explained at A-level by using d orbitals, but most chemists now think the d orbitals contribute very little and the bonding is better described in other ways.
• Noble gases aren’t completely inert. In 1962 Neil Bartlett made the first noble gas compound, and xenon forms stable fluorides like $\text{XeF}_2$. A full shell makes reacting very hard, not impossible.

How it went#

I found the approach mostly successful. The analogies let me get the ideas across without leaning on equations, and afterwards I was able to clear up the points people had found confusing.

What I liked most about the project is that a rule I had been using for years without questioning turns out to rest on physics I only met much later: thermodynamics explains why atoms settle into low energy states, and quantum mechanics explains why those states so often hold exactly eight electrons.