Why Chemistry Runs on Eight

Picture a diner window at night. A salt shaker sits on the table, and above it, in the glass, a neon sign buzzes: OPEN.

Two ordinary things. Each one is a small puzzle, and it's the same puzzle.

Start with the salt. Salt is two elements stuck together, sodium and chlorine. Sodium is a soft silver metal that fizzes, spits and sometimes explodes when it touches water. Chlorine is a yellow-green gas that was used as a weapon in World War I. Neither one can leave other atoms alone. Put them together and you get the stuff you sprinkle on fries.

Now the sign. Neon is a gas. Run electricity through it and it glows. The rest of the time it just sits there, and it doesn't react with the glass, the air or the wires. Neon doesn't need anything from anybody.

School gave you the reason, or half of it: atoms settle down when they have eight electrons in their outer shell. Sodium has one electron to spare and chlorine is one short, so they trade. Neon already has eight. That's the octet rule, and it's one of the most useful facts in chemistry.

But school hands it over as a rule and moves on. Why eight? Not six, not ten. And why does carbon, the atom that life is built on, make exactly four bonds?

The short answer

Eight is a count of seats.

In the first few rows of the periodic table, a full outer shell is four seats: one round seat, and one seat for each of the three directions in space (that's what "three dimensions" means). Each seat holds two electrons. Four seats, two electrons each: eight.

So the eight comes from the three. Here's the part I like best. Part 1 of The Universe Writes in Pen has an eight in it too: reality's record counts time in cycles of eight ticks, and a moment of consciousness is one unbroken cycle. That eight is built from the same three directions: 2 × 2 × 2. The shell's eight is 2 × (3 + 1). Those are two different calculations, and they give the same number for exactly one number of directions: three. Ours.

Here's the whole answer in six steps. The rest of this post takes them one at a time and shows how we know each one.

1. The eight in your kitchen. Atoms near the top of the periodic table settle with eight electrons in their outer shell, and they trade or share electrons to get there.
2. Counting the seats. The outer shell has one round seat plus one seat for each direction in space, and two electrons fit in a seat: 2 × (3 + 1) = 8.
3. The other eight. The tick cycle from Part 1 has one tick for every pattern of signs on the three directions: 2 × 2 × 2 = 8.
4. The crossing. In a space with D directions, the shell holds 2(D + 1) electrons and the tick cycle has 2^D ticks. They're equal at D = 3 and at no other number of directions.
5. Four pairs, four directions. Eight electrons make four pairs, and four pairs point four ways. That's carbon's four bonds, the tetrahedron of methane and the four neighbors of every water molecule in ice. Oxygen's eight protons have nothing to do with it.
6. The bigger shells. The same seat counting gives shells of 2, 8, 18 and 32 electrons.

Why this matters: chemistry is atoms trading and sharing outer electrons, so the number of seats is the first rule it plays by. Every molecule in your body was built by atoms filling these seats, and the number of seats comes from the three directions of space. You also get a test for telling a real link between two numbers from a lookalike. Step 5 runs the test on a lookalike that has the same eight.

Here's the route. I'll stop after each step to zoom out, so you always know where we are, what we've got, and why the next step comes next. The Ledger, Part 1's drawing of reality's permanent record, keeps the map. It's a thick book with reading glasses and exactly one response to anything that happens: Noted.

Step 1: The eight in your kitchen

Atoms near the top of the periodic table settle down when their outermost shell holds eight electrons. They get there by trading electrons or by sharing them. Here's how that works for the salt and the sign.

An atom is a nucleus with electrons around it, and the electrons live in layers called shells. Count the electrons shell by shell, from the inside out:

Look at the outermost number in each row.

Neon has 8 in its outer shell. It's full. It holds those eight tightly and has no room for a ninth, so it takes nothing and gives nothing. That's the sign in the window.

Sodium has one electron alone in its outer shell, sitting on a full layer of 8. Lose that one electron and the layer of 8 is on the outside.

Chlorine has 7. It's one short of 8.

So the two make a deal. Sodium hands its spare electron to chlorine, and now both show eight on the outside. Sodium has lost a negative electron, so it's positively charged. Chlorine has gained one, so it's negative. Opposite charges attract, so the two stick. Billions upon billions of them stack into a crystal, every sodium surrounded by chlorines and every chlorine by sodiums, and that crystal is the salt on the table.

(The Ledger noted the trade. The Ledger notes everything. Neon was in the window with its sunglasses on, not involved.)

Atoms can also share. Two atoms put one electron each into a shared pair, and the pair counts toward both atoms' totals. Water and methane are made that way, and we'll build them in Step 5.

When chemists say an atom "wants" eight, they mean the atoms together end up with less energy once each one shows eight. Sodium on its own has to be paid to give up its electron. Chlorine taking that electron, and the opposite charges pulling together in the crystal, pay back more than that. A full shell is a settled arrangement and a gap isn't. This post is about the number: why eight?

Zoom out. Step 1 is done. Atoms near the top of the periodic table settle with eight electrons in the outer shell, and they trade or share electrons to get there. That's why salt forms and why neon sits in its tube. Next, and why: we have the rule, and the rule has a number in it. To find where the eight comes from, we count the seats inside a shell.

Step 2: Counting the seats

A full outer shell holds eight electrons because it has four seats and each seat holds two. The seats are one round seat and one seat for each direction in space.

Here's how we know. An electron in an atom isn't a tiny ball on a track. It spreads out into a cloud, and each cloud has a shape. Chemists call each shape an orbital. I'll call it a seat, because the part of its behavior that matters here is the seating rule: an orbital holds at most two electrons, and the two have to spin opposite ways, one up and one down. There's no standing room, and a third electron has to find another seat. The seat is a drawing of that rule. The clouds are the real thing.

Two shapes make up the seats that an atom like carbon, oxygen or chlorine fills in its outer shell:

• The round seat (chemists call it s). A ball around the nucleus. It points nowhere, so there's exactly one.
• The dumbbells (p). Two lobes on opposite sides of the nucleus. A dumbbell can point left and right, up and down, or forward and back: one dumbbell for each direction in space. In our space, that's three.

One round seat plus three dumbbells makes four seats. Two electrons in each: eight.

Look at where the three comes from. It isn't a number about atoms. It's the number of directions in space, and the dumbbells are counted by it, one for each direction.

So the count depends on how many directions there are. Here's the same count in other spaces:

With D directions that's 1 + D seats and 2(D + 1) electrons. We live at D = 3, and that's where the eight comes from: 2 × (3 + 1).

The very first shell is the small one: only the round seat, so it fills at two electrons. Helium has exactly those two. A hydrogen atom in a molecule gets to two by sharing a pair.

Zoom out. Step 2 is done. The outer shell fills four seats, one round and one for each of the three directions, with two electrons in each seat. That's eight, and the eight is built from the three. Next, and why: Part 1 has an eight in it too, and it's also built from the three. Before we compare the two, we need to see how the second one is built.

Step 3: The other eight

Part 1's eight is built differently. Quick recap of Part 1: reality is a permanent record of every event, and one simple process writes it, one step at a time. Each step is a tick. The record counts time in cycles of eight ticks, and a moment of consciousness is one unbroken cycle. The cycle has eight ticks because it has one tick for every pattern of signs on the three directions.

Here's what that means. Each direction has two signs: plus or minus, like left or right, up or down, forward or back. Pick a sign for each of the three directions and you've picked one corner of a cube. Two choices, times two, times two:

2 × 2 × 2 = 8

A cube has eight corners, and the cycle has one tick for each. Each tick moves along one edge of the cube, which flips one sign, and the cycle visits every corner before it comes back to the start.¹ Nobody skips a corner. (The Ledger would notice.)

With D directions it's D twos multiplied together, 2^D. Two directions give a cycle of 4 ticks. Four directions give 16.

Now look at how the two eights are built. The seats were built by adding: one round seat, plus one dumbbell for each direction, then two electrons in each. The ticks are built by multiplying: a doubling for each direction. Adding and multiplying grow at different speeds. The question is where they meet.

Zoom out. Step 3 is done. The tick cycle has 2^D ticks, one for each sign pattern on D directions, and 2³ = 8. The shell holds 2(D + 1) electrons, and 2 × (3 + 1) = 8. Two counts, built from the same directions in different ways. Next, and why: the two counts agree at three directions. Do they agree anywhere else? That decides whether the match belongs to our space or to the number eight itself.

Step 4: The crossing

The two counts agree when space has three directions, and in no other case. Put them side by side for every number of directions:

Run your eye down the last two columns like a race. With one and two directions the shell is ahead. At three they tie. From four on the tick cycle is ahead, and its lead grows with every direction you add.

Here's why it can't happen twice. Start at three, where both are 8, and add one direction. The tick cycle doubles: 8 becomes 16, a gain of 8. The shell gains one dumbbell: 8 becomes 10, a gain of 2. The ticks are ahead now, and every direction after that doubles them again while the shell adds just two electrons. The gap keeps growing: 6, then 20, then 50.

Go down from three and it runs the other way. Losing a direction halves the tick cycle, from 8 to 4, a loss of 4. It removes one dumbbell, so the shell goes from 8 to 6, a loss of 2. The shell stays ahead: 6 against 4, then 4 against 2. The two meet at three and nowhere else. You can check it on a napkin. The diner has plenty.

A second way to count. The table gave every seat two electrons in every space. But the two comes from spin, and spin depends on the number of directions too: it comes in more kinds as directions are added, and the number of kinds doubles every two directions. Let each seat hold that many electrons and the shell holds 6, 8, 20, 24, 56 and 64 electrons for 2 through 7 directions, against tick cycles of 4, 8, 16, 32, 64 and 128. The numbers change and the match doesn't move: 8 and 8 at three directions, and nowhere else. (This count starts at two directions. With one direction or none, nothing can turn, so spin has no meaning there.)

Think about what just happened. One count is about electrons sitting around a nucleus. The other is about how many steps the process that writes reality's record takes to come back to where it started. Neither one mentions the other. Both are built from the number of directions space has, and they agree only when that number is three. Space has three.

"Yeah, but eight is a small number," says Yeah-But, the voice in the back of your head from Part 1. "Things land on small numbers all the time. Spiders have eight legs."

They do, and nobody thinks atoms run on spiders. Step 5 has a better lookalike, and a way to tell it from the real thing.

Zoom out. Step 4 is done. The tick cycle and the outer shell are both counts built from the directions of space, and they're equal in exactly one case: three directions, eight and eight. Next, and why: the seats don't only set a number. They set the shapes of molecules. Eight electrons are four pairs, and four pairs push each other into four directions.

Step 5: Four pairs, four directions

Eight electrons around an atom make four pairs, and four pairs point four ways. That one sentence gives you carbon's four bonds, the tetrahedron of methane, the bent shape of water and the four neighbors of every water molecule in ice. Here's how.

Four pairs push apart. Each seat holds a pair of electrons, so eight electrons make four pairs. Electrons repel each other, and so do pairs. Four pairs around one atom push apart until they're as far from each other as space allows. Tie four balloons together by their necks and you'll see where they end up: at the corners of a tetrahedron, a pyramid with a triangle for every face (the four-sided die from tabletop games). A birthday party turns into a chemistry lesson if you tie the balloons right. The balloons are a drawing of the pairs, which push each other apart the same way. Any two of the four directions are about 109.5 degrees apart.

Four ways isn't four dimensions. The four corners of a tetrahedron fit in ordinary three-dimensional space, the way the three corners of a triangle fit on a flat page.

Carbon. Carbon has 4 electrons in its outer shell. Eight minus four is four, so carbon takes four more, and four hydrogens each bring one. Each carbon-hydrogen pair is shared. It counts toward carbon's eight and toward hydrogen's two (hydrogen's shell is the small first one, full at two). Carbon now has four pairs around it, pointing at the four corners of a tetrahedron, with a hydrogen at each corner. That's methane, CH₄. And that's why carbon makes four bonds: it has four outer electrons, and the shell wants eight.

The same subtraction works along the row. Nitrogen has 5 outer electrons, so it makes 3 bonds. Oxygen has 6, so it makes 2. Fluorine has 7, so it makes 1. Hydrogen needs 2 and has 1, so it makes 1 too. Hydrogen 1, oxygen 2, nitrogen 3, carbon 4: counting seats gives you the bond counts of the four atoms that proteins, DNA and sugars are mostly made of.

Water. Oxygen has 6 outer electrons. Two hydrogens bring 2 more: 6 + 2 = 8. That's four pairs again, but only two of them are shared with hydrogens. The other two belong to the oxygen alone, and chemists call them lone pairs. All four pairs push apart like the balloons, so a water molecule has a hydrogen at two of the four corners and nothing but a pair at the other two. That's why water is bent instead of straight. The two lone pairs push a little harder than the shared ones, so the hydrogens sit 104.5 degrees apart, a little closer than the balloons' 109.5.

Ice. Freeze water and each molecule ends up with four neighbors, one in each direction its pairs point. Each of its two hydrogens points at a neighbor's oxygen, and each of its two lone pairs takes in a neighbor's hydrogen.

The decoy. Yeah-But is back, arms crossed. "But oxygen has 8 protons!"

It does. Eight divided by two is four, the number of directions around oxygen in water. Tempting. Is that the reason?

Test it. The rule is "directions = protons ÷ 2". Carbon has 6 protons, so the rule gives 3. But carbon points four ways in methane. Now sulfur. It sits right under oxygen in the periodic table and builds a molecule like water: hydrogen sulfide, H₂S, with two hydrogens, a bent shape like water's and four pairs around the sulfur. (Its bend is tighter: the hydrogens sit about 92 degrees apart.) Sulfur has 16 protons, so the rule says 16 ÷ 2 = 8 directions, twice what water has. Wrong again.

The pair rule (eight electrons, four pairs, four directions) gives four for carbon, oxygen and sulfur alike.

Oxygen has 8 protons because oxygen is the eighth element. That's the whole story of that eight. The protons decide which atom you're holding, and with it how many outer electrons the atom brings to the deal. How many fit around it, eight, comes from the seats.

Here's how to tell a link from a lookalike. Ask where an eight comes from, then change its source and see whether it moves. The seat count and the tick count both come from the number of directions in space, and both change by a formula when that number changes. The table in Step 4 is that experiment. The proton count comes from the nucleus. It has nothing to do with directions, and it equals eight at oxygen because oxygen is the eighth element. That's a lookalike. Change the atom and the seat count still gives four pairs, while the proton rule breaks.

Zoom out. Step 5 is done. Eight electrons are four pairs, and four pairs point four ways: carbon's four bonds, methane's tetrahedron, the bend in water, the four neighbors in ice. Oxygen's eight protons are a lookalike. Next, and why: shells bigger than eight exist, and you've seen their effects in the wide middle of the periodic table. The same seat counting gives them.

Step 6: The bigger shells

The seat counting doesn't stop at eight. The round seat and the dumbbells are only the first two shapes.

After them comes a shape with 5 seats, which chemists call d, and then one with 7 seats, called f. In our three directions the counts go 1, 3, 5, 7: each shape has two more seats than the one before. The first shell has room for only the round seat. The second shell adds the dumbbells, the third adds the 5-seat shape, and the fourth adds the 7-seat shape.

Shell number n has n × n seats and holds 2 × n × n electrons: 2, 8, 18, 32.

Now look at a periodic table. The same counts are written across it. The block on the far left is 2 columns wide, the block on the right is 6 wide, the block in the middle is 10 wide and the block hanging underneath is 14 wide: two electrons for each of 1, 3, 5 and 7 seats. The rows end with the noble gases, helium, neon, argon, krypton, xenon and radon, at 2, 10, 18, 36, 54 and 86 electrons. From neon on down, each one has eight in its outermost shell.

This post counts seats. It doesn't derive the order the seats fill in. That order is read off the measured periodic table, and it's why the table's third row stops at 8 instead of 18: the fourth shell's round seat fills before the third shell's 5-seat shape does.

Zoom out. Step 6 is done. Shells hold 2, 8, 18 and 32 electrons, which is two electrons for each of 1, 3, 5 and 7 seats, shape by shape. The wide blocks of the periodic table are the same counting. Next, and why: nothing is left but to say the whole trip in one breath.

The whole trip, in one breath

Atoms near the top of the periodic table settle with eight electrons in their outer shell, and they trade or share electrons to get there. Eight is four seats of two: one round seat and one seat for each of the three directions of space, 2 × (3 + 1). The tick cycle from Part 1 is eight because of the same three directions, 2 × 2 × 2, and the two counts come to the same number for exactly one number of directions: three. Eight electrons are four pairs, and four pairs point four ways, which is carbon's four bonds, methane's tetrahedron and the four neighbors of every water molecule in ice. Oxygen's eight protons are a lookalike. The same three directions that fix the number of ticks in every moment of your consciousness fix how many electrons fill an outer shell.

Look at the salt shaker once more. Every sodium and chlorine in it shows eight electrons on the outside, counted out by left and right, up and down, forward and back.

(The neon atom would like you to know it's full, thanks. The Ledger noted that.)