Part A · What stuff is

1 · Atoms and the sticky force

If you could pass only one sentence about science to the next generation, Richard Feynman said, it should be this one:

All things are made of atoms: little particles that move around in perpetual motion, attracting each other when they are a little distance apart, but repelling upon being squeezed into one another. Feynman, The Feynman Lectures on Physics, Vol. I, ch. 1

That sentence makes four claims. Stuff is made of particles. The particles never stop moving. They pull on each other when they are close but not touching. They push hard if you squeeze them together. This chapter is about the last two claims. Almost everything else in chemistry is a long footnote to them.

How small, how many

An atom is about a tenth to a third of a nanometre across. One drop of water (0.05 mL) holds about 1.7 × 1021 water molecules. If you take the number of stars in the Milky Way as roughly 2 × 1011, that drop holds the star count of about ten billion Milky Ways.

Another way to feel the scale: blow the drop up to the size of the Earth. Each water molecule would then be about 80 cm across, the size of a beach ball.

There are 118 known kinds of atom, called elements. About ninety of them occur in nature. Everything you have ever touched is some arrangement of those few dozen kinds of particle.

The sticky force

Take two atoms and ask a simple question: how much energy does the pair have, as a function of the distance r between them? The answer, the potential energy curve, is the most important graph in chemistry. Drag atom B below and watch where it sits on the curve.

We hold atom B at rest, a little way out, with nothing else around: no air, no walls, no other atoms. Then we let go. What happens?

It swings back out to exactly where it started, forever. B falls into the dip and speeds up. It bounces off the steep wall on the left and climbs back out. Energy is conserved, so it reaches the same height it started from. The dashed line in the plot is B's total energy, and it never changes.

To stay stuck, B has to get rid of the energy it gained by falling in. In real life a third atom carries it off in a collision, or light carries it away. Now switch Surroundings on and release again. B settles, and the readout shows the energy it handed over.

Top: atom A is held still and atom B can move along the line. Bottom: the potential energy U(r) with B's position marked, plus B's total energy (dashed). "Shake" is the temperature of the surroundings. It does nothing while the surroundings are off.

For simplicity, B moves only along the line toward or away from A, and A never moves. Real pairs also tumble and rotate. That doesn't change the story about energy.

Read the curve from right to left:

The release experiment you just did shows the most misunderstood fact in chemistry. Falling into a well releases energy. Climbing out costs energy. When atoms stick, the energy they gave up goes to the surroundings as heat. To separate them, something has to pay that energy back. You will see this again in every chapter: a bond is a well. Making one releases heat. Breaking one costs energy, never the reverse.

Common trap

"Energy is stored in bonds and released when they break." About half of chemistry students still believe this after being taught otherwise. It's backwards. Breaking a bond always costs energy. When a fuel burns and gives off heat, that's because the new bonds that form (in CO2 and H2O) are deeper wells than the old ones. Chapter 9 does the bookkeeping.

Shake it

Now switch the surroundings on and slowly raise Shake. The surroundings kick B at random. With gentle shaking B rattles around the bottom of the dip. With harder shaking it occasionally gets a run of lucky kicks, climbs out, and wanders off: the bond breaks.

The deciding number is the ratio of the shaking energy to the depth of the well. Temperature times Boltzmann's constant, kT, is the typical size of a kick. When kT is much smaller than ε, things stay stuck; when kT is comparable to ε, they come apart. Chapter 7 makes this precise: the chance of being out of the well goes like e−ε/kT.

U(r) = 4ε [ (σ/r)12 − (σ/r)6 ]

This is the Lennard-Jones potential, the classic textbook model of two neutral atoms such as argon. The 1/r6 term is the attraction and the 1/r12 term the wall. Here σ sets the size of the atom and ε the depth of the well. All the simulations in this course use units where σ = 1 and ε = 1.

The 1/r6 attraction has a real physical origin (see below). The 1/r12 wall is just a convenient steep function. Real chemical bonds have wells of a different shape and are hundreds of times deeper. The overall shape, a pull at medium range, a wall up close and a dip in between, is universal.

Where does the pull come from?

It's all electrical. An argon atom is neutral overall: a positive nucleus inside a cloud of negative electrons. But the cloud fluctuates. For an instant it is a little lopsided, and that tiny lopsidedness nudges the neighbouring atom's cloud into lining up the same way. The two flickers attract. This is the London dispersion force, and it gives the 1/r6.

For argon the well is shallow, ε ≈ 0.01 eV. That's why argon only liquefies at −186 °C, once kT has fallen below the well depth. Chapter 7 explains why it has to fall below it, not just equal it. A real chemical bond, where atoms share electrons, is far deeper: the H–H bond is 4.5 eV. Same curve, different depth. That difference is the subject of chapter 5.

From two atoms to many

Here are 140 atoms with the same sticky force, all connected to surroundings at temperature T. Slide the shaking down slowly and watch.

The same Lennard-Jones atoms as above, in a box, in contact with surroundings at temperature T (shown in units of ε/k). "Not sticky" deletes the 1/r6 attraction and keeps only the wall.

Cool it and the atoms clump into droplets, and then into a neatly packed solid. Heat it and they fly apart into a gas. Switch off the stickiness and they never clump, at any temperature. Every liquid and solid in the world exists because of the sticky force. Without it, everything would be a gas.

Notice one more thing. "Solid", "liquid" and "gas" describe the crowd, not any one atom. A single argon atom isn't liquid. The same goes for colour, shininess, hardness and smell: they belong to many atoms together, never to one. That trips up many beginners, and we'll lean on it often.

Physics bridge

In 1929 Paul Dirac wrote that "the underlying physical laws necessary for the mathematical theory of a large part of physics and the whole of chemistry are thus completely known, and the difficulty is only that the exact application of these laws leads to equations much too complicated to be soluble."

He meant the Schrödinger equation for nuclei and electrons interacting only through Coulomb's law. That really is the whole rulebook. Everything in this course, from the dip you just played with to the octet rule and acids, is an approximation to that one Hamiltonian, chosen because the exact solution is out of reach. A useful way to read chemistry is as a collection of very good cheap approximations.

Dirac himself added that this holds "if one neglects relativity". For heavy atoms it doesn't: relativistic effects are why gold is yellow and mercury is a liquid. And the chemist Roald Hoffmann argues that many of chemistry's most productive ideas, the chemical bond included, can't usefully be reduced to physics, even if they're consistent with it.

Tell it to a 6-year-old
Look
Here's a glass of water and a little bottle of food colouring.
Wonder
"If we put one drop of colour in the glass, and then pour it into a big jug of water, will the colour disappear?"
Try
One drop in the glass. Stir. Pour into the jug. Then pour the jug into a bucket or the bath. Look closely each time.
True sentence
"Everything is made of tiny pieces, much too small to see. The pieces aren't in the water: they are the water. The colour is made of tiny pieces too. They spread out, so the colour gets paler, but they're all still there."

We never said the pieces are balls, and we didn't say what's between them: both would be claims to take back later. "Too small to see" is true for any ordinary microscope; special microscopes can image single atoms. Next time, add the sticky part: "the pieces pull on each other when they're close; that's why water stays together in a drop."

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