Internal Energy

Rub your hands together hard and they warm up. Wrap them round a mug of tea and they warm up again — this time the warmth flows the other way, out of the tea and into your fingers. Where does that warmth actually live while it waits? Not in some invisible fluid sloshing about. It lives in the tiny particles the tea and your skin are made of — in their ceaseless, restless jiggling.

Every material, however cold and still it looks, is quietly buzzing on the inside. The total energy tied up in all that hidden particle activity has a name: the material's internal energy. It is the energy stored inside the stuff, and getting a clear picture of it is the key to understanding heating, cooling, and why boiling water refuses to get any hotter no matter how fierce the flame.

Two kinds of energy hiding inside

Zoom right in on the particles of any substance and you find them holding energy in two different ways at once:

Kinetic energy — the energy of their random motion. Particles are never truly still: they vibrate, tumble and dash about. The faster they move on average, the more kinetic energy they carry, and — as we'll see — the hotter the substance is.
Potential energy — the energy stored in the bonds and spacing between particles. Particles pull on their neighbours like tiny springs. Pulling them further apart (or snapping a bond entirely) stores energy, exactly the way stretching a spring or lifting a weight does.

Add the two together, for every single particle in the material, and you have the internal energy:

U = E_{k,\text{total}} + E_{p,\text{total}}

This is why the thermal (internal) store is filled just by warming something up — heating hands energy to the particles, and it lands in one or both of these two pockets.

The internal energy of a substance is the total of all the energy stored by its particles:

the kinetic energy of the particles' random motion, plus
the potential energy stored in the bonds and spacing between them.

The particle model: solid, liquid, gas

You already met the three states when you studied changing state. The particle model tells us how the particles move in each — and that motion is exactly the kinetic half of internal energy:

In a solid, particles are locked in a fixed pattern by strong bonds. They cannot travel, but they still vibrate on the spot. Low kinetic energy, tightly bound.
In a liquid, the bonds have loosened. Particles stay close but can slide and tumble past one another. More kinetic energy, weaker bonds.
In a gas, the bonds are broken. Particles zoom about freely, far apart, bouncing off the walls. High kinetic energy, almost no bonds left.

Notice that both pockets change as you go up the ladder: the particles speed up (more kinetic energy) and the bonds get pulled apart (more potential energy). A gas therefore carries far more internal energy than the same mass of cold solid.

Temperature: the average, not the total

Here is the idea students most often blur. Temperature is a measure of the average kinetic energy of the particles — how fast they are jiggling on average. It says nothing about how many particles there are, and nothing about the bond (potential) energy at all.

Internal energy, by contrast, is the grand total — every particle's kinetic energy and every bond's potential energy, all added up. So temperature is a kind of average-per-particle; internal energy is the whole sum.

That single distinction explains a surprise: a small object can be scorching hot yet hold very little internal energy, while a huge object can be merely warm yet hold an enormous amount — simply because it has vastly more particles to add up.

A spark flying off a grinding wheel can be over 1000 °C — its few particles are jiggling ferociously, so its temperature is huge. Yet it lands on your arm and does almost nothing: there are so few particles that the total energy it carries is tiny.

A warm bath sits at a gentle 40 °C — each particle jiggling far more calmly, so a much lower temperature. But a bath holds unimaginably many particles, so its internal energy dwarfs the spark's. High temperature does not mean high internal energy: you have to count the particles too. This is exactly why a tiny spark won't warm a cold room, but a lukewarm radiator full of water will.

Heating does one of two things

Heat a substance and you pour energy into its internal store. That extra energy has to go somewhere — and it does exactly one of two jobs:

It raises the temperature. The energy goes into the particles' kinetic pocket: they move faster, so the substance gets hotter. No change of state — just warmer.
It changes the state. At a melting or boiling point, the energy instead goes into the potential pocket — breaking the bonds between particles. The particles do not speed up, so the temperature does not rise at all while the state change is under way.

Follow it through with a block of ice. Start heating and the temperature climbs steadily as the frozen particles vibrate faster — energy into kinetic. Reach 0\,^\circ\text{C} and the climb stops dead: for a while the thermometer reads a flat 0\,^\circ\text{C} even though the flame is still roaring, because now every joule is prising bonds apart to turn ice into water — energy into potential. Only once the last crystal has melted does the temperature start rising again. The internal energy climbed the whole time; the temperature paused.

This is why the energy that goes into a state change is sometimes called hidden or latent — it swells the internal energy without ever showing up on the thermometer.

See it happen

Drag the energy slider to pour heat into the box of particles. Watch three things at once: the speed arrows on the particles (their kinetic energy), the Internal energy bar, and the Temperature bar. As you cross the melting and boiling marks, keep your eye on the two bars: internal energy keeps climbing the whole way, but the temperature bar stalls flat during each change of state, because there the energy is going into breaking bonds, not speeding particles up.

If temperature is the average jiggling of particles, then cooling something means slowing that jiggle down. So what happens if you keep cooling, and cooling, and cooling? The particles move less and less — until they reach the least motion nature allows and simply cannot slow any further. That floor is absolute zero: about -273\,^\circ\text{C}, or 0 on the kelvin scale that scientists use to count temperature from the very bottom.

You can never quite reach it — the closer you creep, the harder the last sliver becomes — but laboratories have chilled atoms to less than a billionth of a degree above it, colder than anywhere in deep space. There is no such thing as "more cold" below absolute zero, because there is no motion left to remove. It is the one temperature the whole Universe has a hard limit on.

Three slippery traps that catch nearly everyone:

Temperature, heat and internal energy are three different things. Internal energy is the total energy stored in a material's particles. Temperature is only the average kinetic energy per particle. Heat is energy on the move — flowing from a hotter object to a cooler one. Don't use the three words as if they all mean the same thing.
During melting or boiling the temperature stays constant. It feels wrong — you're still heating hard — but the thermometer holds dead level, because every joule is breaking bonds (raising potential energy) instead of speeding particles up (raising temperature). The internal energy is rising the whole time even though the temperature isn't.
Hotter does not mean more internal energy. A tiny spark at 1000\,^\circ\text{C} holds far less internal energy than a cool lake, because internal energy is a total over all the particles — and the lake has astronomically more of them. Temperature is an average; internal energy is a sum.