Grab the metal leg of a chair and then the wooden seat beside it. The metal feels colder — yet both have been sitting in the same room all day, so they are at the same temperature. Your hand is a terrible thermometer: it doesn't read temperature, it reads how fast heat flows out of your skin, and metal drains heat faster than wood. To do physics we need something better than a feeling. We need to say precisely what temperature is, why a thermometer works at all, and how to build a scale everyone can agree on. Remarkably, the whole edifice rests on one almost embarrassingly obvious statement — so obvious it was named the zeroth law, because it had to sit underneath the first, second and third laws that were already famous by the time anyone noticed it needed stating.

This is the foundation stone of thermal physics. Everything else in this module — heat, work, engines, entropy — assumes we already know what it means for two things to be "at the same temperature." That is exactly what the zeroth law pins down.

Thermal equilibrium: the end of all change

Drop a hot spoon into a mug of cold water. At first everything changes: the spoon cools, the water warms, you could feel the difference second by second. But wait long enough and the changes stop. The spoon and the water settle at one common, unchanging temperature and stay there forever (until something else disturbs them). Two objects that have stopped exchanging any net heat, so that none of their large-scale properties drift any more, are said to be in thermal equilibrium.

Equilibrium is not stillness at the molecular level — molecules are still jostling and swapping energy furiously across the contact. It is a balance: energy crosses in both directions at equal rates, so nothing changes on average. "Being at the same temperature" and "being in thermal equilibrium" turn out to be two names for the same condition — and that identity is what we are about to make rigorous.

The zeroth law, and why a thermometer is even possible

Here is the statement, in full:

It sounds like nothing. But watch what it buys us. Let C be a thermometer — a little tube of mercury, say. Touch it to object A; the mercury settles to some length. Touch the same thermometer to object B; it settles to the same length. The zeroth law now guarantees that A and B would be in equilibrium if you put them in contact — without ever putting them together. The number the thermometer shows is a genuine, shared property. That number is what we call temperature: the thing that is equal for any two bodies in thermal equilibrium. Reveal the figure to follow the go-between argument.

Temperature is what a thermometer measures

That last line is worth taking literally. A thermometer is any device with a property that changes smoothly and repeatably with hotness — call it a thermometric property:

• the length of a column of mercury or alcohol (it expands when heated);
• the electrical resistance of a platinum wire (it rises with temperature);
• the voltage across a thermocouple junction;
• the pressure of a fixed volume of gas — the one we will build a scale from.

Each of these gives a working temperature scale: pick two reference points, mark them, divide the gap into equal steps. The old Celsius scale did exactly this with water — 0\,^\circ\text{C} at the ice point, 100\,^\circ\text{C} at the steam point, a hundred equal divisions between. Fahrenheit did the same with different pegs. These scales are useful, but they are conventional: they depend on a particular substance and a particular pair of reference temperatures. Two different mercury-and-glass thermometers agree at the fixed points but can disagree slightly in between. Is there a temperature scale that doesn't depend on which substance you pick? There is — and gases point the way.

The ideal-gas temperature scale and absolute zero

Seal a fixed amount of gas in a rigid container and measure its pressure as you change its temperature. You find a beautifully straight line: pressure rises steadily with temperature. Now do the same with a different gas, or a different amount — you get a different line, but a straight one. And here is the magic: every one of those lines, extended backwards, hits zero pressure at the same temperature, near -273.15\,^\circ\text{C}. Drag through the chart and watch the two gas samples plunge to a common point.

That common intercept can't be an accident of one particular gas — all gases agree on it. It marks the coldest temperature there can be: absolute zero, the point where the pressure of an ideal gas would vanish because its molecules would have no thermal jostling left to push with. It is a natural, substance-independent zero — so let's put the origin of our scale there.

Worked examples

Example 1 — a simple conversion. A comfortable room is at 21\,^\circ\text{C}. In kelvin that is

Example 2 — doubling the absolute temperature. A rigid canister of gas sits at 27\,^\circ\text{C} with a pressure of 100\ \text{kPa}. To what temperature must it be heated to double the pressure to 200\ \text{kPa}? First go to kelvin: 27\,^\circ\text{C} = 300\ \text{K}. Since P \propto T at fixed volume, doubling P means doubling T:

Note the trap this avoids: doubling the Celsius reading (to 54\,^\circ\text{C}) would barely change the pressure. Proportionality lives on the absolute scale, never on Celsius.

Example 3 — the zeroth law in the kitchen. A cook checks a roast with a thermometer, reads 75\,^\circ\text{C}, then checks the oven air and reads 75\,^\circ\text{C} too. Without ever comparing roast and air directly, the zeroth law tells the cook they are in thermal equilibrium — the roast has stopped heating up, it is done as it will ever get at this setting.