The Zeroth Law and Temperature
Three bodies drift into equilibrium, and the law that sounds trivial turns out to be the only reason a thermometer works.
Skip to the animationTwo bodies each in thermal equilibrium with a third are in thermal equilibrium with each other — an experimental fact that neither of the other laws implies, and the sole justification for the existence of thermometers.
Equilibrium, before any instrument exists
Hotness is obvious to the touch and useless as a measurement. The only operational test available before instruments exist is to place two bodies in thermal contact and see whether anything changes.
Energy flows from the hotter to the colder until it stops. That final state is thermal equilibrium — not stillness at the molecular level, but the absence of any *net* transfer.
The law itself
- 1Bring A into contact with C and wait. They reach equilibrium.
- 2Separately, bring B into contact with C and wait. They also reach equilibrium.
- 3Now put A and B together — bodies that have never met.
- 4Nothing happens. No heat flows in either direction.
So "is in thermal equilibrium with" is a transitive relation. It looks trivial and it is not derivable: neither the first law nor the second implies it, which is exactly why it needed stating separately — and why it ended up numbered zero, after the others were already named.
Why this licenses a thermometer
A thermometer is the third body. It is made small so it barely disturbs what it measures, and given a readable property — a mercury column, a resistance, a voltage.
Touch it to A and read it; touch it to B and read it. If the readings agree, A and B are at the same temperature — without ever having met. That inference is invalid without transitivity, and comparing two bodies would otherwise require physically bringing them together.
Temperature is *defined* by this law rather than measured by it. It is the property that two bodies in thermal equilibrium share.
Thermometric properties, and the problem with them
| Property | Instrument | Note |
|---|---|---|
| Length of a liquid column | Liquid-in-glass | Cheap, limited range |
| Electrical resistance | Platinum resistance thermometer | The laboratory standard |
| Thermoelectric EMF | Thermocouple | Wide range, small, robust |
| Pressure at constant volume | Gas thermometer | The one that matters — see below |
| Radiated energy | Pyrometer | Non-contact, for very high temperatures |
Each defines its own scale, and they disagree between their fixed points. Calibrate mercury and platinum at 0 °C and 100 °C and they will differ at 50 °C. So the number depends on the substance you happened to use, which is intolerable for a physical law.
The absolute scale
As pressure is reduced, all gases converge on the same behaviour. Plot the pressure of a fixed volume against temperature and extrapolate to zero pressure: the line reaches zero at −273.15 °C for every gas.
That substance-independence is what makes it a genuine absolute zero rather than an artefact of one material. The kelvin scale starts there, and it is why the second law, the ideal gas equation and every radiation formula are written in kelvin rather than celsius.
The numbers you will be asked for
- Zeroth law
A ≡ C and B ≡ C ⟹ A ≡ B
Transitivity of thermal equilibrium.
- Ideal gas scale
T = 273.16 × (p / p_triple)
Extrapolated to zero pressure, so it is substance-independent.
- Kelvin and celsius
T(K) = t(°C) + 273.15
Same size of degree, different zero.
- Absolute zero
0 K = −273.15 °C
Where the extrapolated pressure line reaches zero.
Advantages and disadvantages
Advantages
- Makes temperature a well-defined property rather than a sensation.
- Justifies measuring two bodies with one instrument instead of comparing them directly.
- The gas scale is independent of the working substance.
- Gives absolute zero a physical meaning rather than a conventional one.
Disadvantages
- Cannot be derived — it must be taken as an experimental fact.
- Every practical thermometer disturbs the system it measures, if only slightly.
- Substance-based scales disagree between their fixed points.
- A gas thermometer is accurate and far too cumbersome for routine use.
Watch it work
Check yourself
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One question at a time. Pick an answer to see why it is right or wrong, then move on — there is no score to keep and nothing is saved.