Temperature, Equilibrium & Thermometry

Two bodies are in thermal contact when heat can pass between them without either doing work on the other — a copper wall, not a piston. Bring them together and their measurable properties change, then stop changing: that final state is thermal equilibrium, and reaching it can take a second for a thermocouple bead or an hour for a stirred bath. The zeroth law adds transitivity: if A and B are each in equilibrium with C, they are in equilibrium with each other. That is a physical claim, not a triviality — nothing in mechanics forces it — and it is what makes a thermometer possible. C can be a small probe carried from A to B; if its reading is unchanged, A and B need never touch. It also makes equilibrium an equivalence relation, so the states of any system split into classes and each class can be labelled with one number. Everything else in thermometry is the choice of that labelling.

A thermometric property is anything that changes monotonically with hotness and can be read: the length of a mercury column, the resistance of a platinum wire, the emf of a thermocouple, the pressure of a gas held at fixed volume. Choosing one and decreeing how its value maps to temperature defines an empirical scale. The historical Celsius recipe uses two fixed points and a straight line: t = 100 °C × (X − X₀)/(X₁₀₀ − X₀). A Pt100 element with R₀ = 100.00 Ω and R₁₀₀ = 138.51 Ω that reads 111.67 Ω is assigned t = 100 °C × 11.67/38.51 = 30.30 °C. Notice what has and has not been done. The reading is exactly right at the two fixed points because it was made so; between them it is a definition, and a mercury thermometer built by the same recipe will not agree with it, because mercury length is not a linear function of platinum resistance. Each empirical scale is only as linear as its working substance.

One property behaves better than the rest. Hold a gas at constant volume — a bulb joined to a manometer whose mercury is brought back to a fiducial mark after every change — and record its pressure p in the bath and pₜₚ at the triple point of water. The ratio p/pₜₚ depends slightly on which gas and how much of it was used, but reduce the filling and every gas converges on one value. The reason is that the correction to p ∝ T is proportional to the gas density, so a plot of p/pₜₚ against pₜₚ is a straight line whose intercept at zero filling is gas-independent. Defining T = 273.16 K × lim(p/pₜₚ) therefore erases the working substance. For the steam point the intercept is 1.365954, giving 373.124 K, or 99.974 °C: once 273.16 K was fixed at the triple point in 1954, the boiling point of water became a measured quantity, and it is no longer exactly 100 °C.

Plot the pressure of a dilute gas at fixed volume against Celsius temperature and the points fall on a line, p = p₀(1 + t/273.15 °C), where p₀ is the ice-point pressure. Change the filling and the slope changes with p₀, but the intercept does not: every line reaches p = 0 at t = −273.15 °C. No gas gets there — it condenses first — so the intercept is a limit of dilute-gas behaviour, not a state anything reaches. Moving the origin to that limit gives T/K = t/°C + 273.15 and turns the line into a proportionality, p ∝ T, the form the kinetic model will later explain. Two nearby numbers do different jobs. 273.15 K is the ice point, used for conversion; 273.16 K is the triple point, chosen as the calibration state because one pressure and temperature fix it, reproducible to 0.1 mK, whereas the ice point drifts with air pressure and dissolved gas. Since 2019 the kelvin is defined by fixing k = 1.380649 × 10⁻²³ J K⁻¹ exactly, so the triple point is now a measured 273.16 K, and the gas thermometer is one primary way to realise the unit rather than its definition.

How far is a two-point scale wrong between its fixed points? Take the platinum relation R = R₀(1 + At + Bt²) with A = 3.9083 × 10⁻³ °C⁻¹ and B = −5.775 × 10⁻⁷ °C⁻². The linear scale reads tₗᵢₙ = 100 °C × (R − R₀)/(R₁₀₀ − R₀) = (At + Bt²)/(A + 100 °C × B), so the error is δ = tₗᵢₙ − t = B t (t − 100 °C)/(A + 100 °C × B). It vanishes at both fixed points, as it must, and dδ/dt = B (2t − 100 °C)/(A + 100 °C × B) = 0 puts the extreme at t = 50 °C, where δ = −2500 °C² × B/(A + 100 °C × B) = +0.37 °C. A Pt100 read linearly therefore shows 50.37 °C in a bath the gas scale calls 50.00 °C. The cure is not a purer wire but a better interpolation: add a third fixed point and fit the quadratic. That is how practical scales such as ITS-90 work — many fixed points, a named interpolating instrument, and a specified formula between them.

A property can be sensitive and still fail as a thermometer. A thermocouple's emf against a 0 °C reference junction is close to E = αt + βt² over a wide range; for α = 14.0 μV °C⁻¹ and β = −0.020 μV °C⁻² the emf peaks at the neutral temperature tₙ = −α/(2β) = 350 °C, and E(100 °C) = E(600 °C) = 1200 μV. One reading, two temperatures: past its neutral point this couple is useless, because its property is no longer monotonic. The second demand is thermal. The probe reaches the bath's temperature by exchanging heat with it, so its heat capacity must be small next to the bath's or the act of measuring moves the answer — a 2 g bead dropped into 5 g of liquid reports the mixture, not the liquid. Sensitivity, range, monotonicity and size are the checklist, and no single instrument passes it everywhere, which is why a laboratory keeps several.