Temperature and thermal equilibrium

Two systems are in thermal contact when energy can pass between them without work being done — through a conducting (diathermal) wall rather than an insulating (adiabatic) one. Press a warm copper block against a cool one and their measurable properties change: lengths, resistances, pressures. After long enough they stop. That final, unchanging macroscopic state is thermal equilibrium, and it is a state, not a process. Nothing has stopped at the molecular level; molecules still collide across the boundary and energy still crosses it in both directions. What has vanished is the net flow. The whole point of temperature is that a single number, assigned to each body separately, decides in advance whether that net flow will be zero, and if it is not, which way it goes.

Suppose block A is in equilibrium with thermometer C, and block B is separately in equilibrium with that same C. Nothing so far logically forces A and B to be in equilibrium with each other — and yet every time it is tested, they are. That experimental fact is the zeroth law of thermodynamics, and it does real work. It says thermal equilibrium is transitive, so it sorts all systems into disjoint classes, each class holding everything that would sit unchanged beside everything else in it. One label per class is therefore enough, and that label is temperature. Without transitivity a thermometer would be useless: you could only ever learn about the object it was touching, never compare two objects by comparing two readings. Recognised last but numbered zeroth, the law is logically prior to the first and second laws, which is exactly how it got its name.

To attach numbers you need a thermometric property X: something measurable that changes smoothly and monotonically with hotness. A mercury column's length, a platinum wire's resistance, a gas pressure at fixed volume, a thermocouple emf and the spectral radiance of a hot surface all qualify. Monotonic matters — a property that rose and then fell would hand one reading two temperatures. Now declare the scale linear in X and pin it with two reproducible fixed points, the ice point at 0 °C and the steam point at 100 °C at 1 atm, so that t = 100 °C × (X − X₀)/(X₁₀₀ − X₀). A mercury column standing 24.0 mm at the ice point and 212.0 mm at the steam point reads 118.5 mm in a bath, giving t = 100 °C × (118.5 − 24.0)/(212.0 − 24.0) = 50.27 °C. Notice what just happened: the linearity was chosen, not discovered.

Because linearity was a choice, the scale belongs to the substance. Calibrate a platinum resistance thermometer and a mercury thermometer at the same two fixed points and they agree exactly there, by construction, and disagree everywhere in between. Platinum's resistance really follows R = R₀(1 + At + Bt²) with A = 3.9083 × 10⁻³ °C⁻¹ and B = −5.775 × 10⁻⁷ °C⁻², so a scale that pretends R is linear reads 50.37 °C when the true temperature is 50.00 °C. The discrepancy is only 0.37 °C, but it is systematic, it vanishes at both fixed points, and it is largest halfway between them. Adding a third fixed point patches this but does not cure it. What is needed is a thermometric property whose linearity is not a convention about one particular material.

The dilute gas supplies exactly that. Hold a fixed amount of gas at fixed volume and plot its pressure against Celsius temperature: the graph is a straight line. Take p₀ = 1.000 × 10⁵ Pa at the ice point, and the same sample reads 1.36610 × 10⁵ Pa at the steam point, a slope of 366.10 Pa per °C. Run the line backwards and it reaches p = 0 at t = −(1.000 × 10⁵ Pa)/(366.10 Pa °C⁻¹) = −273.15 °C. Repeat with helium, nitrogen or argon, at any starting pressure, and the intercept is unchanged: the lines fan out but share one zero. That common intercept is absolute zero, and putting the origin of the scale there gives T/K = t/°C + 273.15, on which p ∝ T outright. Absolute zero is where the extrapolation lands, not a state any gas survives to occupy — every real gas condenses long before it arrives.

A thermometer never reports the bath's temperature; it reports its own, and the zeroth law then transfers the number. Two practical consequences follow. First, the probe must be small in heat capacity compared with whatever it measures, or inserting it shifts the very temperature it was sent to find. Second, you must wait. Write the remaining gap as ΔT = T_probe − T_bath, and let t stand here for elapsed time rather than a Celsius temperature: the gap closes exponentially, ΔT = ΔT₀ e^(−t/τ), with a time constant τ set by the probe's heat capacity and by how well it couples to the bath. A probe with τ = 4.0 s starting 60 K below an 80.0 °C bath is still 60 × e⁽⁻²⁾ = 8.1 K low after 8.0 s, and only after about 20 s — five time constants — has the gap shrunk to 60 × e⁽⁻⁵⁾ = 0.40 K. A reading taken too early is an accurate measurement of the wrong thing.