The Zeroth Law and Thermodynamic Coefficients

What Makes a Thermometer Possible

The zeroth law looks trivial and is not: if A is in thermal equilibrium with C, and B is in thermal equilibrium with C, then A and B are in thermal equilibrium with each other. Thermal equilibrium is transitive.

That transitivity is exactly what licenses temperature. It means all mutually equilibrated systems share one common property, so a single number can label the equivalence class, and it means a thermometer works, since C can be a small instrument brought to each body in turn.

Empirical scales rely on some property that varies with heat: the length of a mercury column, a resistance, a gas pressure. These agree at their calibration points and drift between them, because different substances vary differently. The ideal gas scale is better behaved: at low density all gases converge on the same reading, giving $T$ from $\lim_{P\to 0} PV$. The thermodynamic scale, defined later from Carnot efficiencies, turns out to coincide with it while depending on no substance whatever.

A substance's response to changes is captured by three coefficients:

$$\alpha = \frac{1}{V}\left(\frac{\partial V}{\partial T}\right)_P, \quad \kappa_T = -\frac{1}{V}\left(\frac{\partial V}{\partial P}\right)_T, \quad \beta = \frac{1}{P}\left(\frac{\partial P}{\partial T}\right)_V$$

the thermal expansivity, the isothermal compressibility, and the pressure coefficient. The minus sign in $\kappa_T$ makes it positive, since raising the pressure always reduces the volume, a substance with negative $\kappa_T$ would be mechanically unstable and would collapse.

The three are not independent. Because any two of $P$, $V$, $T$ fix the third, the chain rule ties them:

$$\alpha = \kappa_T \beta P$$

Common pitfall: treating the zeroth law as an empty statement. Transitivity is a genuine empirical fact about nature, not a logical necessity, and without it "temperature" would not be a well-defined property at all, since two bodies matching a third could still disagree with each other.