Zeroth law of thermodynamics

Two systems are in thermal equilibrium when, placed in contact through a wall that allows heat to pass (a diathermic or conducting wall), no net heat flows between them. Practically, this means they have settled to the same temperature — nothing more is happening thermally, even though at the microscopic level molecules are still colliding and exchanging energy in both directions; it is only the net flow that has stopped. An adiabatic wall, by contrast, blocks heat flow altogether (an idealised insulator), so two systems separated by one can sit at different temperatures indefinitely without ever equilibrating. Thermal equilibrium is only meaningful across a diathermic boundary, or when a boundary is briefly opened to let two systems interact and then settle.

Zeroth law of thermodynamics: if a system A is separately found to be in thermal equilibrium with a system C, and a system B is also separately found to be in thermal equilibrium with C, then A and B are in thermal equilibrium with each other — even if A and B are never placed in direct contact.

where: $\leftrightharpoons$ denotes "is in thermal equilibrium with," and A, B, C are any three thermodynamic systems (solids, liquids, gases, or a measuring instrument such as a thermometer). Mathematically, thermal equilibrium behaves like an equivalence relation — reflexive (a system is trivially in equilibrium with itself), symmetric (A in equilibrium with B implies B in equilibrium with A), and, thanks to the zeroth law, transitive (A equilibrium C and B equilibrium C together force A equilibrium B). Transitivity is not automatic from the definition of equilibrium alone — it had to be stated as a separate physical law, verified by experiment, because "no net heat flow between two bodies in contact" says nothing by itself about a relationship between two bodies that were never in contact. That transitivity is exactly what makes temperature possible as a single number attached to a system. If A, B, C, D, ... all separately equilibrate with the same reference system (a thermometer), the zeroth law guarantees they are all mutually in equilibrium too, and it is consistent to assign them all the same temperature value. Temperature is thus a well-defined state property — a single number describing a system in thermal equilibrium, independent of the path by which that state was reached. The same idea of a state property returns later with internal energy in the first law.

The zeroth law is not a mathematical necessity — it is an experimental fact about how real bodies exchange heat, and understanding why it needed stating (rather than assumed) is the heart of this topic. Consider what happens without assuming the zeroth law. A thermometer C touched to block A settles at a fixed reading — A and C are in thermal equilibrium. The same thermometer touched to a completely different block B settles at the same reading — B and C are also in equilibrium. But nothing in this procedure ever brought A and B into contact. Logically, one could imagine a universe where A and B, despite matching the same thermometer separately, would still exchange net heat if actually placed together — perhaps "temperature" was not one consistent number but depended on which pair of bodies you compared. The zeroth law is the experimental statement that this pathological possibility never occurs: matching a common reference is enough to guarantee mutual equilibrium. This is precisely why a thermometer works as a scientific instrument: a convenient system C, calibrated against fixed points, is brought into contact with different objects A, B, ... one at a time, instead of every pair of objects needing to be tested directly against each other. Without the zeroth law, "the thermometer reads the same for A and B" would say nothing about A and B's relationship to each other — only their separate relationships to the thermometer. With it, equal readings on a common thermometer become a rigorous, transitive proof of equal temperature. This is also why the law was identified only after the first and second laws: those already used "temperature" freely, and it took later scrutiny to notice the logical foundation for that usage had never been stated as its own law — hence the honest but awkward name "zeroth."

A metal block A is placed in contact with a mercury thermometer C, and after some time the reading settles at $45^\circ\text{C}$ with no further change. The same thermometer is then placed in contact with a wooden block B, and its reading again settles at $45^\circ\text{C}$. A third block, D, is separately tested and settles the thermometer at $50^\circ\text{C}$. What can be concluded about A, B and D, and why?

Step 1. Identify the equilibrium relations established. The thermometer C, after each steady reading, is in thermal equilibrium with the object it touched: $A \leftrightharpoons C$ at $45^\circ\text{C}$, $B \leftrightharpoons C$ at $45^\circ\text{C}$, and $D \leftrightharpoons C$ at $50^\circ\text{C}$ (three separate contacts, C never touching two blocks at once).

Step 2. Apply the zeroth law to A and B. Since $A \leftrightharpoons C$ and $B \leftrightharpoons C$ at the same reading, the zeroth law guarantees $A \leftrightharpoons B$: A and B are in thermal equilibrium with each other and share the same temperature, $45^\circ\text{C}$, even though they were never brought into direct contact.

Step 3. Check D against A and B. D equilibrates the thermometer at a different reading, $50^\circ\text{C}$, so D is not at the same temperature as A or B. The zeroth law only forces equilibrium between systems that match the same reference reading — it does not force D into equilibrium with A or B, and in fact if D were placed in contact with A or B, net heat would flow from D (hotter) into A or B (cooler) until a new common equilibrium temperature was reached.

Final answer: A and B are at the same temperature, $45^\circ\text{C}$, and are in thermal equilibrium with each other by the zeroth law; D, at $50^\circ\text{C}$, is not in equilibrium with either A or B, and would exchange heat with them if brought into contact.

Competitive stretch

If four calorimeters P, Q, R, S each equilibrate a reference thermometer at the same reading, the zeroth law lets you conclude, in one step, that every pair among them is mutually in thermal equilibrium — six pairwise conclusions from four measurements, not six direct contacts. This is transitivity's practical power: the comparisons needed across $n$ objects drop from $\binom{n}{2}$ pairwise tests to just $n$ tests against one thermometer.

• Treating the zeroth law as being about how much heat is transferred or how fast — it is purely about the logical basis for comparing temperatures, saying nothing about heat quantity or rate.
• Dismissing the law as "too obvious to matter" — it is intuitive, but it is precisely what turns thermometry from an assumption into a scientifically justified procedure; exam answers should state why it is needed, not just that it "seems true."
• Confusing thermal equilibrium (equal temperature, no net heat flow) with zero total heat energy or zero internal energy — two systems in thermal equilibrium can hold very different amounts of internal energy; only their temperatures match.
• Mixing up the zeroth law with the first law — the zeroth law establishes temperature as a state property; the first law is a separate statement about conservation of energy involving heat, work and internal energy.

Ask the tutor: if the "equilibrium with a third system" idea still feels abstract, ask the AI tutor to walk through more everyday A-B-C equilibrium scenarios or extra reasoning practice.