Zeroth law of thermodynamics

The zeroth law formalizes the idea that “being in thermal equilibrium” is a consistent notion across many systems, and it provides the conceptual foundation for defining temperature .

Definition (transitivity of thermal equilibrium). If system $A$ is in thermal equilibrium with system $B$, and $B$ is in thermal equilibrium with system $C$, then $A$ is in thermal equilibrium with $C$.

This is often summarized as: thermal equilibrium is transitive.

Physical interpretation. The zeroth law justifies thermometry: if a thermometer (system $B$) reaches thermal equilibrium with a system of interest ($A$), and the same thermometer reading is obtained when it equilibrates with another system ($C$), then $A$ and $C$ are mutually in thermal equilibrium. Operationally, this means a single scalar label—temperature—can characterize the “thermal state” shared by systems in mutual thermal equilibrium.

Equivalence-class viewpoint. Define a relation “$\sim$” on equilibrium states by declaring $A\sim B$ if they are in thermal equilibrium. The zeroth law asserts that, on equilibrium states, this relation behaves like an equivalence relation (in particular, it is transitive). The corresponding classes are sometimes emphasized explicitly as zeroth-law equivalence classes .

Consequences in thermodynamics.

• Temperature as a state variable. The zeroth law supports the existence of a state variable $T$ that is equal for systems in mutual thermal equilibrium, fitting into the general framework of thermodynamic state descriptions.
• Thermal contact and walls. Whether two systems can approach thermal equilibrium depends on the boundary : a diathermal wall permits energy exchange as heat, while an adiabatic wall blocks it.
• Reservoir idealization. A thermal reservoir is an ideal system whose temperature is effectively unchanged by exchanges, making the zeroth-law notion of “same $T$” especially operational.

To turn “equal temperature” into a specific numerical scale requires additional structure; see absolute temperature scale .