= Zeroth law of thermodynamics
{wiki}
If two <thermodynamic systems> are each in <thermal equilibrium> with the same third system in the same equilibrium state, they are in <thermal equilibrium> with each other. Write $A\sim B$ for this relation. The <zeroth law of thermodynamics> supplies <transitivity>:
$$
A\sim C\quad\text{and}\quad B\sim C\quad\Longrightarrow\quad A\sim B.
$$
Its <symmetric relation> property and self-equilibrium (<reflexivity>), together with <transitivity>, make <thermal equilibrium> an <equivalence relation>. Its <equivalence classes> can be labelled by a common <empirical temperature>; choosing an absolute numerical scale requires the further structure of <thermodynamic temperature>.<Statistical Physics by David Tong/4 Classical Thermodynamics>[§4.1, printed pp. 109–110: thermal contact, transitivity and temperature labels]{ref}
The comparison assumes internally equilibrated macroscopic bodies, unchanged composition and mechanical constraints, and contact permitting only <heat> exchange. Use rigid, impermeable <diathermal walls> so that neither matter transfer nor boundary <work> spoils the test. The same state of $C$ matters: letting $C$ warm while measuring $A$, then cool while measuring $B$, does not supply the premise. Ordinary nonrelativistic laboratory <thermodynamics> is assumed; the <Tolman temperature law> treats gravitational qualifications. Equal <temperatures> establish the thermal condition, without requiring <mechanical equilibrium> or <chemical equilibrium>.
\Image[/zeroth-law-thermal-contact-comparison.png]
{title=<Thermal equilibrium> is transitive; an <adiabatic wall> supplies no equality test}
{id=zeroth-law-thermal-contact-comparison}
{description=The top row compares $A$ and $B$ separately with the same equilibrium state of reference $C$. The lower left shows the predicted absence of net <heat transfer> when $A$ and $B$ are put in <thermal contact>. The lower right is a counterexample to inferring equality from insulation: a rigid <adiabatic wall> can separate bodies at different <temperatures>. All numerical labels are illustrative.}
This explains why a <thermometer> can compare unlike materials: it supplies a reusable third system. A <temperature> is a property of an equilibrium state, while <heat> is <energy> transferred because of a <temperature> difference. Two bodies at the same <temperature> can have different <internal energies> and <heat capacities>. A <thermometer> therefore does not measure a body's stored amount of <heat>; the finite-body disturbance and waiting-time calculations in <thermometer> describe how making the comparison can change the state being measured.<Statistical Physics by David Tong/4 Classical Thermodynamics>[§4.2, printed p. 111: heat as energy transfer, rather than a stored state quantity]{ref}
The name “zeroth” reflects logical order rather than the date of the first <thermometer>: this premise about <thermal equilibrium> precedes the already numbered <first law of thermodynamics>, <Second law of thermodynamics> and <Third law of thermodynamics>. This is the naming explanation given in <University Physics Volume 2 by Samuel J. Ling and others> (§1.1, “Thermal Equilibrium”).<University Physics Volume 2 by Samuel J. Ling and others>[§1.1, paragraph introducing the zeroth law: retrospective numbering; the textbook credits the name to Ralph Fowler in the 1930s]{ref}
The corresponding <Zeroth law of black-hole mechanics> states constancy of <surface gravity> over an equilibrium horizon; <Hawking radiation> identifies that quantity with physical temperature up to its universal conversion factor.
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