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For two length-$L$ slabs with zero exterior temperatures and perfect thermal contact, put ${\alpha }_i=\sqrt{{\rho }_i{c}_i/k_i}$, $u=\omega {\alpha }_{1}L$, and $v=\omega {\alpha }_{2}L$. A separated solution with time factor $e^{-{\omega }^{2}t}$ and $\omega \ne 0$ exists precisely when the displayed interface determinant vanishes. Dividing by $\cos u\cos v$ gives a tangent characteristic equation only away from its poles. Common zeros of both cosines are legitimate normal modes and must be retained. The zero-frequency case gives only the trivial solution of a differential equation with these positive material parameters and exterior boundary conditions.

Conservation of thermal energy and Fourier's law give the heat equationsThe density times the specific heat capacity, ${\rho }_i{c}_i$, is the volumetric heat capacity, and $k_i$ is the thermal conductivity.

The condition $T_1(-L,t)=0$ holds the left exterior face at a fixed zero reference temperature; $T_2(L,t)=0$ does the same for the right exterior face. The condition $T_1(0_{-},t)=T_2(0_{+},t)$ expresses perfect thermal contact, with no temperature jump at the interface. Finally,expresses continuity of the signed heat flux in the positive $x$ direction: the interface neither stores nor produces heat.

The common boundary between two conducting materials in a thermal interface problem. Perfect thermal contact imposes continuity of temperature and of normal heat flux.

A heat equation problem in adjacent materials with separate thermal conductivities and volumetric heat capacities. Perfect thermal contact imposes continuous temperature and continuous normal heat flux at the interface. Multiplying each material's heat equation by its temperature cancels the interface boundary terms in a weighted energy estimate.

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: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.^[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.

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.^[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”).^[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.

Put a contact thermometer $C$ in thermal contact with $A$, wait for thermal equilibrium, and record response $X_A$. Repeat with $B$ under the same operating constraints. If $X_A=X_B$, and the calibrated response is an injective function of temperature on this range, $C$ occupies the same thermal reference state in both comparisons. The zeroth law of thermodynamics then predicts $A\sim B$. Equality of raw numbers from different uncalibrated thermometers supplies no such premise.^[ref]

Use a probe whose participating heat capacity is small relative to each sample, or use maintained heat baths. Otherwise the matching readings describe the disturbed final states. Also hold the probe's immersion and mechanical conditions fixed: its response can depend on more than temperature. Matching readouts only tests agreement within resolution and repeatability, rather than proving exact equality of temperatures.

Agreement within a tolerance is not itself an equivalence relation. With a $0.10{}^{\circ }\mathrm{C}$ tolerance, readings $25.00$ and $25.16{}^{\circ }\mathrm{C}$ each agree with $25.08{}^{\circ }\mathrm{C}$, yet disagree with each other. This is a limitation of the measurement rule, not a failure of the exact transitivity asserted by the zeroth law of thermodynamics.