Math 2.0

Classical thermodynamics, heat transfer Laws of thermodynamics Fact

Zeroth law of thermodynamics

Builds on Thermodynamic systemThermodynamic equilibriumEquivalence relationTemperature

Statement

Let AA, BB and CC be thermodynamic systems, each in an equilibrium state. If AA is in thermal equilibrium with CC, and BB is in thermal equilibrium with CC, then AA is in thermal equilibrium with BB.

Remarks

The law is a postulate drawn from experience, not a theorem. Together with the convention that each system is in thermal equilibrium with itself, it makes thermal equilibrium an equivalence relation on equilibrium states. Its classes can then be labelled by real numbers, and such a labelling is an empirical temperature. A thermometer is the system CC: if it shows the same reading in AA and in BB, then AA and BB placed in contact exchange no net heat. Without the law, a thermometer reading would not decide whether two bodies are in balance with each other.

Going deeper

For a student

In the axiomatic treatment of thermodynamics, the zeroth law is the reason that temperature exists as a state function. Together with reflexivity, it makes the relation "in thermal equilibrium with" symmetric and transitive, so the equilibrium states split into classes of equal temperature. Choosing a real-valued label for these classes gives an empirical temperature; the second law later selects the thermodynamic temperature among them.

Reading path

To understand this page from the ground up, read these 5 pages in order. The foundations come first.

Show the path as a graph
Equivalence relation, concept, no evidence levelEquivalence relationconceptTemperature, concept, no evidence levelTemperatureconceptThermodynamic equilibrium, concept, no evidence levelThermodynamic equilib…conceptThermodynamic system, concept, no evidence levelThermodynamic systemconceptZeroth law of thermodynamics, statement, no evidence levelZeroth law of thermod…statement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Equivalence relation concept
  2. Temperature concept
  3. Thermodynamic equilibrium concept
  4. Thermodynamic system concept
  5. Zeroth law of thermodynamics statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Zeroth_law_of_thermodynamics
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Kindstatement, fact
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MSC80A05, 80A10
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AboutThermodynamic equilibrium, Temperature
UsesThermodynamic system, Thermodynamic equilibrium, Equivalence relation, Temperature
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Changed3 November 2026

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