Zeroth law of thermodynamics

The Zeroth law of thermodynamics states that if two bodies are individually in thermal equilibrium with a separate, third body, the first two are also in thermal equilibrium with each other, or in other words, thermal equilibrium is transitive. It thus provides the technical definition of a thermometer and verifies that it works.

It is used to answer questions about thermodynamic equilibrium, heat and energy transfer between systems, and temperature.

Thermal Equilibrium is a state in which neither matter nor energy has a net flow within a system or between systems. A single system can be in this state with itself or with other systems.

For example, a glass of ice water that has just been poured is not in thermodynamic equilibrium with itself. The ice and water are at different temperatures, so there is a net flow of energy from the warmer water to the cooler ice. After several minutes, once the ice has melted fully, the glass of chilled water is in thermodynamic equilibrium with itself: the (now melted) ice has warmed up and the water has cooled down.

On the other hand, the glass of chilled water and the room it is sitting in are two separate thermodynamic systems. They are connected because energy is allowed to flow between them. They are not in thermodynamic equilibrium because they are at different temperatures. Energy will flow from the room to the water, raising the temperature of the water a lot and decreasing the temperature of the room a little until the two are equal. The systems are now in thermodynamic equilibrium.

A beaker of hot water and a beaker of cold water are placed on a table in a room. After a very long time, which beaker is at a higher temperature?

Solution: “After a very long time” implies that the beakers are allowed to reach thermal equilibrium with the room. Since they are both in thermal equilibrium with the room, they must be in thermal equilibrium with one another. (This is an illustration of the equivalence relation property.) They are the same temperature.

One precise statement of the Zeroth Law is that the relation of thermal equilibrium is an equivalence relation on pairs of thermodynamic systems. In other words, the set of all systems each in its own state of internal thermodynamic equilibrium may be divided into subsets in which every system belongs to one and only one subset, and is in thermal equilibrium with every other member of that subset, and is not in thermal equilibrium with a member of any other subset.

This means that a unique "tag" can be assigned to every system, and if the "tags" of two systems are the same, they are in thermal equilibrium with each other, and if different, they are not. This property is used to justify the use of empirical temperature as a tagging system. Empirical temperature provides further relations of thermally equilibrated systems, such as order and continuity with regard to "hotness" or "coldness," but these are not implied by the standard statement of the zeroth law.