The term zeroth law was coined by Ralph H. Fowler. In many ways, the law is more fundamental than any of the others. However, the need to state it explicitly as a law was not perceived until the first third of the 20th century, long after the first three laws were already widely in use and named as such, hence the zero numbering.A system in thermal equilibrium is a system whose properties (like pressure, temperature, volume, etc.) are not changing in time. A hot cup of coffee in your kitchen is not at equilibrium with its surroundings because it is cooling off and decreasing in temperature. Once its temperature stops decreasing, it will be at room temperature, and it will be in thermal equilibrium with its surroundings.

Two systems are said to be in thermal equilibrium when 1) both of the systems are in a state of equilibrium, and 2) they remain so when they are brought into contact, where ‘contact’ is meant to imply the possibility of exchanging heat, but not work or particles. And more generally, two systems can be in thermal equilibrium without thermal contact if one can be certain that if they were thermally connected, their properties would not change in time.

Thus, thermal equilibrium is a relation between thermodynamical systems. Mathematically, the zeroth law expresses that this relation is an equivalence relation. (Technically, we would need to also include the condition that a system is in thermal equilibrium with itself.)

Mathematically, the zeroth law expresses the transitivity of the equilibrium relationship. Two other trivial properties of the equilibrium relationship are often included in the statement of the third law:
• reflexivity - a system is in thermal equilibrium with itself.
• symmetry - if system A is in thermal equilibrium with system B, then system B is in thermal equilibrium with system A.

These are included because, together with the transitivity statement, the equilibrium relationship between two systems becomes what mathematicians call an equivalence relation which allows a number of mathematical theorems to be easily proven. Also, we will not need to make statements like “if A and B are in equilibrium with C then they are in equilibrium with each other” which does not follow from transitivity alone, but does follow from transitivity and symmetry.

Description
Two systems are said to be in thermal equilibrium when 1) both of the systems are in a state of equilibrium, and 2) they remain so when they are brought into contact, where ‘contact’ is meant to imply the possibility of exchanging heat, but not work or particles. Thus, thermal equilibrium is a relation between thermodynamical systems. In the following we will write A ~ B for ‘A is in thermal equilibrium with B’.

The zeroth law states that this relation is transitive, which means that whenever system A is in thermal equilibrium with B, and B is in thermal equilibrium with system C, then A and C are also in thermal equilibrium. Formally:
The zeroth law: A ~ B and B ~ C => A ~ C