Zeroth law of thermodynamics

The zeroth law of thermodynamics is the law which states that thermal equilibrium is transitive. 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 $A\sim B$ 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: $A\sim B\wedge B\sim C\Rightarrow A\sim C$

Temperature and the zeroth law

It is often claimed, for instance by Max Planck in his influential textbook on thermodynamics, that this law proves that we can define a temperature function, or more informally, that we can 'construct a thermometer'. Whether this is true is a subject in the philosophy of thermal and statistical physics. We will look at it in a formal way.

Definition: If we can assign to the state spaces of all thermodynamical systems $A,B,C,\ldots$ functions $\Theta _{A},\Theta _{B},\Theta _{C},\ldots$ such that $A\sim B\Leftrightarrow \Theta _{A}(s_{A})=\Theta _{B}(s_{B})$, where $s_{X}$ is the state of system $X$, we can define a temperature function.

Claim: The zeroth law of thermodynamics implies that we can define a temperature function.

It is easy to see that the zeroth law is a necessary condition for the existence of a temperature function. The '=' in $\Theta _{A}(s_{A})=\Theta _{B}(s_{B})$ is, of course, a transitive relation, so $\sim$ should be so as well. The '=' relation is not just transitive, it is an equivalence relation, which means that it is reflexive, symmetric and transitive. We would need two additional 'laws of thermodynamics' to express this:

Reflexivity: $\forall A:A\sim A$

Symmetry: $A\sim B\Rightarrow B\sim A$

The relation "is in equilibrium with" is symmetric by definition. It is trivial to extend the relation "is in equilibrium with" so that A~A.

The temperature so defined may indeed not look like the Celsius temperature scale, or even be continuous, but it is a temperature function.

In the space of thermodynamic parameters, zones of constant temperature will form a surface, which provides a natural order of nearby surfaces. It is then simple to construct a global temperature function that provides a continuous ordering of states. Note that the dimensionality of a surface of constant temperature is one less than the number of thermodynamic parameters (thus, for an ideal gas described with 3 thermodynamic parameter P, V and n, they are 2D surfaces).

For example, if two systems of ideal gas are in equilibrium, then P₁V₁/N₁ = P₂V₂/N₂ where P_i is the pressure in the i^th system, V_i is the volume, and N_i is the 'amount' (in moles, or simply number of atoms) of gas.

The surface $PV/N=const$ defines surfaces of equal temperature, and the obvious (but not only) way to label them is to define T so that $PV/N=RT$ where R is some constant. These systems can now be used as a thermometer to calibrate other systems.

The name

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. There is still some discussion about its status in relation to the other three laws.

References

• Jos Uffink, J. van Dis, S. Muijs; Grondslagen van de Thermische en Statistische Fysica; Utrecht University