# Materials Science and Engineering/List of Topics/Thermodynamics/Zeroth Law of Thermodynamics

The zeroth law of thermodynamics is a generalized statement about bodies in contact at thermal equilibrium and is the basis for the concept of temperature. The most common enunciation of the zeroth law of thermodynamics is:

"If two thermodynamic systems are in thermal equilibrium with a third, they are also in thermal equilibrium with each other."

In other words, the zeroth law says that if considered a mathematical binary relation, thermal equilibrium is transitive.

## 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.

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). The temperature so defined may indeed not look like the Celsius temperature scale, but it is a temperature function.

For example, if two systems of ideal gas are in equilibrium, then ${\displaystyle P1V1/N1=P2V2/N2}$  where ${\displaystyle P_{i}}$  is the pressure in the ith system, ${\displaystyle V_{i}}$  is the volume, and ${\displaystyle N_{i}}$  is the 'amount' (in moles, or simply number of atoms) of gas.

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