Use the definition of temperature to prove the zeroth law of thermodynamics, which says that if system A is in thermal equilibrium with system B, and system B is in thermal equilibrium with system C, then system A is in thermal equilibrium with system C. (If this exercise seems totally pointless to you, you're in good company: Everyone considered this "law" to be completely obvious until 1931, when Ralph Fowler pointed out that it was an unstated assumption of classical thermodynamics.)

Temperature is a measure of hotness or coldness represented on one of several arbitrary scales that describes the natural flow of heat energy.

If two items are in thermal equilibrium, their temperatures are said to be the same. The entropy and energy of a system can be used to express temperature. The gauge of randomness is entropy. The Zeroth law states that if a system A and a system B are in thermal equilibrium with another system C individually, then systems A and B will be in thermal equilibrium as well.

Mathematically, temperature can be stated as:

$\frac{1}{T}=\left(\frac{\partial S}{\partial U}\right)$

By rearranging the terms,

$T=\left(\frac{\partial U}{\partial S}\right)$

The entropy vs energy graph will have equal slopes if the system is in thermal equilibrium. Because any two systems in thermal equilibrium have the same $\frac{\partial S}{\partial U}$values, system B and C must have the same slope as system A, resulting in their slopes being identical. The zeroth law, which asserts that a system A can be set in thermal equilibrium with any other systems that are all in thermal equilibrium with each other, is the foundation of thermodynamics.

As a result, if the slopes of all three systems A, B, and C are similar, the systems are said to be in thermal equilibrium, or at the same temperature.