Explain and write the Zeroth Law of Thermodynamics.

(N/A) To determine whether a system is in thermal equilibrium with its surroundings, a third system (body) can be used.
In figure $(a)$, systems $A$ and $B$ are separated by an adiabatic wall, and both are in contact with a third system $C$ separated by a conducting (diathermic) wall. This entire assembly is enclosed by an insulating wall.
The states of the systems (macroscopic variables) do not change until systems $A$ and $B$ attain thermal equilibrium with $C$.
After this, suppose the adiabatic wall between $A$ and $B$ is replaced by a conducting wall, while $C$ is insulated from $A$ and $B$ by an adiabatic wall, as shown in figure $(b)$. It is observed that the states of systems $A$ and $B$ do not change.
This implies that they are in thermal equilibrium with each other. This is the basis of the Zeroth Law of Thermodynamics. Hence, this law is stated as follows:
"Systems that are each in thermal equilibrium with a third system are in thermal equilibrium with each other." This is the Zeroth Law of Thermodynamics.
$R. H. Fowler$ formulated the Zeroth Law of Thermodynamics in $1931$, long after the first and second laws of thermodynamics were stated and numbered.
The Zeroth Law clearly suggests that when two systems $A$ and $B$ are in thermal equilibrium, there must be a physical quantity that has the same value for both.
This variable, whose value is equal for two systems in thermal equilibrium, is called temperature $(T)$.
Thus, if $A$ and $B$ are separately in equilibrium with $C$, then $T_{A} = T_{C}$ and $T_{B} = T_{C}$. This implies that $T_{A} = T_{B}$. This means that systems $A$ and $B$ are also in thermal equilibrium.
Thus, the following conclusion can be drawn from this law: "there exists an important physical quantity called temperature."