Let us consider a system requiring a pair of independent co-ordinates X and Y for their complete description. If the values of X and Y remain unchanged so long as the external factors like temperature also remains the same, then the system is said to be in a state of thermal equilibrium.
Two systems A and B having their thermodynamic co-ordinates X and Y and X1 and Y1 respectively separated from each other, for example, by a wall, will have new and common co-ordinates X? and Y? spontaneously, if the wall is removed. Now the two systems are said to be in thermal equilibrium with each other.
Zeroth law of thermodynamics
If two systems A and B are separately in thermal equilibrium with a third system C, then the three systems are in thermal equilibrium with each other. Zeroth law of thermodynamics states that two systems which are individually in thermal equilibrium with a third one, are also in thermal equilibrium with each other.
This Zeroth law was stated by Flower much later than both first and second laws of thermodynamics.
This law helps us to define temperature in a more rigorous manner.
If we have a number of gaseous systems, whose different states are represented by their volumes and pressures V1, V2, V3 ... and P1, P2, P3... etc., in thermal equilibrium with one another, we will have φ1 (P1,V1) = φ2 (P2, V2) = φ3 (P3, V3) and so on, where φ is a function of P and V. Hence, despite their different parameters of P and V, the numerical value of the these functions or the temperature of these systems is same.
Temperature may be defined as the particular property which determines whether a system is in thermal equilibrium or not with its neighbouring system when they are brought into contact.