# Thermal Equilibrium and Zeroth Law of Thermodynamics

Thermal equilibrium is the preamble to Zeroth law of thermodynamics. In mechanics, we refer to equilibrium with the conservation of linear momentum or angular momentum depending upon the type of motion. That means Net external force and torque on a system are zero.

But in thermodynamics, if we say that the state of a system is in equilibrium state then, the macroscopic variables that characterize the system do not change in time.

Example
A gas inside a closed rigid container completely insulated from its surroundings, with fixed values of pressure, volume, temperature mass and composition that do not change with time, is in a state of thermodynamic equilibrium.

It is observed that a higher temperature object which is in contact with a lower temperature object will spontaneously transfer heat to the lower temperature object.

In general, whether or not a system is in a state of equilibrium depends on the surroundings and the nature of the wall that separates the system from the surroundings.

Now it is better to introduce some important terms which are related to Zeroth law of thermodynamics.
It is an insulated wall from which neither energy nor matter can enter or leave from the container which contains gas.
1. Diathermic Wall (In Greek Diathermic means "heat passing through")
It is a wall that permits the flow of energy in the form of heat.

From our common experimental experience we can list out the following points:

If two systems, separately in equilibrium, are characterized by the same value of the parameter, then the systems will remain in equilibrium when brought into thermal contact with each other.

If the two systems are characterized by different values of the parameter then they will not remain in equilibrium when brought into thermal contact with each other.

Now we will start to discuss about the Zeroth law of thermodynamics and how it is related to thermal equilibrium and temperature.

Let us consider the two systems A and B and both the systems are isolated by adiabatic wall. That means neither energy nor matter can enter or leave either system. In this case both the systems cannot communicate through this wall since it is an adiabatic wall.
Now we replace the adiabatic wall with diathermic wall, which permits the flow of energy in the form of heat and finally it will reach thermal equilibrium.

Now instead of two systems, consider three systems A, B and C. The systems A and B are separated by an adiabatic wall, while each is in contact with a third system C, through a conducting wall.

The states of the systems which is characterized by certain macroscopic variables, will change until both A and B come into thermal equilibrium with C.

After this stage the adiabatic wall between the systems A and B is replaced by a conducting wall and the system C is insulated from A and B by an adiabatic wall.

Observation: We found that the states of A and B remains as in the previous case. That means both the systems are in thermal equilibrium with each other.

Observation Leads to Defining Law: "If two systems are at the same time in thermal equilibrium with a third system, they are in thermal equilibrium with each other. If A and C are in thermal equilibrium with B, then A is in thermal equilibrium with B". This statement is called as Zeroth Law.

Practically this means that all three are at the same temperature, and it forms the basis for comparison of temperatures.

The Zeroth law is more fundamental than first and second law even though actually it was stated much later than both the First and Second Laws of thermodynamics. It is so named because it logically precedes the First and Second Laws of Thermodynamics.

The Zeroth Law clearly suggests that when two systems A and B, are in thermal equilibrium then there must be a physical quantity that has the same value for both the systems.

This physical quantity is a new thermodynamic parameter whose value is equal for two systems in thermal equilibrium and is called as temperature (T). This is the only property which allows us to think of the possible make and use of thermometer.

If A and B are separately in equilibrium with C, then
TA = TC and TB = TC
This leads to TA = TB

i.e. the systems A and B are also in thermal equilibrium.