

As indicated before, thermal
equilibrium, a thermodynamic system at an equilibrium
state is characterized by the value of the thermodynamic coordinates. The
following notation is used in this notes to represent the different
thermodynamic coordinates for a given system, A: a_{1}, a'_{1},
a"_{1}... where a, a', a" ... represent the different thermodynamic
coordinates and the sub index 1 identified a particular value for these
coordinates.
Thus, when two thermodynamic systems, A and B, are in thermal equilibrium with each
other, the thermodynamic coordinates of the systems will have a well defined
value such as a_{1}, a'_{1}, a"_{1}... for system A and
b_{1}, b'_{1}, b"_{1}... for system B. After the
systems have reached the equilibrium state, defined by the previous sets of
thermodynamic coordinates, systems A and B can be insulated and separated from
each other. When separated, suppose that system A is maintained unchanged; that
is, it is surrounded by adiabatic walls and all its thermodynamic coordinates
are kept fixed. On the other side, system B is also surrounded by adiabatic
walls but some of the thermodynamic coordinates are changed to be a new set
b_{2}, b'_{2}, b"_{2}.... Next, systems A and B are
again placed in thermal contact with each other while both are insulated from
the surroundings. At this point, their thermal evolution is studied
experimentally. Under the previous conditions, it is found that the two
systems do not undergo further thermal exchange indicating that the two
thermodynamic systems are still in thermal equilibrium.
