The diagnostic is a new thermodynamic state function called the entropy. Just as any thermodynamic equilibrium state of a gas (for example) can be characterized by a temperature, pressure, volume and internal energy we can derive from these an additional characteristic property or diagnostic which will prove to be helpful in systematically predicting the direction of spontaneous thermodynamic processes.
To within an arbitrary constant we define entropy in terms of the difference
between the entropy of two different thermodynamic states:
where the integral is taken over any reversible path which connects
the two thermodynamic equilibrium states A and B. (Note that to be
consistent with the book I am now writing
for a change in entropy and will also use
for a change in internal energy).
In order for S to be a thermodynamic state function
the difference in entropy between states A and B must be independent
of the path between these states.
We can show that this is so for all reversible paths in the special case
of an ideal gas. In that case we have
Dividing this equation by T and introducing the equation of state for the
ideal gas we obtain
Irrespectively of the reversible path taken the integral of dQ/T depends
only on thermodynamic properties of the initial and final states:
We have thus proved that S is a thermodynamic state function for an ideal gas and
Eq. 4 can be used to calculate entropy for this simple thermodynamic system just
as we used 
to calculate the change in
internal energy between states A and B.
.