The concept of entropy is introduced at a very early stage in both
physics and chemistry education so it makes a lot of sense to discuss
entropy's definition and applications. My analysis will be restricted
to the classical interpretation - one that deals with dQ - the heat
absorbed.
Definition 1: dS = dQrev/T
where "rev" means that the heat is absorbed in a reversible process.
Here lies the first confusion. "Reversible" may mean:
A) The system passes through a succession of equilibrium states
B) The system passes through a succession of equilibrium states and,
in addition, is only allowed to absorb heat from surroundings at the
SAME temperature.
Most textbooks define A and apply A; some (whose authors read Carnot
and other classical sources) define B but then again apply A. This
problem does have a scientific solution which is not given in
textbooks and hardly anybody knows about it. So if a clever student
has mastered the definition B and sees dS=dQ/T applied to heating of
a gas at constant volume, his/her interest in thermodynamics will
probably disappear forever.
In chemistry, there is a famous equation known as the
fundamental equation of chemical thermodynamics:
dU = TdS - PdV + SUM mu_i dn_i
where mu_i and n_i are the chemical potential and amount of the ith
component respectively. Some textbooks pretend they deduce the
fundamental equation, others are fair enough to declare it as an
AXIOM. In any case, the dS in the fundamental equation cannot be the
dS in Definition 1 since the fundamental equation is applied to
systems DISTANT from equilibrium (i.e. heat is NOT absorbed in a
reversible process). So the fundamental equation in fact DEFINES dS
for non-equilibrium chemical processes:
Def 2 can be specified by substituting the first law dU=dQ-dWexp into
it:
dS = (1/T)(dQ - dWexp + PdV - SUM mu_i dn_i)
where Wexp is work of expansion done by the system. Most textbooks
make the (not very correct) assumption dWexp=PdV and then the
definition gets simpler:
dS = (1/T)(dQ - SUM mu_i dn_i)
This definition has existed for a century perhaps but was IMPLICIT.
So when a scientist (I. Prigogine) made it explicit and presented it
as his own discovery, he was awarded the Nobel prize for this
discovery.
I think it is very important for students to know about Def 1
and Def 2 and also to know that the two definitions have nothing to
do with one another.