The statistical definition of the entropy appears in many forms; almost every textbook finds its own version - and all versions are equally correct. You will always find the definition | ||||||||

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But the meaning of may be quite different on a first glance.
Let's look at a few examples:P | ||||||||

"
means the probability of a macrostate, where P in turn is proportional to the number of microstates
accessible to the system contained within that macrostate".P | ||||||||

The quote is
from: R.P.
Baumann; Modern Thermodynamics with Statistical Mechanics, p. 337. | ||||||||

Note that a probability is a number £ 1; thus would
be always negative.S | ||||||||

is
the volume in Pphase space occupied by the system. | ||||||||

Becker, Theorie der Wärme, S. 117 (That's what I had as a
student). | ||||||||

Note that this looks like a number with a dimension! | ||||||||

Now some random finds without the detailed quote. | ||||||||

is the number of indistinguishable microstates belonging to one macrostate. P |
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That is the definition we used in the script. Note that this is a pure, and mostly very large number. | ||||||||

is the probability
for a macrostate, i.e. the number P of microstates belonging to a certain macrostate P_{i}i
divided by the sum over all possible . P_{i} | ||||||||

Note that
than is a pure number between P0 and 1. | ||||||||

What is correct? The
numerical value of obviously could be positive or negative and generally very different depending on
which definition one uses.S | ||||||||

The answer, of course, draws on the old fact that in
classical physics (including thermodynamics) there is no absolute scale for
energies (and entropy times temperature is a form of energy). | ||||||||

We thus can
always use a instead of P*, defined by P
with P* = P/P_{0} arbitrary constant factor (that does not depend on the variables of the system under
consideration). All that happens is that you add a constant factor to the entropy or free energy of a system; i.e. you
change the zero point of the energy scale.P_{0} = | ||||||||

If we replace by
P, we obtain for the entropy .P* | ||||||||

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For the free enthalpy we then simply have G* = G
– kT · ln P_{0} = G – const | ||||||||

Moreover,
since in practice most applications contain the derivative of with respect to some variable
S of the system, constant factors will disappear, i.e..x | ||||||||

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In short, all definitions are equivalent and you don't have to worry about the additional constant factors that may appear. Feel free to use the definition that is most easily applied to the problem under consideration. | ||||||||

However, if you like to worry,
or noticed that there was a little disclaimer above, read on in the
advanced section. | ||||||||

2.1.1 Simple Vacancies and Interstitials

Internal Energy, Enthalpie, Entropy and Free Enthalpie

Exercise 2.1-7 Quick Questions

Internal Energy, Enthalpie, Entropy and Free Enthalpie

Boltzmann's Constant and Gas Constant

Quantum Mechanical Concept of Entropy

Alternative Derivations of the Mass Action Law

© H. Föll (Defects - Script)