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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 P
may be quite different on a first glance. Let's look at a few examples: |
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"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". |
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The quote is from:
R.P. Baumann; Modern Thermodynamics with
Statistical Mechanics, p. 337). |
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Note that a probability is a number £ 1; S thus would be always
negative. |
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P
is the volume in phase space occupied by the
system.
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Becker, Theorie der
Wärme, S. 117 (That's what I had as a student). |
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Note that this looks like a number with a dimension! |
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Now some random finds without the detailed
quote. |
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P is the number of
indistinguishable microstates belonging to one macrostate. That is the
definition we used in the script. |
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That is the definition we used in the script. Note that this
is a pure, and mostly very large number. |
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P is the
probability for a macrostate, i.e. the number Pi of
microstates belonging to a certain macrostate i divided by the sum over
all possible Pi. |
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Note that P than is a pure number between
0 and 1. |
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What is correct? The numerical value of
S obviously could be very different depending on which definition
one uses. |
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The answer, of course, draws on the old fact that in classical
physics (including thermodynamics) there is no
absolute scale for energies (and entropy is a form of energy). |
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We thus can always use a P* instead of
P, defined by P* = P/P0
with P0 = 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: |
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If we replace P by P*, we obtain
for the entropy . |
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S* |
= |
k · ln P* = k · ln |
P
P0 |
= k · ln P k · ln
P0 = S const. |
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For the free enthalpy we then simply have G* =
G kT · ln P0 = G
const |
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Moreover, since in practice most applications contain the
derivative of S with respect to some variable x of
the system, constant factors will disappear, i.e.. |
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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. |
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However, if you like to
worry, or noticed that there was a little disclaimer above,
read on in the advanced section. |
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