last modified 11/2/2009
© M. S. Shell 2009
1/9
The third law
ChE210A
Absolute entropies and absolute zero
In this lecture we address two questions:
•
Is there such thing as an absolute value of the entropy?
That is, is there a way and
does it make sense to identify an exact numerical value of
g1845
for a particular system and
a particular state point, rather than a change in entropy between two state points?
•
What is the behavior of the entropy and other thermodynamic functions as
g2176
ap
proaches absolute zero?
Recall the microscopic definition of the entropy,
g1845 g3404 g1863
g3003
ln Ω
As we have seen, the interpretation of
Ω
is that it counts the number of microscopic states of a
system.
The behavior of
Ω
depends on whether or not we have a quantum or classical view
point of the world:
•
classical description
– In this approximation, we cannot count exactly the number of
microstates, since we can continuously change the positions and velocities of all the
atoms.
Therefore, there are an infinite amount of configurations.
While we cannot
count absolute numbers of configurations, we can count relative numbers, and hence
obtain entropy differences.
For example, if a volume
g1848
containing a single particle is
expanded to
2g1848
, the number of microstates doubles such that
Δg1845 g3404 g1863
g3003
lng46662g1848
g1848
⁄
g4667 g3404
g1863
g3003
ln 2
.
•
quantum description
– Quantum mechanics says that there are discrete quantum states
a system can occupy.
For example, for a particle in volume
g1848 g3404 g1838
g2871
, only states whose
energy is given by
g2035 g3404
g3035
g3118
g2876g3040g3013
g3118
g3435g1866
g3051
g2870
g3397 g1866
g3052
g2870
g3397 g1866
g3053
g2870
g3439
are allowed, where
g1866
g3051
, g1866
g3052
, and g1866
g3053
are all
positive integers.
Therefore, in a quantum description and in reality, it is possible to
count microstates exactly.
These considerations show that a quantum description is required if we are to have an absolute
entropy, whereby we can count
Ω
exactly.
Is this enough to give us an absolute entropy?
What if, in contrast to the above equation, we
had defined the entropy using
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last modified 11/2/2009
© M. S. Shell 2009
2/9
g1845 g3404 g1863
g3003
ln Ω g3397 g1845
ref
where
g1845
ref
is some universal, materialindependent constant? Would this redefining of the
entropy imply any
physical
changes to the equilibrium behavior of a system, as given by the
relationships that the entropy function implies?
It turns out that this addition would not
change any aspect of the physical behavior of our system.
Recall that
g1846, g1842,
and
g2020
all derive from
derivatives of
g1845
; therefore, the
g1845
ref
constant vanishes in the relationship between
g1831, g1848, g1840
and
these properties, for any substance.
It is for the same reason, in fact, that classical descriptions,
which cannot have an absolute entropy, can describe the properties of many systems.
There is no reason why we could not define the entropy with a nonzero value for
g1845
ref
.
Recall
that the original justification for the existence of the entropy was that the rule of equal a priori
probabilities said that the macrostate with the most microstates dominated the equilibrium
behavior.
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 Fall '08
 Staff
 Thermodynamics, Absolute Zero, Entropy, M. S. Shell

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