In my copy of Atkins' book, Physical Chemistry (c.1978, revised and corrected
printing), there is a discussion on pp.693-695 of residual entropy. He presents
the notion as one of two alternative hypotheses which can be used to explain
why calorimetric measurements of entropy of a substance might be lower than
the value computed from spectroscopic data, the other hypothesis being that
the calorimetric measurement failed to detect a phase change.

As an example of the latter, he cites the case of carbon monoxide, CO,
where calorimetric entropy was measured as 194 J/(K-mol) while the
spectroscopic value is 197.9 J/(K-mol). The difference is close enough
to R ln 2 to suggest that it might be due to residual entropy involving
two possible orientations in the CO crystal, but Atkins says it was found
that there was actually an undetected low temperature phase change in CO
which is the source of the discrepancy.

I'm a little confused by his discussion for various reasons. First, I was
under the impression that the 3rd law of thermodynamics required that
entropy go to zero at absolute zero. If so, than to entertain the idea
of residual entropy is really to consider a way of disproving the 3rd
law of thermodynamics, a point that Atkins doesn't seem to mention in
his discussion. If I'm interpreting the 3rd law correctly, apparently
there would always have to be a phase change of some kind en route
to absolute zero to eliminate residual entropy.

The other point I am confused about is the following.

He gives the example of the NO, which he presents as have more
secure evidence of residual entropy, and the example of H2O, which
is the culmination of some numerical examples worked out throughout
chapter 21. In regard to the latter, he notes (pp.694-5) "the hydrogen-bonded
nature of the ice crystal and the tetrahedral arrangement of hydrogen atoms
around every oxygen atom." and claims: "Two of the hydrogen atoms are attached
by normal, short, sigma-bonds, and the two others attached by long
hydrogen bonds. There is a randomness in which two of the four bonds are
short, and an approximate analysis of the problem leads to the prediction
of a residual entropy of about R ln (3/2) J/(K-mol), in good agreement
with the experimental value."

I really need to see past the handwaving in order to understand why
the hydrogen bonding of the ice crystal, the tetrahedral arrangement of
hydrogen atoms around every oxygen atom, two or which are sigma bonds and
two of which are hydrogen bonds, lead to the number 3/2 from first principles.
Can someone please explain that "approximate analysis of the problem"?

After this remark, Atkins then presents what he touts as "a rather roundabout
way of arriving at the residual entropy", based on the numerical examples
involving water worked out in the chapter. His calculations lead him to
a value of -206.8 J/(K-mol) for the free energy F_0(1500) of water at
1500 degrees Kelvin, as compared with the experimentally measured value
of -211.71. The difference is about 4.9 J/(K-mol) which he happily notes
is about R ln 1.8 and that 1.8 is close to his theoretically predicted 3/2.
The trouble with this otherwise neat example is that his numerical example
computing F_0(1500) contains an error: on pp.684-685, he quotes the
formula for the partition function incorrectly, leaving out a factor
of sqrt(pi), causing his value for F_0(1500) on p.685 to be off by
R ln sqrt(pi), and this accounts for most of the discrepancy that
he uses to support the case for residual entropy of water by his
"rather roundabout way".

In effect, therefore, his "rather roundabout way" actually tends to prove
there is no residual entropy in water. But that contradicts his other argument
from his a priori 3/2 and from the discrepancy of 3.4 J/(K-mol) between
calorimetric and spectroscopic entropy, given on p.692 as the real reason
to believe in residual entropy of water.

So, that is why I find his discussion confusing and I would appreciate it
if someone could clarify the matter. I would also like to know, since there
have been more editions of Atkins' book since my copy was published, whether
his numerical errors have persisted into current editions.

Ignorantly,
Allan Adler [Only registered users see links. ]

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I see you are going through Adkins. Frankly, I don't like the book
much, preferring McQuarrie and Simons, although that too is
far from perfect.

Allan Adler wrote:

The key is that the third law states that entropy goes to zero at
absolute zero for PERFECT CRYSTALS. In the case of, for
example CO you can have COCOCO or COOCOC etc.

This problem exists in spades for proteins.

Now, in a way this is defining your way out, ie a perfect crystal
is a crystal that has zero entropy at zero K, but there is a
statistical approach, ie, how many equivalent arrangements
of a crystal at 0 K are possible. If it is one, then you have
a perfect crystal, if more you can define the residual entropy
as k ln W.