1b) However, people should not try to pass off their
opinions as laws of nature.
2a) I know how to do Legendre transformations.
2b) Doing Legendre transformations does not require
having a set of "natural variables".
I mention this because I recently encountered the
following article:
USE OF LEGENDRE TRANSFORMS IN CHEMICAL THERMODYNAMICS
(IUPAC Technical Report)
Prepared for publication by
ROBERT A. ALBERTY
Pure Appl. Chem.,
Vol. 73, No. 8, pp. 1349–1380, 2001.
That article states, as if it were an established fact,
that the equation
dU = TdS - PdV ... [1]
is "the" fundamental equation for the energy U. As a
matter of opinion, anyone is entitled to opine that
equation [1] seems "fundamental" to them, but in my
opinion it is not particularly fundamental, not more
fundamental than several other energy-related equations
that come to mind. In particular it is IMHO much less
fundamental than a plain, simple, local statement of
the law of conservation of energy.
As Daniel Patrick Moynihan liked to say, people are
entitled to their own opinions, but they are not entitled
to their own facts.
======
The paper goes on to assert that
"The variables in the differentials on the right-hand
side of the fundamental equation have a special
significance and are referred to as natural variables"
and cites almost a dozen references.
I disagree with this conclusion even more strongly than
I disagreed with "the fundamental equation".
I have never seen any law of nature that makes expressing
U as a function of (S,V) more "natural" than innumerable
other ways of expressing U. Sometimes (S,V) is convenient,
and sometimes not. For example, in the heat capacity
measurements that are commonly done in high school and
elsewhere, it is convenient to think of U as a function
of (T,V). Trying to use (S,V) instead would be grossly
unnatural in this situation.
Sometimes (S,V) is convenient, but convenience is not
a law of nature.
I am quite aware that the following four familiar
energy-related equations are connected via Legendre
transformations:
In particular, if you _choose_ to start with any these
four equations, you can generate the others by means
of simple Legendre transformations. But what if you
_choose_ to start elsewhere????
Those four familiar equations are fine as far as they
go, but please keep in mind that the starting point is
a choice, not a law of nature. You have the right to
choose whatever you like, but you must respect the
right of others to choose differently.
The paper goes on for 32 pages, "proving" and "explaining"
a load of "facts" that are not actually true.
As far as I can tell, the whole notion of "natural
variables" is profoundly unsophisticated, ill-founded,
impractical, and pedagogically unhelpful. It is
disappointing that:
-- the idea could become somewhat widespread in the
community;
-- someone would write a paper about it;
-- which would be published in a refereed journal; and
-- would be accepted as a canonical IUPAC document.
It might be amusing to assign this paper as an exercise
in critical thinking, to see how many fallacies the
students could spot.
-- proof by bold assertion
-- appeal to authority
-- circular argumentation
-- failure to consider competing hypotheses
-- et cetera.
=================
If anybody sees any redeeming social importance in the
notion of "natural variables", please let me know.