[Phys-L] counterintuitive Gibbs-Duhem

[John Denker <jsd@av8n.com>]

In another context, on 8/8/21 12:27 AM, I wrote:

> Saying "nothing else changes" is a non-starter. You cannot
> change one thing without affecting lots of other things. The
> number of plausible variables vastly exceeds the number of
> degrees of freedom.

This happens All.The.Time in thermodynamics.

Suppose we have a certain system, and we wish to explore how
it behaves as a function of the intensive variables pressure (P)
temperature (T), and chemical potential (μ).

A good first step would be to make a 3D plot, where the dP
direction, the dT direction, and the dμ direction are all
mutually perpendicular and independent......

... except you can't. The parameter space is inherently two-
dimensional. Any one of the aforementioned basis directions
is linearly independent on the other two, as indicated in
this 2D diagram:

  https://www.av8n.com/physics/wpage/gibbs-duhem.png

Let's be clear: You can't say "let's take a step in the dT
direction, and nothing else changes. You just can't. There
has to be some associated dP or dμ or both. There's a constraint
that must be satisfied. There are too many variables and not
enough degrees of freedom.

How do I know this is the case? How do I know this absolutely
must be the case?

Hint #1: Y'all know me. You know how much I love scaling arguments.

Hint #2: Look at the subject line.

Give-away: This is worked out in lurid detail here:
  https://www.av8n.com/physics/scaling.htm#sec-thermo