dE = T ds - P dV [1a]
= "heat" + "work" [1b]
= thermal + mechanical ☠ [1c]
There are at least a dozen worse ways of writing that equation,
about 100 ways of misinterpreting it ... but that is not the
topic for today. I would like to point out that even under
the best and most generous interpretation, equation [1a] is
still not the whole story ... and the correspondence between
[1a] and [1c] is either wrong or highly misleading.
In particular, we must not equate "work" with "mechanical
transfer of energy".
Perhaps the simplest illustration is /hoisting/ a parcel of
fluid in a gravitational field. This can easily be done in
such a way that it does not change the temperature, entropy,
pressure, or volume of the fluid ... even though it does
change the energy. It is conventional and reasonable to
include the gravitational potential energy in the definition
of E.
The idea that we can raise the energy without doing work
on the parcel is consistent with the fact that there is a
work/kinetic-energy theorem ... not a work/total-energy
theorem.
Let's be clear: work is *not* synonymous with mechanical
transfer of energy.
It works better to write something like this:
dE = T ds - P dV + m g dh [2a]
= "heat" + "work" + "hoist" [2b]
= thermal + \---mechanical---/ [2c]
There are two "mechanical" terms. And we haven't even touched
on non-mechanical non-thermal terms such as µ dN.
This is a Big Deal because a lot of people take equation [1a]
to be "the" first law of thermodynamics, as if it were the 11th
commandment. Also, they define work to be mechanical energy
change. All that kinda maybe sorta works for selected examples,
but usually just spherical-cow-in-the-ivory-tower examples. If
you extend it to real-world situations, even rather simple real-
world situations, it's a disaster.
Partially tangential remark: None of the equations presented
above are the first law of thermodynamics. Not in practice,
not in principle, not even close. The first law should be
stated as conservation of energy, pure and simple.
Equations such as [1a] and [2a] are corollaries that depend
on a boatload of assumptions.
Furthermore, it is usually a mistake to focus on «heat» and
«work» anyway. If in doubt, if there is even the slightest
risk of confusion, forget about those concepts and instead
focus on energy and entropy.