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> Can you give me an example or two of a reversible process for which
this division into W and Q fails?
For starters, it fails for anything that isn't
at (or very near) thermal equilibrium.
It also fails in a grand-canonical situation;
you would need additional terms on the RHS of
the "E=W+Q" expression.
I do not define S in terms of "Q" or T.
I consider entropy to be primary and fundamental.
Entropy is well-defined even in cases where the
temperature is zero, unknown, irrelevant, or
undefinable.
Let me chime in here: Can _you_ give an example or two where the First Law
as stated _is_ helpfully applied? -- ie a process where Q and W are both
present _and_ are both identifiable? _Except_ of course the ubiquitous
adiabatic cylinder and piston
Of course there's more, but let's pause for any comments.