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# Re: [Phys-l] Thermodynamics question

• From: John Denker <jsd@av8n.com>
• Date: Mon, 11 Jan 2010 03:05:04 -0700

On 01/10/2010 09:19 PM, LaMontagne, Bob wrote:
... The equation T ds = du + Pdv is a standard state variable
equation for a control mass. I thought it was a universal equation
that was not tied to a particular process, but maybe that's where my
thinking is wrong.

There's definitely a problem there. The equation
mentioned above falls into the category of being
only sometimes true.

Tangential remark: That's why my little quiz was
not a T/F quiz but rather a T/N quiz, where
T means "reliably true" and
N means "not reliably true" which includes things
that are only sometimes true.

The aforementioned equation can be rewritten as
dE = T dS - P dV [1]
and if we write it that way we ought to be able to
get past all questions of terminology, notation, and
interpretation ... and focus on the meaning.

Even so, the meaning of equation [1] is not universally
true ... and it is certainly not the best way to express
the first law of thermodynamics. For multiple practical
as well as pedagogical reasons, the first law should
be stated as

_Energy obeys a strict local conservation law._

That's it. Pure and simple.

If/when the energy can be expressed as a function of
V and S alone, and is differentiable, then equation [1]
follows immediately. It must be emphasized that this
equation does not express conservation of energy. It
does not depend on the fact that energy is conserved;
it only requires that energy be a differentiable function
of state. A similar equation applies to any other
differentiable function of state, including things like
the temperature and molar volume, which are obviously
not conserved. For more on this, see the newly revised
and reorganized section:
http://www.av8n.com/physics/thermo-laws.htm#sec-state-function

In the case where electricity is flowing, the energy
is definitely not a function of V and S alone, so
equation [1] needs to be repaired. If energy can be
expressed as a function of V, S, and charge (which
might make sense for a battery) then we can immediately
write

dE = T dS - P dV - voltage d(charge) [2]

Tangential remark: The minus signs in equations [1]
and [2] are somewhat arbitrary. The choices shown
here are conventional, although the conventions
changed in the not-too-distant past.

To say more-or-less the same thing in another way, you
need to make sure that your notion of thermodynamic
state includes enough components to span the state
space. If you leave off one of the variables, then
E and all the other things that are supposed to be
functions of state won't be functions of state.

Your state vector needs many enough components to
span the entire space, and few enough components
so that they are linearly independent.

When you expand the gradient vector as in
equation [1] or equation [2], there will be one
term on the RHS for each dimension, i.e. one for
each component of the state vector.