Re: [Phys-L] correction: heat content

[John Denker <jsd@av8n.com>]

On 02/10/2014 03:18 PM, Jeffrey Schnick wrote:

> I don't think you have to restrict the work term to zero to talk 
> about increasing the thermal energy

That's true.

Previously I disagreed with this, but I shouldn't have.

I finally got around to checking the math, and it doesn't say
what I previously said it says.  Sorry.

> In terms of your example of the baby bottle, with some water in it,
> standing at rest on a table, if you tilt the nipple over by pushing
> the tip of it to one side without changing the orientation of the
> rigid part of the baby bottle you increase the elastic energy of the
> system,

That's an excellent explanatory example.  That shows that the
phenomenon is not just mathematically possible; it is physically
realizable.  The requirements are restrictive and a bit peculiar,
as we shall see, but not unphysical.

Let me clarify the example:  To make it work, we need the hitherto
tacit assumption that the nipple is an ideal spring, with no
thermodynamics of its own.  In particular, changing the temperature
of the nipple is assumed to have no effect on the spring constant.
  (The first time through I didn't realize this was the intent, 
  so I didn't benefit from the example.  My bad.)

By the same token, we keep the important stated requirement that 
the nipple does not affect the "rigid" parts of the bottle.  We 
clarify that the rigid part of the bottle is where the milk is, 
i.e. where virtually all of the heat capacity is.

So .... we have constructed a system where we can write

       dE = - P dV + T dS                       [1]

however, the V and the S are completely decoupled!  There's nothing
wrong with this, but it is a bit peculiar.

I claim that about 98% of what I said previously was correct.  It
remains true that you cannot integrate T dS in order to define
"heat content" as a function of state ... *except in trivial cases*
by which I mean situations so cramped that it is impossible to
build a heat engine.  I stand by that.

Separately, by way of /example/ the /easiest/ way to construct a
cramped system is to require the P dV term to be zero.  Usually
that means holding V constant.  That is easy and fairly common, 
but as Professor Schnick points out, it is not the only way to
construct a cramped system.

The math is ridiculously easy if you use the right approach.
This afternoon I went for a bike ride and ran through about
two dozen wrong approaches, but I will not bore you with that.

Here is one nice way of looking at it:  For a cramped system,
we want
      T dS = dQ                          [2]

applying the exterior derivative to both sides, we get

      dT ∧ dS + T ddS = ddQ              [3]
hence
      dT ∧ dS = 0                        [4]

since d(d(anything)) is necessarily zero.  As the needlepoint
sampler says:  The boundary of a boundary is zero.

That tells us that T can be a constant, or it can have a slope
in the same direction a S does ... but all the other directional
derivatives of T must vanish.  In the simple system described by
equation [1], T must be independent of V.

  The aforementioned trick of holding V constant also works, 
  but if V is not constant then T must be independent of V.

Now that our attention has been attracted to dT, we can have all
kinds of fun.  There is nothing to stop us from writing

            ∂T               ∂T
      dT = -------- dV  +  --------- dS     [5]
            ∂V | S           ∂S | V

where the second term on the RHS obviously plays no role in 
equation [4], and the first term is the one we need to go to
zero.  Since T is already a derivative of the energy, we can
state the key requirement in terms of the mixed partial:

            ∂∂E
         ---------  =  0                     [6]
           ∂S ∂V

It should now be obvious in multiple ways that if one term
on the RHS of equation [1] is grady then the other one must 
also be.  So we can define a "work content" function as well
as a "heat content" function.

IN ANY CASE the point remains that you can only get away with
"heat content" in a cramped system.  To say the same thing the
other way:  In any system that allows construction of a heat 
engine, you cannot distinguish "heat content" from "work content".
Indeed, neither one of those things is definable.  This is not
a problem with the terminology;  since the thing does not exist,
nobody cares what you call it.

One last remark:  This peculiar system must be considered
a compound system, consisting of two cramped subsystems, i.e.
the nipple and the milk.  In other words:
 -- The milk is thermal but not dynamic.
 -- The nipple is dynamic but not thermal.
So neither subsystem exhibits uncramped thermodynamics.  If you 
just looked at the form of equation [1] you might have guessed 
that we have a non-cramped system, but that's not the case.
It's cramped for peculiar reasons that equation [1] will not 
tell you.

The thing that I learned in the last couple of days is that 
for a cramped compound system, the notion of "heat energy" is 
not trivial, because it can /sometimes/ be used to distinguish
one subsystem from the other.  This stands in contrast to a
simple (non-compound) cramped system, where the "heat energy"
is for all practical purposes the same as the plain old energy.

I suspect this is a fairly common path whereby students invent
and solidify misconceptions.  In a cramped compound system they 
can define a W-function and a Q-function, which is OK and indeed
useful, but then they try to generalize this to uncramped systems,
which is not OK.  Really not.

This gets back to the article that provoked this long discussion.
I reckon the ocean can be considered a compound system consisting
of cramped subsystems.
 -- the surface waves are dynamic but not thermal;
 -- the bulk heat capacity is thermal but not dynamic ... or 
   more precisely, coupled only verrry loosely to the dynamics.

So the more I think about it, the article authors were in every
way correct to talk about the "heat energy" in the ocean.  Leaving
out the word "heat" and just talking about the energy would have 
been equally correct but less clear and less informative.  Changing 
the terminology to something like "thermal energy" would leave
the meaning unchanged.