Re: CONSERVATION OF ENERGY

[John Mallinckrodt <ajmallinckro@CSUPomona.Edu>]

Warning: Long!

A couple of recent postings induce me to comment again on the concept of 
internal versus "thermal" or "mechanical" energy:

Most of us would agree that an ideal vibrating two-mass-and-spring 
system has a total (internal) energy which can be determined by adding 
the energy of vibration (which alternates between potential and kinetic 
modes) to the standard thermodynamic internal energies of the two masses 
and of the spring considered as subsystems.  If this system were 
isolated and if there were some form of coupling between the 
macroscopically observable vibration and the internal modes (for 
instance, flexing dissipation in the spring), then, eventually, the 
oscillations would die out, the temperatures of the masses and the 
spring would increase, and the total energy of the system would remain 
unchanged.

In this specific, oversimplified situation I can understand and, 
perhaps, even live with the idea that the entire system has an internal 
energy composed of the sum of a thermal component and a mechanical 
component.  The second law allows the mechanical component (which really 
simply represents an overconcentration of energy in one mode) to be 
reduced in favor of its redistribution over the other modes, but not 
vice-versa.

Now consider a more realistic situation in which a large chunk of matter 
is struck and then vibrates in a complex and macroscopically observable 
manner.  Let's think about how we would apportion its internal energy 
between mechanical and thermal components.  We can (in principle) do a 
spectral analysis to determine the amount of vibrational energy in each 
vibrational mode.  But where do we cut off the sum?  At what wavelength 
or frequency does a vibrational mode cease to be "macroscopically 
observable?"  Since, in principle, the precise motion of every atom can 
be specified by a sufficiently extended spectral analysis, isn't *all* 
of the energy vibrational (e.g., "mechanical") and, therefore, is there 
really *any* "thermal" energy at all?  I simply don't see *any* rational 
way of deciding how much of the internal energy here is "mechanical" and 
how much is "thermal."

Perhaps one could say that the "thermal" energy is the sum of the 
energies in all modes that have energies within some expected deviation 
of that specified by equipartition, but we are still faced with an 
arbitrary--and, ultimately, not very meaningful (or useful)--decision 
about what that cutoff will be.

My conclusion from considering such examples has been that we should not 
deceive ourselves into thinking that internal energy can be divided into 
thermal and mechanical portions.  We should be content to say that a 
system is or is not "in thermal equilibrium" and leave it at that.  

I hasten to add however that I *am* willing to separate out bulk (i.e., 
"center of mass") translational kinetic energy and distinguish it from 
internal energy because it--in sole contrast to all other forms of 
system energy--is frame-dependent and, therefore, clearly *not* a 
property "of" the system.

In light of the above, I will make some remarks on recent contributions.

In Ludwik's "summary" posting he wrote:

> Macroscopic work is not energy, it is a process by which mechanical energy 
> (kinetic, elastic or gravitational) can be either gained or lost by a 
> system. 

You need to lose (or at least reposition) the word "mechanical" in the 
above--e.g., "Work is a mechanical process that changes the internal 
energy of a system."

> It is also the name of a quantity (force times distance) which is 
> used to know (to measure) how much mechanical energy was lost or gained.

This is an oversimplification and, therefore, misleading without a clear 
specification of *what* force(s) and *what* distance(s).  Furthermore, 
the word "mechanical" is particularly problematic here.  There are many 
possible definitions for work and the specific system energy that 
changes depends critically on the choices for the force(s) and 
distance(s).  [At the risk of sounding like a broken record I refer you 
to my paper with Harvey Leff, "All About Work," AJP, Vol. 60, 356-365, 
(1992).]  Generally in thermodynamics we use the external forces only 
and the motions of the points of application *with respect to* 
(something like) the system's center of mass frame.  This definition 
manages to exclude the effects of net forces which accelerate the center 
of mass.  For instance, we don't usually care if a volume of gas is 
accelerated as long as it maintains the same pressure and temperature.  
(After all, a volume of gas "at rest" on the surface of the earth is 
"equivalent" to ...  Oh never mind; I *really* don't want to open that 
one up again!)  

> Likewise, heat is not energy. The word heating refers to a macroscopic 
> process through which thermal energy is either added or removed from a 
> system. 

Heat, just like work, changes the *internal* energy of a system.  The 
primary difference is that it is a thermal rather than mechanical 
*process.*  Heat can just as easily change the "mechanical component" 
(whatever that means) of a system's internal energy as can work.  
Consider what happens when you heat a system which consists of a volume 
of gas AND a confining but free-to-move, massive piston.

> It is also the name of a quantity (c*m*dT) which is used to know 
> (to measure) how much thermal energy was lost or gained.

Not in general.  The heat added to a system is not always measurable in 
this fashion (consider the isothermal expansion of a gas) and, again, 
the *form* in which that energy "shows up" is just as arbitrary as when 
it is added via work.  As shown in the gas plus piston example, heat 
does not necessarily change the "thermal" energy even if one *could* 
make a rational definition of that term.

> I heard somewhere a statement that "how much heat in a body?" is like 
> asking how much rain is in the ocean." ...

Good example!

Then Emilio said:

>    My brief comment to the opinion that "heat" should be reserved to the
> "transmission of energy by the effect of a difference in temperature"
> (as stated by Al Bachman).

Except that processes which, I think, might properly go by the name 
"heating" also occur in situations in which the temperature is simply 
not defined on either side of the system boundary.  We need to 
distinguish between reversible and irreversible heating.  Both are what 
I would call "thermal" processes but only the second kind involves 
"temperature differences."  Yes, a little picky.

Then Ludwik responded:

> Emilio, I suspect (withou being sure) that John M. would refer to the
> temperature increase of food in the microwave oven as warming rather than
> heating. Would the Marsian word gaming be acceptable to you?

Well, I think I'd call it "warming due to heat" as opposed to "warming 
due to work."  I don't intend "warming" to refer to some "third kind" of 
thermodynamic process, it simply means the internal energy has increased 
and rethermalized.

Then Jim Green chimed in with some good points and also said:

> 4) The First Law is a good guide to this usage:  If W+Q=dU and a speaker
> wants to maintain consistent usage in a discussion, then just what meaning
> should be assigned to each term?  Well, there is the Second Law to deal with
> as well: if dS=Q/T, then those things assigned to Q should change S.  
> =name for Q needs to be agreed upon.  Usually "Q" is named "heat" -- this is
> just fine, but then "heat" is what is done *to* the system -- and in the
> process changes the S of the system. ...

But don't forget that the relationship dS == (i.e., "is defined to be") 
Q/T applies only to reversible heating processes.  In irreversible 
processes there is no general relationship between Q (if any) and the 
change in S (if any).

> 5) There should be a sharp distinctions between "Q" and "W"; that
> distinction should be that "Q" changes S and "W" does not.  Some examples
> might help here: a  Bunsen burner does "Q", pdV does not, it therefore is
> "W".  An egg beater *does* give rise to "Q" ie it *does* change S, but there
> is no "dT" between the system and the universe.  Are we willing to call what
> the egg beater does "heat".  If we are being consistent with our First Law
> and Second Law criteria, we must.  If we do not want to be consistent, we
> just don't understand thermodynamics and our students will be mightily
> confused.

Here you are running into the difficulties associated with forgetting 
the special role of reversible processes.  Maybe the real point is that 
in irreversible processes all that we can really determine (and all we 
really care about) is how much the internal energy of the system 
changed.  (The first law for irreversible processes should probably 
simply read: "dU = how much energy was added to the system.")  *Then* we 
construct a *reversible* path with well defined heat and work in order 
to calculate the change in entropy.

John
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