Re: [Phys-l] thermo differential and extensive/intensive variables

[John Denker <jsd@av8n.com>]

On 06/15/2007 12:40 AM, Stefan Jeglinski wrote:

> I'm looking for a better discussion than I've found regarding 
> thermodynamic functions. In particular, in reading various 
> treatments, it would appear that E = E(S,V) and E = E(V,T) are valid, 
> but E = E(S,T) is not. 

Yeah, I've seen that sort of treatment.  It has trivial
advantages and horrific disadvantages.

> I was thinking originally that there was some 
> rule about mixing intensive and extensive variables, but the 2nd of 
> the above 3 functional forms seems to kill that idea.

Agreed, there is no such rule.  Your skepticism is well-aimed.

  Some peculiar things happens if *all* of the variables are intensive,
  but other than that, you can choose any combination of intensive and
  extensive variables.  Even the all-intensive case makes sense if you
  are careful, and is actually kinda elegant.

> But why can't I use the 3rd functional form, E(S,T)? 

You can.  No problemo.  Again, your skepticism is well-aimed.

> It would seem no 
> more or less valid than E(V,T) from an intensive/extensive 
> standpoint, but I'd get
>
> 	dE = (dE/dS)dS + (dE/dT)dT = TdS + CvdT
>
> which is clearly at odds with the above.

> [1] Interpret the parenthetical quantities as garden variety partial 
> derivatives with the appropriate variable held constant.

Well, that's the *ENTIRE* issue.  What are the "appropriate"
things to hold constant?  The widely-used and wildly-used
notation e.g. dE/dT is ambiguous.

Introducing the idea that E is "naturally" a function of V and S
is a lame attempt to solve some problems with the notation for
partial derivatives ... but it is usually neither necessary nor
sufficient to solve the problems.

You can solve and/or prevent a huge range of problems by
explicitly writing
   dE/dT (at constant P)
or
   dE/dT (at constant V)
i.e. being super explicit about what is being held constant.

There is an ever-present temptation to leave out the statement
of what is being held constant, and in *some* cases you can get
away with it, especially in introductory "end-of-chapter question"
situations ... but in real-world thermo problems, the usual result
of taking the "short cut" is complete disaster.

This is mentioned at
  http://www.av8n.com/physics/thermo-laws.htm#sec-efgh
and discussed with more formality and more detail at
  http://www.av8n.com/physics/partial-derivative.htm#sec-complete

> I'm looking for a better discussion 

Here's my attempt to formulate thermodynamics without the usual
hogwash:
  http://www.av8n.com/physics/thermo-laws.htm