My  $.02 follows.  Hopefully this does not turn into a 3-way public 
conversation as before :-(

> 
> Permit me to offer my views on equation (1) and equations related  
> to it.  Thermodynamics haters (whom I will never understand) may  
> wish to tune out at this point! 

I'll never understand them either!

> 
> Equation (1) dU = TdS - pdV is indeed obtained by combining the  
> First and Second Laws with the following constraints: (a) The  
> system is closed (a First Law requirement), (b) The work is  
> mechanical only, and (c) The process is performed reversibly, ie,  
> infinitesimally close to equilibrium--thermal, mechanical, chemical,  
> physical.  The system is not constrained to being a single  
> phase or a single component and system chemical change is 
>  permitted.  One could relax non-mechanical reversibility by  
> replacing the = sign by a < sign or by then restoring the = by  
> inclusion on the right of an internally created entropy term.   One  
> can add other work terms if appropriate; electrical and surface work 
>  are not uncommon; I will not include such terms here though. 

Hmm, perhaps the the following  argument

(1) does not require the restriction (b)
(2) shows where (c) comes in

The first law for a differential process in a closed system can be 
written in general as

dE = \delta Q + \delta W

where E is the total system energy

E = U + grav pe + kin e + ...

and \delta W includes ALL types of work interaction.  Regarding E 
as a state function allows us to choose any path we wish for the 
RHS.  Choosing a reversible path (this is where "reversible" 
comes in),

dE = \delta Q_{rev} + \delta W_{rev}

Now we can replace \delta Q_{rev} by TdS to give

dE = TdS + \delta W_{rev}

The possible forms of work can be divided into pV work and 
"everything else", giving

dE = TdS - pdV + \delta W_{rev, other}        (A)

I would claim that eqn (A) is the most general form of "the first 
fundamental equation".  It includes all possible types of energy and 
(reversible) work terms (the latter being defined at fixed S and V in 
this case).  I think that Pentcho's piston-cylinder problem involving 
liquid and vapor phases and a floating block inside the cylinder 
can be handled with this equation with the last term set to 0, and 
with gravitational terms included in E.

As for the explicit inclusion of consideration of chemical change, let 
me try the following:
- this part is a little shakier :-(

One of the possible forms of work in the closed system is the work 
due to to chemical change (chemical reaction).  One way to find the 
form of this term is to consider the dependence of U on n_i, S, and 
V - getting the additional term given by Lee's eqn. (2).  I would claim 
that Lee's eqn. (2) is valid for a _closed system_ undergoing 
chemical change at constant S and V, i.e. we can write

dE = TdS - pdV + \sum \mu_i dn_i + \delta W_{rev, *} (B)

where the * now means all work other than pV work _and_ that 
given by the 3rd term on the RHS.   This 3rd term is  "the work due 
to chemical change in the system at constant S and V".  Of course, 
the most convenient way to reformulate this in terms of constant P 
and T - in terms of G.

It seems to me that eqn (B) still applies to a closed system (even 
tho with the dn_i terms it is typically referred to as applying to an 
_open_ system).  It applies to a closed system in which the 
species within it are undergoing chemical change.

Comments. Joe, Pentcho, or others?

> 
> The relationship, equation (2), 
> dU = (dU/dS)sub(V, all nsubi)dS + (dU/dV)sub(S, all nsubi)dV +  
> Sum(musubi dnsubi)       [d = partial d] 
> has a different origin, simply that U = f(S,V,nsub1,nsub2, - - -), ie,  
> U is a function of state, ie, of a set of independent variables; the  
> mu are chemical potentials and the n are the amounts of  
> substance of the various components in the various phases   
> (Pentcho quite improperly calls these mole numbers; I wish  
> he would come up to date with proper IUPAC names!)   
> Equation (2) assumes that the given variables on the right are  
> sufficient; one would not add a charge term because charge  
> depends on the n, but one could add an area or a length term (eg,  
> for a stretched rubber band) if appropriate; I will not include such  
> terms here though.  Equation (2) is perfectly valid for a single  
> component-single phase system, incidentally. 
> Comparison of equations (1) and (2), under the constraints of  
> equation (1), together with requirements for functions of state, give 
> (dU/dS)sub(V, all nsubi) = T universally, 
> (dU/dV)sub(S, all nsubi) = -p universally, 
> Sum(musubi dnsubi)) = 0 under the constraints of equation (1) 
> 
> Thence, equation (3) 
> dU = TdS - pdV + Sum(musubi dnsubi)   
> universally true so long as there are sufficient independent  
> variables here.  Furthermore this will reduce to equation (1) in  
> situations where all dnsubi are zero and I can think of twol ways of  
> doing this: (i) System closed + no chemical reaction in the  
> system, (ii) System is the main part of a (theoretical) van't Hoff  
> equilibrium box in which reaction can occur but for which constant  
> nsubi are maintained by transfer to or from the attached chambers  
> replete with semi-pemeable membranes.  So here we have an  
> alternative set of conditions to those originally given for equation  
> (1). 
> 
> All these relationships can of course be reformulated in terms of  
> enthalpy, Gibbs energy and Helmholtz energy. 
> 
> I agree with Pentcho that textbooks can be very confusing on these 
>  key equations.  No doubt thermodynamics is a complicated  
> philosophy, fascinating for all that.  E A Guggenheim said he never  
> completely understood it! 

And Arnold Sommerfeld said (I have been looking for the exact 
reference and can't find it - can anyone help?)

1.	The first time you study thermodynamics, it is 
incomprehensible.

2.	The second time you study it, you _think_ you understand 
most of it, except for some parts.

3.	The third time you study it, you _know_ you don't 
understand it, but since you can readily carry out the manipulations 
involved, this doesn't bother you any more.


Best Regards,


W. R. Smith, Professor
Dept. of Mathematics and Statistics and School of Engineering
University of Guelph, Guelph, Ontario, CANADA N1G 2W1
Tel: 519-824-4120, ext. 3038; FAX: 519-837-0221; 


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