# Derivation of Gibbs-Duhem relation: physical explanation?

I am a 4th-year undergraduate in materials
engineering taking a course based on Reed-Hill and
Abbaschian, Physical Metallurgy Principles (1991).
Usually, the first time I read a chapter, I suspend
my disbelief and take all of the equations at face
value. Then, when I read the chapter again, I try
to follow all the derivations line-by-line.
The more times I read the derivation of the
Gibbs-Duhem equation in Reed-Hill, the more dubious
it seems.
Mathematically, it makes sense (although not quite
the way Reed-Hill does it), and intuitively, it
seems like it ought to be true. The problem is with
the physical explanation.
The Gibbs-Duhem equation says that the sum over all
components in a solution of the product of the number
of moles of a component times the differential in
partial molar free energy for that component is equal
to zero. It is actually applicable to any partial
molar quantity, but Reed-Hill chooses to prove it for
free energy.

In Reed-Hill and Abbaschian (1991), it appears on pp.
302-305. In the earlier version of Reed-Hill (1964),
it is on pp. 311-314. A very similar derivation,
using partial molar volume instead of partial molar
free energy, is found in Moore, Physical Chemistry,
3rd ed. (1963), pp. 118-120.
Reed-Hill says this:
form it by simultaneously adding infinitesimal
quantities of the three components [...] Each time
that we add the infinitesimals, however, let us
make the amounts of the components in the
infititesimals have the same ratios as the final
numbers of moles of the components [...] If the
solution is formed in this manner, its composition
at any instant will be the same as its final
composition. In other words, the composition will
be CONSTANT at all times, and since the partial-
molal free energies are functions of only the
composition of the solution (at constant temperature
and pressure, they will also be CONSTANT during the
formation of the solution." [my emphasis]
This leads following expression for the differential
in free energy resulting from the addition of these
infinitesimal quantities:
- - -
dG = G dn + G dn + G dn
A A B B C C
Where "G bar" is partial molal free energy, and the
subscripts refer to components. (Hopefully my attempt
to render this in ASCII is legible to everyone).
Integrating the above, under the condition that all of
the partial molal free energies are constant, yields:
- - -
G = G n + G n + G n
A A B B C C
Reed-Hill then re-differentiates (!) this expression
to obtain:
- - - - - -
dG = G dn + G dn + G dn + n dG + n dG + n dG
A A B B C C A A B B C C
He then subtracts the previous expression for dG from
this one, and obtains the Gibbs-Duhem relationship:
- - -
n dG + n dG + n dG = 0
A A B B C C
But since he started from the assumption that the
partial molal free energies were CONSTANT, this
conclusion would appear to be trivial, since if
G is constant for each component, then dG for each
component must be zero, and obviously 0 + 0 + 0 = 0!
This derivation, as it stands, does not convincingly
show (to me, anyway) that the Gibbs-Duhem equation
ought to hold for situations in which dG is not
equal to zero for all components - and clearly such
situations are of interest.
Reed-Hill refers to Darken and Gurry, Physical
Chemistry of Metals (1953), in regard to this
derivation. It appears in that book on pp. 238-240.
Darken and Gurry actually give a slightly different
derivation, but they mention the approach which
Reed-Hill takes (i.e. forming a solution by a series
of infinitesimal constant-composition steps) in a
footnote, and claim that it is valid because free
energy is a state function, so only the initial and
final states matter.
I can ALMOST buy this explanation, except that, as
far as I can tell, in this case the initial and final
states are the same in terms of composition!
As I mentioned, Darken and Gurry's derivation is
different, but is similarly confusing - if not more
so. It involves adding a certain number of moles of
each of the components and then removing an equal
number of moles from the solution. If anyone is
interested, I will go through it, but I have a
feeling that this message is much too long already.
Gokcen, Statistical Mechanics of Alloys (1986), has
a much more convincing mathematical derivation which
relies on the properties of homogenous equations,
but he does not attempt any sort of physical
explanation.
I think my main difficulty is that I do not have a
very clear physical picture of what a differential
of a partial molar quantity is.
Anyway, if anyone could take the time time to address
this, or else refer me to a book which explains it in
a more clear manner, I'd really appreciate it.
Thanks for taking the time to read this.
Dave Palmer
snipped-for-privacy@NOSPAMiit.edu (remove NOSPAM to contact)
(708) 236-5360 x220 (work)
(773) 955-2223 (home)
Well, Dave, you have certainly done some digging in different books.
Sometimes you can focus too much on a physical intrepretation when the math will do alone. But, I too had some dissatisfaction with the Gibbs Duhem equation.
You might find this powerpoint presentation of interest.
pages.pomona.edu/~TCC04747/20040225.Lecture.ppt
I think it would be neat to find the remaining power point lectures for this activity.
Jim
Dave Palmer wrote:
The whole slide collection is here:
Physical Chemistry
Thermodynamics and Chemical Kinetics
Professor Tim Corcoran Professor Chris Stromberg
These are the authors.
Dave Palmer wrote:

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