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:

"Let us start with zero quantity of the solution and

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)

- posted 18 years ago