# Derivation of Gibbs-Duhem relation: physical explanation?

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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)

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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:

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The whole slide collection is here: