Donald G. Miller Lawrence Radiation Laboratory University of California Livermore
Duhem and the Gibbs-Duhem Equation
I.,
I n recent issues of THIS JOURNAL,i s homogaeous of degree m with ~espectto the variables 11, . . ., then one has the identity the Gibbs-Duhem equation has been the subject of aF bF aF numerous comments (e.g., J. CHEM.EDUC.,39, 527 zl-+zn-+ . . . +zn-ax" =mF ax, ax* (1962); ibid., 40, 225-8 (1963)). However, the role played by Duhem has not been mentioned. By hypothesis, one has the identity I am not as yet sure when or by whom Duhem's name F(Xn, . . ., As,) = AnF(z, . . ., 2,) was added to the equation. But it appears that Duhem The derivatives with respect to A of each member of this identity was the first to use Euler's theorem explicitly to prove must also be identical with one another the Oibbs-Duhem equation. The first references appear in his book "Le Potentiel Thermodynamique"' which was published in 1886.2 On pages 32-3 Duhem applies Euler's theorem to a two-component system, and on pages 141-2 he discusses the general case. If the various terms of the first member are transformed by This whole subject is also given in his "MBcanique means of Theorem I and the common factor Am-' is suppressed Chimique,"3 which while not rare is usually found only on both sides, one finds the equality given in the theorem. in large libraries. The presentation is so clear and Gibbs Free Energy (Thermodynamic Potential) of a concise that I believe a translation of it using current Homogeneous Solution not,ation would be interesting and informative. A homogeneous solution of fluids 1, . . ., n at a constant presSome Properties of Homogeneous Functions
I t is said, with Euler, that a function F(zj, . . ., z,) of n variables, XI, . ., z, is homogeneo~isqf degree m with respect to these variables if the identity
.
..., As,) = XmF(zl,..., I,,) holds for arbitrary XI, . . ., z, and A. F(Ar,,
Theorem I. If a fundion is homogeneous of degree m with respect lo the vwiables z,, . . .,z., its partial deriuatiues with respect to each of these variables are hoinogeneous functions of degree (rn - 1 ) i n these same variables. The identities F(hz,, F[h(z,
. . ., Ar,,)
+ Az,), Xz2, . . ., Ax.]
=
A'"F(xl,
=
Arnf'(z~
. . ., 2")
+ Am,
Xa,
. .., X d
sure P and constant temperature T has a Gibbs free energy G which depends on the masses mi, . . ., m , of the fluids and on the pressure P and the temperature T: G
=
G(mt,
.. ., m,,
P, T )
At the same P and T, let us mix masses Am,, . . ., Am, of the fluids 1, . . ., n . We obtain a second solution, of the same nature as the preceding, but of mass A times greater. If we neglect the forces that the various parts of a mixture exert on one another, we may attribute to the second solution s. Gibbs free energy A times greater than to the first. We will thus have far arbitrary h G(Am,,
.. ., Am., P, T)
=
XG(rn1,
. . ., m,, P , T)
The Gibbs free a e r g y of a homogeneous solution of n ,fluids is a humogemous function of the fist degree i n the masses of the fluids. Let us set
resulting from the previous definition yield and apply Euler's theorem to the function G. We will have G = mt01 or in passing to the Limit as Az, tends toward zero
which proves the theorem. Theorem 11 (Euler's Theorem). I f the function. F ( a ,
. . .,z,)
This work was performed under the auspices of the U. S. Atomic Energy omm mission. 1 DUHEM, P., "Le Potentiel Thermodynamique et Ses Applications 9. La MBcsniaue Chimiaue et B L'Etude des PhBnomhea , 1886. Eleetriques," A. ~ e r k a n npar&, 3 I t had actually been written in 1884 (when Duhem was 23) and submitted as a thesis a t the Sorbanne. This thesis was refused through the influence of Msrcelin Berthelot, who was still trying to support his erroneous maximum work principle. DUHEM,P., "Trait6 ElBmentaire de la MBcanique Chimique Fondhe sur la Thermodynamique," 4 volumes, A. Hermann, Pwis, 1897-9, Val. 3, pp. 1 4 .
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Journal of Chemical Edumtion
+ . . . + m,C,
(2s
It results from this formula that the functions C r , . ., Gn are quantities of the same type aa the quotient of an energy or a work' by a mass. Hence these are quantities of the same type as a potential function. We will call them partial specific free energies of the e m s t i t u a t s 1, . . ., n zn the solution (thermodynamic potential functions). In vjrtue of theorem I of the preceding paragraph, the func. tians GI, . . ., 6. are homogeneous functions of degree zero in the variables m,, . . ., m,. To each of these functions we apply Euler's theorem, and we will find the identities
ihich result from the definition of the functions GI, . . ., G. iven by equation (I), permit the substitution of equations a GI ... + n h -h a = 0 mth r am,
+
m,-
a C,
+ ... +m.-
hm
a& amn
=0
for equations (3).
Duhem was a great admirer of Gibbs, was his principal champion in France, and extended a number of Gibbs' ideas. It is altogether fitting that his name be appended to the Gibbs-Duhem equation.
Volume 40, Number 12, December 1963
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649
