change and we may not apply Euler's equation for a total differential. I apologize for the omission of a commn in equation (5) and a superscript a in the derivative below equation (6) in the original paper.
then G is homogeneous of degree q in nland nl. Let and G = G(m, a )
C o m h i i g (1)and (2) F
=
ZgG
Partial differentiation of (3) with respect to Z
To the Editor:
Let ml= Zn, and rnz = Znz, then It was with great interest that I read the symposium. on the "Teaching of Thermodynamics" in the October, 1962 issue. I look forward to reading future symposia of this type. and I n the same issue there appeared a most interesting discussion of the Gibbs-Duhem equation (THISJOURNAL, 39, 527 (1962)) (in its most general form as applied to any extensive function of composition a t constant presBy the calculus, since F = G(ml,m,) sure and temperature). Prof. Rogers in discussing this in terms of textbook errors is really writing about errors of omission rather than errors of commission. I n writing my book on thermodynamics (-3,I was Combining (4) and (6) : faced with the problem of whether to use the approach of Gibbs in deriving this equation or the more rigorous method of MacDougall (Q employing the elegant applications of Euler's theorem on homogeneous functions. Let Z = 1, then F = G, m, = nt and m = a 39,491 As pointed out by Prof. Bent (THISJOURNAL, (1962)) in the very first article of the above-mentioned Combining (7) and (8) symposium, Gibbs had the genius of circumventing mathematical difficulties. Because Euler did his work over a century before Gibbs, it is doubtful that Gibbs was not aware of Euler's theorem, yet he chose to avoid (which is Euler's theorem for two variables). the abstract mathematical approach and adhere to a Let ,GI= (a?/&,) and G2= (bG/anz) then partial difphysical interpretation. Being a concise and sophistiferentiation of (9) yields cated writer, Gibbs' derivation (1) is so simple that it usually escapes comprehension a t the first reading (it took me a t least ten readings to fully appreciate his approach). Since G is a continuous function: I therefore chose to use Gibbs' method, but where 36, Gibbs used one sentence, I ended up with several paraG _ -b2G= - a3c = an, an,& a n 3% (11) c r a ~ h s . His conce~tof "inteaation through the mass'' not only gives us an extremely simple approach to the Combining (10) and (11) derivation of the Gibbs equation of existence for homogrnrons substances in tyuilibrium (Gibbs-Duhern rquntion) but hrter swvesas a basis for drri\.ing hisequations for stable phase conditions (2). and similarly for G2. Therefore, GIand GZ are homoIt is my feelmg that the Euler theorem should not be geneous of degree (q - 1). For q = 1from (9) introduced until the student has obtained a full undernlGl + nzGs = G (13) standing of the physical concepts (either in graduate where work or advanced undergraduate work for exceptional students). In fact, the Lewis and Randall (4) apG(Zn1,Znr) = ZG(n1,nn) proach of deriving the Euler equation for extensive funcfrom (12) tions of state without the use of calculus should be introduced at an early level in the teaching of chemistry. Prof. Rogers cites the clever use of a power series polynominal in the derivation of Euler's theorem by Mellor. A simple derivation suggested by Prof. F. H. Miller in his book on the calculus (7) which avoids this limitation may be given as follows: That is, G is homogeneous of degree 1 and G1 is homogeneous of degree 0. This is another way of saying that If G(Zn,,Zm) = Z*G(nl,m)> (1) -
u
.
226
/
Journd of Chemical Education
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G is proportional to mass at constant composition (extensive) and G,is independent of mass at constant composition being dependent on composition only (intensive). By combining (11) and (14), MacDougall(6) derives the generalized Gihbs-Duhem equation in an alternate manner.
The ahove method is essentially that used by Guggenall texts do not heim' and by G i b b ~ . Admittedly, ~ present this method in a satisfactory form, hut such presentations are reflections upon the texts, and not upon the method. The prmf of Euler's theorem given ahove may be extended t o functions which are homogeneous of positive integral degree. If the degree of homogeneity is two, equation (1) still holds, but the f,'s are now homogeneous of degree one. Hence, using (3), fi
Literature Cited (1) GIBBS,J . W., "The Collected Worka of J. Willard Gihba." Vol. 1, Yale University Press, New Haven, Conn., 1948, n R7~ -. --( 2 ) GIBBS,J. W., ibid., p. 106. (3) GILMONT,R., "Thermodynamic Principles for Chemical Engineers," Prentice-Hall, Inc., Englewood Cliffs, N. J., 1959, p. 1 3 H 3 7 . (4) L ~ w r s , G . N., A N D RANDALL, M., "Thermodynamics," MeGraw-Hill Book Co.. Inc., New York, N. Y., 1923, p. 42. (5) MACD~UGALL, F. H.. "Thermodynamics sad Chemistry," John Wiley and Sans, Inc., New York, N. Y., 1939, p. 2.526. F . H., ibid., p. 28. ( 6 ) MACDOUGALL, ( 7 ) MILLER,F . H., "C&lculu~,"John Wiley and Sons, Inc., New York, N. Y., 1946, p. 153 exercise 27.
To the Editor: The total differential of any fimction f of r variables (xl, . . ., x.) i s df = I:fjdxi
where f,
= I: ( 5
wazi)zi
(4)
Introduction of (4) into (1) gives df =
r
(aji/az,)zidzi
[,J
This may he integrated as before, since the second partial derivatives df&rj are homogeneous of degree zero. The result is f ( z , , . . . ,&I = '/* I: (aj
