To the Editor:
LETTERS
To the Editor: In "Textbook Errors, 40" (J. CHEM.EDUC., 39, 527 (1962)),D. W . Rogers discusses the derivation of the Gibbs-Duhem equation and asserts that it is "not correct" to integrate the equation so as to obtain the equation G = G,n,
+ &rb
I should like to point out that his assertion is not correct. In equations ( 1 ) and (2), G is an extensive thermodynamic property of a mixture, GI and G2 are partial molal quantities of components 1 and 2 , and nl and nr are the number of moles of components 1 and 2, respectively. Now partial molal quantities are intensive properties and are independent of any actual values of n, and nz, but do depend on the composition of the mixture, that is, upon the ratio n1/n2. It is to be understood that nl/n2 is constant during the integration of equation ( 1 ) . Lewis and Randall put it this way ("Thermodynamics," McGraw-Hill Book Co., Inc., New York, 1923, p. 42): If, therefore, to a given solution we add the several constituents simultaneously, keeping their ratios constant, these partial molal quantities will remain oanstant. We may therefore integrate equation (I), keeping nl, m, . in constant proportions, and find equation (Z)."
Since GI and G2 are intensive, under the conditions stated previous to equation ( I ) , i.e., G = j(nl, n,), they are functions only of n1/n2andimust be constant if a!n2 is constant. As the paper shows, this leads to a trivial result for the Gibhs-Duhem equation,
The significance of the &bbs-Duhem equation is not that n,dG, and ndG2 are zero which, in general, they are not, but that they are equal and opposite. Further reflection shows that the "simple proof" of Euler's theorem given by Mellor is itself not general but can easily be extended to cover the present case. Consider equation (6)
If rr and p can only be zero or one and the sum of a and p must be one for a first degree homogeneous function,
and we see the identity between a,, az and spectively. For the differentiation shown
GI, GZ re-
and
a must not be a functiou of nl or nz. Since we are not willing to allow a to be constant we must show that an identical result to my equation (7) is obtained when a is taken to be a function of ndm. Differentiating equation (6) under the condition that nl/nz is not constant
..
Also Rogers' equation ( 1 ) would not be correct for a system of more than two components.
and
F. E. CONDON THECITYCOLLEGE NEWYORK,NEWYORK
EDITOR'S NOTE: This and the following pages present only s part of the correspondence generated by the article which zppeared in October as "Textbook Errors, 40." We rtpologize to the authors of letters other than those appearing,notably Professor S. D. Christian of Oklahoma, who stressed essentially the same points as those made here. The published letbrs have been abbreviated in the interest of saving apace. Readers should blame any abruptness or seeming lack of transitions on the editnr'h blup pencil, not on the correspondent's lack of style.
by nl and (dG/dn~) by n2 Upon multiplying (dG/bn~) and adding their products, the second and fourth terms drop out and the identity is proven. The second criticism is, "Equation ( 1 ) would not he correct for a system of more than two components." Please note the sentence before equation ( 1 ) which begins, "For simplicity, the equation will be worked out for a system of two compositionvariahles, G = j(m, nz)". . .. The continuation of the same sentence ". . . that is, T, P , n8,nl . . . n, are constant." leads t o a digression wh~chmay be worth mentioning. If na,nn . . . n, are zero, the system is binary and equation 1 holds. If na,n4 . . . n, are not zero, but constant, equation 1 still holds because it refers to an in$nitesimal change in n, and n2. If there is finite change in nl and nz which takes place in a finite system (not infinitely large) then the ratios of na,n4 . . n, to the total number of moles
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Volume 40, Number 4, April 1963
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225
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
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Journd of Chemical Education
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