G constant in one step and then treat i t as a variable in the next (Gibbs did), the derivation through Euler's theorem is straightforward and rigorous. which is Euler's theorem. For the special case of first order homogeneous functions, q = 1 and Euler's theorem becomes SG
n -
On,
3G +n,= G
nd6i + Gini + n&
+ Gdn,
(8)
Subtracting equation (2) from (8) gives the GibbsDuhem equation, equation (4), without the onerous restriction of constant partial molal quantities. Many authors appear to be aware of this difficulty and get around it by integrating a t constant (7, and Gi then stating that the equation so obtained is general and by "general differentiation" equation (8) may be obtained. This is a debatable mathematical technique and is confusingto the student While, from the physical aspects of the problem, it may be defensible to hold
VM
The author wishes to thank Prof. K. J Mysels for help in preparation of the manuscript and Mr. A. Lazaridis for suggesting this problem.
bna
which is identical to equation (3). Differentiating equation (3) now gives dG =
Acknowledgment
/ Journal ot Lhemicol tducation
Literature Cited
(1) GIBES,J. W., "The Collected Works of J Willard Gibba," Vol I, Yale University Press, New Haven, 1960, pp. 87-88. (2) DUHEM,P , "Le Potential Thermodynamique et ses Applications " A Hermann. Pans, 1886 (3) GUGGENHEIM, E. A., "Thermodynamics," North-Holland Publishing Company, Amsterdam, 1950, p. 25. (4)REID, C., "Principles of Chemical Thermodynamics," Reinhold Publishing Co., New York, 1960, pp. 151-153. (5) (a) MACDOUGAL, F. A , "Thermodynamics and Chemistry," John W h y and Sons, Inc., New York, 1939 pp 25-29. (b) EPSTEIN,P. S , "Textbook of Thermodynamics," John Wilev and Sons., Inc ,.New York. 1937., DO. .. 102-104. (6) REID,C., o p . c k , Chapter 2. (7) MELLOR, J. W., "Higher Mathematics for Students of Physics and Chemistry," Dover Publications Inc., New York, 1955, pp. 75-76. ~~
