Consider a simple thermodynamic system which is described by ~* extensive varia- bles, E i, and (n-- 1) intensive variables, I~-. The Gibbs-Duhem relation for ...
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B A Consequence of the Gibbs-Duhem Relation. J. DC~N~N~-DAvrns Department o/ A p p l i e d Mathematics. The University - Hull
(ricevuto il 16 0 t t o b r e 1967)
Consider a simple t h e r m o d y n a m i c system which is described by ~* e x t e n s i v e variables, E i, and ( n - - 1) intensive variables, I~-. The G i b b s - D u h e m relation for this system m a y be written n-1
(1)
E¢ dIj = 0 . j=l
T h e relation n
(2)
~ E~L = o,
where /1 = 1, also holds for the system. The derivation of these two e q u a t i o n s is well-known (~,~). Differentiate (2) w i t h respect to E r, keeping I i , i ~ k, constant to give
where x ~ {I2 . . . . . Ik-1, Ik+l . . . . , I~}. F r o m (1) it is seen t h a t =0 ~Er] x
Hence, using (2), (3) leads to
,~r
[FEdx
(1) H . B . CALLEX: Thermodynamics (New Y o r k , 1960), p. J,7. (3) p . T . LANDSBERG: Thermody~amicswith Quantum-Statistical Illustrations ( N e w Y o r k , 1961), p. 133.
A CONSEQUENCE OF T l I E GIBBS-DUttEM RELATION
181
In this equation, which is true for all r, the I~'s are independent; and so ~Ei~ E~ [ ~ E , ] . -- E,"
(4)
To illustrate the usefulness of this relation, consider a single-component simple system and suppose that P ~ P (V, N, T) and tt ~ t t ( V , N, T). The Maxwell relation
may then be written
By using (4), this becomes
This result is useful when discussing particle number fluctuations in a grand canonical ensemble (a).
The author wishes to t h a n k Prof. P. T. LANDSBERG for his help and encouragement and for many useful discussions.
(s) T. L. HILL: Statistical Mechanics (New York, 1956), p. 105.
