102 Prof. W« E. Story on .Partial 



3. For any liquid mixture that satisfies conditions a-g of 

 paragraph 1, we have the generalized Duhem-Margules 

 equation 



n 1 (/lnp 1 + n 2 rflnp2 + W3^1np 3 + . .. + n K d\np K -• 0, (18) 



where "In" denotes "natural logarithm/' that is logarithm 

 to the base e— 2*718 . . ., and the differentials refer to any 

 infinitesimal changes in the molar proportions of the com- 

 ponents and the corresponding infinitesimal changes in their 

 partial pressures, at the given temperature. _ This equation 

 was originally given * only for a binary mixture, but the 

 method of proof is applicable to any number of components. 

 This equation may or may not hold for other mixtures, but 

 Me regard it as proved only for mixtures that satisfy con- 

 ditions a— a. 



On account of (2), equation (18) may be written 



x 1 d\np 1 + x % d\np i + .'e i d\np z +... + x K dlnp K = 0, (19) 

 or, by (3), 



e. d\n& +x 2 dln& + x s dln& + . . . +w mm . l d\n?s=l +dlnp = 0, 

 P K P< Pk P K 



which is a convenient form to use when x l} x 2 , .r 3 , . . ., x _ t 

 are taken for the independent variables. This equation, being 

 a homogeneous linear differential equation in the k functions 

 Pu P21 Pzi ' • -> P K °f K ~~ 1 independent variables, is equivalent 

 to a system of k — 1 homogeneous linear partial differential 

 equations in these functions and therefore suffices, with the 

 conditions (10) and (17), to determine all the k functions 

 when one of them is known. But we shall find it more 

 convenient to derive them all from another function to be 

 determined by experiment. 



Because p s = for x s = 0, by (10), there exists a definite 



positive power of x g} say ./*, — where e g is a positive integer 



or fraction, — such that p s : x € s s is neither nor infinite for 

 t v s — ; so that, by conditions c and d, this ratio is neither 



* Duhem, Ami. de VEcole normale sup. (3) vol. iv. p. 9 (1887) ; "Disso- 

 lutions et Mesures, 3 m£m., Les melanges doubles," 7rav. et Mem. de la 

 Faculte de Lille, iii. d (1894) : Traite elementaire de mecanique chimique, 

 vol. iv. book 8, chap. 7 (1899). 



Margules, Sitzungsber. der Wiener Akad. vol. civ. p. 1243 (1895). 



See also Ostwald, Allgemeine Chemie (2 Aufl.), p. 636 ff., and JNernst, 

 Theoreti>che Chemie (2 Aufl.), p. 118. 



A simple deduction of the equation by Luther is given in Ostwald's 

 work, p. 639. 



