H 



THE GRAPHICAL REPRESENTATION OF EQUI- 

 LIBRIA IN BINARY SYSTEMS BY MEANS OF 

 THE ZETA (FREE ENERGY) FUNCTION 



[Gibbs, I, pp. 115-129] 



F. A. H. SCHREINEMAKERS 



I. Introduction 



1. In the section entitled Geometrical Illustrations (pp. 115- 

 129 of the "EquiUbrium of Heterogeneous Substances") Gibbs 

 indicates how a general geometrical treatment of phase equilib- 

 ria can be based on the properties of the thermodynamic func- 

 tions. A full account of this geometrical method and its sub- 

 sequent developments would require an exposition of the whole 

 subject of generalised graphical thermodynamics. Since such a 

 comprehensive treatment is not possible in this Commentary, 

 it is hoped that the following discussion of certain equilibria in 

 two-component (binary) systems will serve to illustrate and 

 explain the important geometrical method initiated by Gibbs, 

 and introduce the student to the study of graphical thermo- 

 dynamics based on the properties of the free energy function ^. 



II. The r-x Diagram and the f -Curve (Free Energy Curve) 



2. Let us represent the composition of a phase containing the 

 two components W and X thus: x mols X + (1 — x) mols W. 

 We shall call this quantity, which contains in toto 1 mol, the 

 unit quantity of the phase. Then m unit quantities of the phase 

 contain mx mols X and m{l — x) mols W. Now the f-value of a 

 phase is determined by its temperature t, its pressure p, its 

 composition x, and its quantity m (units). Unless mentioned 

 otherwise, however, we shall mean by the f of a phase the ^ of 

 unit quantity of this phase. The f of w units will then be m^, 

 provided that the total energy, total entropy and total volume 



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