152 J.W. Gibhs — Equilibrium of Heterogeneous Substances. 



mass, first to any specified state of the same temperature, and then 

 into combination with the given mass. In the first part of the pro- 

 cess the work expended is evidently represented by the value of y.' 

 for the unit of the substance in the state specified. Let this be 

 denoted by </'', and let /< denote the potential in question, and W the 

 work expended in bringing a unit of the substance from the specified 

 state into combination with the given mass as afoi-esaid ; then 



lx=ip'-^W. (123) 



Now as the state of the substance for which 6=0 and ?/ = is 

 arbitrary, we may simultaneously inci-ease the energies of the unit of 

 the substance in all possible states by any constant C, and the 

 entropies of the substance in all possible states by any constant K. 

 The value of //•, or £ — t //, for any state would then be increased by 

 C -^ t K, t denoting the temperature of the state. Applying this 

 to if:' in (123) and observing that the last term in this equation is 

 independent of the values of these constants, we see that the potential 

 would be increased by the same quantity C — t K, t being the tem- 

 perature of the mass in which the potential is to be determined. 



ON COEXISTENT PHASES OF MATTER. 



In considering the different homogeneous bodies which can be 

 formed out of any set of component substances, it will be convenient 

 to have a term which shall refer solely to the composition and ther- 

 modynamic state of any such body without regard to its quantity or 

 form. We may call such bodies as differ in composition or state dif- 

 ferent phases of the matter considered, regarding all bodies which 

 differ only in quantity and form as different examples of the same 

 phase. Phases which can exist together, the dividing surfaces being 

 plane, in an equilibrium which does not depend upon passive resist- 

 ances to change, we shall call coexistent. 



If a homogeneous body has n independently variable components, 

 the phase of the body is evidently capable of n. -|- 1 independent vari- 

 ations. A system of r coexistent phases, each of which has the same 

 n independently variable components is capable of « + 2 — r varia- 

 tions of phase. For the temperature, the pressure, and the poten- 

 tials for the actual components have the same values in the different 

 phases, and the variations of these quantities are by (97) subject to 

 as many conditions as there are different phases. Therefore, the num- 

 ber of independent variations in the values of these quantities, i. e., 

 the number of independent variations of phase of the system, will be 

 n+2 -r. 



