96 EQUILIBRIUM OF HETEROGENEOUS SUBSTANCES. 



substance. For we may imagine the substance brought from the state 

 in which \{s = and the temperature is the same as that of the given 

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

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

 process the work expended is evidently represented by the value of 

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

 denoted by *}/, and let JUL 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 aforesaid ; then 



fj. = ^'+W. (123) 



Now as the state of the substance for which e = and j/ = is 

 arbitrary, we may simultaneously increase the energies of the unit 

 of the substance in all possible states by any constant (7, and the 

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

 The value of \]s, or e trj, for any state would then be increased by 

 CtK, t denoting the temperature of the state. Applying this to 

 \fs' 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 CtK, 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 

 different 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 

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

 same n independently variable components is capable of n + 2 r 

 variations of phase. For the temperature, the pressure, and the 

 potentials 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. There- 

 fore, the number of independent variations in the values of these 

 quantities, i.e., the number of independent variations of phase of the 

 system, will be ?i-f 2 r. 



