THERMODYNAMIC AL SYSTEM OF GIBBS 147 



shall he zero for the given fluid, and shall he positive for every 

 other phase of the same components, i.e., for every homogeneous hody 

 not identical in nature and state ivith the given fluid {hut composed 

 entirely of [some or all of the substances] Si, Sz, ■ . .»S„), the con- 

 dition of the given fluid will he stable." 



The following proof may be given of this proposition. It is 

 evident that if (201) is positive for every other phase of the 

 components, its value for the whole mass must be positive when 

 the latter is in any other than its given condition. The value 

 of (201) is therefore less when the mass is in the given condition 

 than when it is in any other condition. Since on account of 

 the conditions imposed by the surrounding envelop neither 

 the entropy, volume, or the quantities m^, W2, ...Wnfor the 

 whole mass can change, it follows that the energy in the given 

 condition is less than that in any other condition of the same 

 entropy and volume. The given condition, by (5), is therefore 

 stable. 



Since (201) is zero when applied to the given fluid (i.e., when 

 e is the energy, rj the entropy, v the volume, mi, . . .mn the 

 quantities of the components of the given fluid), it is evident 

 that T is its temperature, P its pressure, and Mi, Mi, . . . Af „ 

 the potentials of its components in the given state. If we wish 

 to test the stability of the fluid with respect to the formation 

 of some other phase we must insert for e, -q, v, mi, etc. the values 

 of the energy, entropy, volume, and masses in a mass of the phase 

 in question (not necessarily at the same temperature and 

 pressure). If there is no other phase of the components for 

 which the quantity so obtained has a positive value the given 

 fluid is stable. 



It has already been shown that the expression (201) repre- 

 sents the reversible work which must be expended in forming a 

 phase of energy e, entropy t], volume v and masses mi, m^,... 

 mn within a medium having the temperature T, pressure P, 

 potentials Mi, Mi, . . . Mn. The condition of stability there- 

 fore amounts to this: the fluid is stable if no other phase can 

 be formed in it without the expenditure of work. 



When the value of the expression (201) is zero for the given 

 fluid and negative for some other phase of the same components 



