J. W. (rlhhH — J^quilibriuin. of Hetero(/en€OHS Substances. 2'.\b 



In the first place, if we consider tlic case of a gas-mixture wliich 

 only diifers from an ordinary ideal gas-mixture for which some of 

 the components are equivalent in that there is perfect freedom 

 in regard to the ti'ansformation of these components, it follows at 

 once from the general formula of equilibrium (l) or (2) that equili- 

 brium is only possible for such phases as we have called phases of 

 dissipated energy, for which some of the characteristic equations have 

 been deduced in the preceding pages. 



If it should be urged, that regarding a gas-mixture which has 

 convertible components as an ideal gas-mixture of which, for some 

 reason, only a part of the phases are actually capable of existing, we 

 might still suppose the particular phases which alone can exist to be 

 determined by some other principle than that of the free convertibility 

 of the components (as if, perhaps, the case were analogous to one 

 of constraint in mechanics), it may easily be shown that such a hypo- 

 thesis is entirely untenable, when the quantities of the proximate 

 components may be varied independently by suitable variations of the 

 temperature and pressure, and of the quantities of the ultimate com- 

 ponents, and it is admitted that 'the relations between the energy, 

 entropy, volume, temperature, pi-essure, and the quantities of the 

 several proximate components in the gas-mixture are the same as for 

 an ordinary ideal gas-mixture, in which the components are not con- 

 vertible. Let us denote the quantities of the n' proximate compo- 

 nents of a gas-mixture A by m^, m^, etc., and the quantities of its n 

 ultimate components by nii, nio, etc. {n denoting a number less than 

 w'), and let us suppose that for this gas-mixture the quantities £, ?/, u, 

 «, /J, >«j, ^2, etc. satisfy the relations characteristic of an ideal gas- 

 mixture, while the phase of the gas-mixture is entirely determined by 

 the values of m-i, mg, etc., with two of the quantities f, 7, w, (?,/). 

 We may evidently imagine such an ideal gas-mixture B having n' 

 components (not convertible), that every phase of A shall correspond 

 yfMh one of B in the values of £, 7, v, t, p, m j , mg , etc. Now let us give 

 to the quantities mj, mg, etc. in the gas-mixture A any fixed values, 

 and for the body thus defined let us imagine the v-7]-e surfiice (see 

 page 1 74) constructed ; likewise for the ideal gas-mixture B let us 

 imagine the v-i]-£ surface constructed for every set of values of 

 m^, m^, etc, which is consistent with the given values of m^, ixi^-, 

 etc. i. e., for every body of which the ultimate composition would be 

 expressed by the given values of m , , mg , etc. It follows immediately 

 from our supposition, that every point in the v-f]-£ surface relating to 

 A must coincide with some point of one of the v-if-e surfaces relating 



