174 EQUILIBRIUM OF HETEROGENEOUS SUBSTANCES. 



the relations (277) and (324). The same cannot be said of the equation 

 (278). But in a very important class of cases it will be sufficient if 

 the applicability of (277) and (324) is admitted. The cases referred to 

 are those in which in certain phases of a gas-mixture the components 

 are convertible, and in other phases of the same proximate composition 

 the components are not convertible, and the equations of an ideal gas- 

 mixture hold true. 



If there is only a single degree of convertibility between the com- 

 ponents (i.e., if only a single kind of conversion, with its reverse, can 

 take place among the components), it will be sufficient to assume, in 

 regard to the phases in which conversion takes place, the validity of 

 equation (277) and of the following, which can be derived from (324) 

 by differentiation, and comparison with equation (11), which expresses 

 a necessary relation, 



[t drj -p dv - ^fernO dt] M = 0.* (325) 



We shall confine our demonstration to this case. It will be observed 

 that the physical signification of (325) is that if the gas-mixture is 

 subjected to such changes of volume and temperature as do not 

 alter its proximate composition, the heat absorbed or yielded may 

 be calculated by the same formula as if the components were not 

 convertible. 



Let us suppose the thermodynamic state of a gaseous mass M, of 

 such a kind as has just been described, to be varied while within 

 the limits within which the components are not convertible. (The 

 quantities of the proximate components, therefore, as well as of the 

 ultimate, are supposed constant.) If we use the same method of 

 geometrical representation as before, the point representing the volume, 

 entropy, and energy of the mass will describe a line in the v-q-e 

 surface of an ideal gas-mixture of inconvertible components, the form 

 and position of this surface being determined by the proximate com- 

 position of M. Let us now suppose the same mass to be carried 

 beyond the limit of inconvertibility, the variations of state after 

 passing the limit being such as not to alter its proximate composition. 

 It is evident that this will in general be possible. Exceptions can 

 only occur when the limit is formed by phases in which the proximate 

 composition is uniform. The line traced in the region of convertibility 

 must belong to the same v-q-e surface of an ideal gas-mixture of 

 inconvertible components as before, continued beyond the limit 

 of inconvertibility for the components of M, since the variations of 

 volume, entropy, and energy are the same as would be possible if the 

 components were not convertible. But it must also belong to the 

 v-ij-e surface of the body M, which is here a gas-mixture of con- 



* This notation is intended to indicate that 7% , w 2 , etc. are regarded as constant. 



