236 J. W. (J^ibbs — Equilibriuiii. of Ileterof/entous Substances. 



to £ not only in respect to position but also in respect to its tangent 

 plane (which represents temperature and pressure) ; therefore the 

 «-//-£ surface relating to A must be tangent to the varioiis v-r,-e sur- 

 faces relating to B, and therefore must be ai] envelop of these sur- 

 faces. P'roni this it follows that the points which represent phases 

 common to both gas-mixtures must represent the phases of dissipated 

 energy of the gas-mixture B. 



The properties of an ideal gas-mixture which are assumed in 

 regard to the gas-mixture of converti1)le components in the above 

 demonstration are expressed by equations (277) and (278) with the 

 equation 



e:=:£A'\>",f-^"'iE,). (824) 



It is usual to assume in regard to gas-mixtures liaving convertible 

 components that the convertibility of the components does not affect 

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

 tion (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 compo- 

 nents 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 Avill 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 d )} —p dv - 2j (c^m^) dt] „. = 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 cal- 

 culated by the same formula as if the components were not conver- 

 tible. 



Let us suppose the thermodynamic state of a gaseous mass J/, of 

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

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

 tities of the proximate components, therefore, as well as of the ulti- 



* This notation is intended to indicate that ?;i|, m.^, etc. are regarded as constant 



