Page:Scientific Papers of Josiah Willard Gibbs.djvu/210

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 components (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,

$[td\eta -pdv-\sum _{1}(c_{1}m_{1})dt]_{m}=0.$[1]

(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 $v-\eta -\epsilon$ surface of an ideal gas-mixture of inconvertible components, the form and position of this surface being determined by the proximate composition 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 $v-\eta -\epsilon$ 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 $v-\eta -\epsilon$ surface of the body $M$, which is here a gas-mixture of con-

1. This notation is intended to indicate that $m_{1},m_{2}$, etc. are regarded as constant.