Page:Scientific Papers of Josiah Willard Gibbs.djvu/208

The equations of the phases of dissipated energy of a binary gas-mixture, the components of which are identical in substance, are comparatively simple in form. In this case the two components have the same potential, and if we write $\beta$ for ${\frac {a_{1}}{a_{2}}}$ (the ratio of the volumes of equal quantities of the two components under the same conditions of temperature and pressure), we shall have

$\log {\frac {m_{1}^{\beta }}{m_{2}v^{\beta -1}}}={\frac {A}{a_{2}}}+{\frac {B}{a_{2}}}\log t-{\frac {C}{a_{2}t}},$

(319)

$\log {\frac {m_{1}^{\beta }p^{\beta -1}}{m_{2}(a_{1}m_{1}+a_{2}m_{2})^{\beta -1}}}={\frac {A}{a_{2}}}+{\frac {B'}{a_{2}}}\log t-{\frac {C}{a_{2}t}};$

(320)

where$A=H_{1}-H_{2}-c_{1}+c_{2}-a_{1}+a_{2},$

(321)

$B=c_{1}-c_{2},$$B'=c_{1}-c_{2}+a_{1}-a_{2},$

(322)

$C=E_{1}-E_{2}.$

(323)

Gas-mixtures with Convertible Components.

The equations of the phases of dissipated energy of ideal gas-mixtures which have components of which some are identical in ultimate analysis to others have an especial interest in relation to the theory of gas-mixtures in which the components are not only thus equivalent, but are actually transformed into each other within the gas-mixture on variations of temperature and pressure, so that quantities of these (proximate) components are entirely determined, at least in any permanent phase of the gas-mixture, by the quantities of a smaller number of ultimate components, with the temperature and pressure. Such gas-mixtures may be distinguished as having convertible components. The very general considerations adduced on pages 138–144, which are not limited in their application to gaseous bodies, suggest the hypothesis that the equations of the phases of dissipated energy of ideal gas-mixtures may apply to such gas-mixtures as have been described. It will, however, be desirable to consider the matter more in detail.

In the first place, if we consider the case of a gas-mixture which only differs from an ordinary ideal gas-mixture for which some of the components are equivalent in that there is perfect freedom in regard to the transformation of these components, it follows at once from the general formula of equilibrium (1) or (2) that equilibrium 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,