Page:Scientific Papers of Josiah Willard Gibbs.djvu/204

168 different kinds of gas, and by $$V$$ as before the total volume, the increase of entropy may be written in the form And if we denote by $$r_{1}, r_{2}$$, etc., the numbers of the molecules of the several different kinds of gas, we shall have  where $$C$$ denotes a constant. Hence and the increase of entropy may be written

We will now pass to the consideration of the phases of dissipated energy (see page 140) of an ideal gas-mixture, in which the number of the proximate components exceeds that of the ultimate.

Let us first suppose that an ideal gas-mixture has for proximate components the gases $$G_{1}, G_{2}$$, and $$G_{3}$$, the units of which are denoted by $$\mathfrak{G}_{1}, \mathfrak{G}_{2}$$ and $$\mathfrak{G}_{3}$$that in ultimate analysis $$\lambda_{1}$$ and $$\lambda_{2}$$ denoting positive constants, such that $$\lambda_{1} + \lambda_{2} = 1$$. The phases which we are to consider are those for which the energy of the gas-mixture is a minimum for constant entropy and volume and constant quantities of $$G_{1}$$ and $$G_{2}$$, as determined in ultimate analysis. For such phases, by (86), for such values of the variations as do not affect the quantities of $$G_{1}$$ and $$G_{2}$$ as determined in ultimate analysis. Values of $$\delta m_{1}, \delta m_{2}, \delta m_{3}$$ proportional to $$\lambda_{1}, \lambda_{2}, -1$$ and only such, are evidently consistent with this restriction: therefore If we substitute in this equation values of $$\mu_{1}, \mu_{2}, \mu_{3}$$ taken from (276), we obtain, after arranging the terms and dividing by $$t$$,  where