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

different kinds of gas, and by $V$ as before the total volume, the increase of entropy may be written in the form

$\sum _{1}(a_{1}m_{1})\log V-\sum _{1}(m_{1}a_{1}\log v_{1}).$

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

$r_{1}=Cm_{1}a_{1},$$r_{2}=Cm_{2}a_{2},$ etc.,

where $C$ denotes a constant. Hence

$v_{1}:V::m_{1}a_{1}:\sum _{1}(m_{1}a_{1})::r_{1}:\sum _{1}r_{1};$

and the increase of entropy may be written

${\frac {\sum _{1}r_{1}\log \sum _{1}r_{1}-\sum _{1}(r_{1}\log r_{1})}{C}}\cdot$

(298)

The Phases of Dissipated Energy of an Ideal Gas-mixture with Components which are Chemically Related.

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

${\mathfrak {G}}_{S}=\lambda _{1}{\mathfrak {G}}_{1}+\lambda _{2}{\mathfrak {G}}_{2},$

(299)

$\lambda _{1}$ 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),

$\mu _{1}\delta m_{1}+\mu _{2}\delta m_{2}+\mu _{3}\delta m_{3}\geqq 0$

(300)

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

$\lambda _{1}\mu _{1}+\lambda _{2}\mu _{2}=\mu _{3}.$

(301)

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 $t$,

$\lambda _{1}a_{1}\log {\frac {m_{1}}{v}}+\lambda _{2}a_{2}\log {\frac {m_{2}}{v}}-a_{3}\log {\frac {m_{3}}{v}}=A+B\log t-{\frac {C}{t}},$

(302)

where

$A=\lambda _{1}H_{1}+\lambda _{2}H_{2}-H_{S}-\lambda _{1}c_{1}-\lambda _{2}c_{2}+c_{3}-\lambda _{1}a_{1}-\lambda _{2}a_{2}+a_{3},$

(303)

$B=\lambda _{1}c_{1}+\lambda _{2}c_{2}-c_{3},$

(304)

$C=\lambda _{1}E_{1}+\lambda _{2}E_{2}-E_{3}.$

(305)