Page:Scientific Papers of Josiah Willard Gibbs.djvu/205

If we denote by $\beta _{1}$ and $\beta _{2}$ the volumes (determined under standard conditions of temperature and pressure) of the quantities of the gases $G_{1}$ and $G_{2}$ which are contained in a unit of volume of the gas $G_{3}$, we shall have

$\beta _{1}={\frac {\lambda _{1}a_{1}}{a_{3}}},$⁠and⁠$\beta _{2}={\frac {\lambda _{2}a_{2}}{a_{3}}},$

(306)

and (302) will reduce to the form

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

(307)

Moreover, as by (277)

$pv=(a_{1}m_{1}+a_{2}m_{2}+a_{3}m_{3})t,$

(308)

we have on eliminating $v$

$\log {\frac {m_{1}^{\beta _{1}}m_{2}^{\beta _{2}}p^{\beta _{1}+\beta _{2}-1}}{m_{3}(a_{1}m_{1}+a_{2}m_{2}+a_{3}m_{3})^{\beta _{1}+\beta _{2}-1}}}={\frac {A}{a_{3}}}+{\frac {B'}{a_{3}}}\log t-{\frac {C}{a_{3}t}},$

(309)

where⁠$B'=\lambda _{1}c_{1}+\lambda _{2}c_{2}-c_{3}+\lambda _{1}a_{1}+\lambda _{2}a_{2}-a_{3}.$

(310)

It will be observed that the quantities $\beta _{1},\beta _{2}$ will always be positive and have a simple relation to unity, and that the value of $\beta _{1}+\beta _{2}-1$ will be positive or zero, according as gas $G_{3}$ is formed of $G_{1}$ and $G_{2}$ with or without condensation. If we should assume, according to the rule often given for the specific heat of compound gases, that the thermal capacity at constant volume of any quantity of the gas $G_{3}$ is equal to the sum of the thermal capacities of the quantities which it contains of the gases $G_{1}$ and $G_{2}$, the value of $B$ would be zero. The heat evolved in the formation of a unit of the gas $G_{3}$ out of the gases $G_{1}$ and $G_{2}$, without mechanical action, is by (283) and (257)

$\lambda _{1}(c_{1}t+E_{1})+\lambda _{2}(c_{2}t+E_{2})-(c_{3}t+E_{3}),$

or⁠$Bt+C,$

which will reduce to $C$ when the above relation in regard to the specific heats is satisfied. In any case the quantity of heat thus evolved divided by a $a_{3}t^{2}$ will be equal to the differential coefficient of the second member of equation (307) with respect to $t$. Moreover, the heat evolved in the formation of a unit of the gas $G_{3}$ out of the gases $G_{1}$ and $G_{2}$ under constant pressure is which is equal to the differential coefficient of the second member of (309) with respect to $t$, multiplied by $a_{3}t^{2}$.

It appears by (307) that, except in the case when $\beta _{1}+\beta _{2}=1$, for any given finite values of $m_{1},m_{2},m_{3},$ and $t$ (infinitesimal values being excluded as well as infinite), it will always be possible to assign such a finite value to $v$ that the mixture shall be in a state