Page:Scientific Papers of Josiah Willard Gibbs.djvu/207

suppositions otherwise the same), we shall have for infinitesimal values of $t$ an infinitesimal value either of $m_{1}$ or $m_{2}$, and for infinite values of $t$ finite or infinitesimal values of $m_{3}$ according as $\beta _{1}+\beta _{2}$ is equal to or greater than unity.

The case which we have considered is that of a ternary gas-mixture, but our results may easily be generalized in this respect. In fact, whatever the number of component gases in a gas-mixture, if there are relations of equivalence in ultimate analysis between these components, such relations may be expressed by one or more equations of the form

$\lambda _{1}{\mathfrak {G}}_{1}+\lambda _{2}{\mathfrak {G}}_{2}+\lambda _{3}{\mathfrak {G}}_{3}+{\text{ etc. }}=0,$

(311)

where ${\mathfrak {G}}_{1},{\mathfrak {G}}_{2}$, etc. denote the units of the various component gases, and $\lambda _{1},\lambda _{2}$, etc. denote positive or negative constants such that $\sum _{1}\lambda _{1}=0$. From (311) with (86) we may derive for phases of dissipated energy,

$\lambda _{1}\mu _{1}+\lambda _{2}\mu _{2}+\lambda _{3}\mu _{3}+{\text{etc.}}=0,$

or$\sum _{1}(\lambda _{1}\mu _{1})=0.$

(312)

Hence, by (276),

$\sum _{1}\left(\lambda _{1}a_{1}\log {\frac {m_{1}}{v}}\right)=A+B\log t-{\frac {C}{t}},$

(313)

where $A,B$ and $C$ are constants determined by the equations

$A=\sum _{1}(\lambda _{1}H_{1}-\lambda _{1}c_{1}-\lambda _{1}a_{1})$

(314)

$B=\sum _{1}(\lambda _{1}c_{1})$

(315)

$C=\sum _{1}(\lambda _{1}E_{1}).$

(316)

Also, since $pv=\sum _{1}(a_{1}m_{1})t,$

$\sum _{1}(\lambda _{1}a_{1}\log m_{1})-\sum (\lambda _{1}a_{1})\log \sum _{1}(a_{1}m_{1})+\sum (\lambda _{1}a_{1})\log p=A+B'\log t-{\frac {C}{t}},$

(317)

where$B=\sum _{1}(\lambda _{1}c_{1}+\lambda _{1}a_{1}).$

(318)

If there is more than one equation of the form (311), we shall have more than one of each of the forms (313) and (317), which will hold true simultaneously for phases of dissipated energy.

It will be observed that the relations necessary for a phase of dissipated energy between the volume and temperature of an ideal gas-mixture, and the quantities of the components which take part in the chemical processes, and the pressure due to these components, are not affected by the presence of neutral gases in the gas-mixture.

From equations (312) and (234) it follows that if there is a phase of dissipated energy at any point in an ideal gas-mixture in equilibrium under the influence of gravity, the whole gas-mixture must consist of such phases.