Page:Scientific Papers of Josiah Willard Gibbs.djvu/192

will also be desirable to consider more rigorously and more in detail the equilibrium of such a gas-mixture with solids and liquids, with respect to the above rule.

By differentiation and comparison with (98) we obtain

${\frac {\eta }{v}}=\sum _{1}\left(\left(c_{1}+a_{1}-{\frac {\mu _{1}-E_{1}}{t}}\right)e^{\frac {H_{1}-c_{1}-a_{1}}{a_{1}}}t^{\frac {c_{1}}{a_{1}}}e^{\frac {\mu _{1}-E_{1}}{a_{1}t}}\right),$

(274)

${\frac {m_{1}}{v}}=e^{\frac {H_{1}-c_{1}-a_{1}}{a_{1}}}t^{\frac {c_{1}}{a_{1}}}e^{\frac {\mu _{1}-E_{1}}{a_{1}t}},$	$\scriptstyle {\left.{\begin{matrix}\ \\\\\ \\\ \\\ \\\ \ \end{matrix}}\right\}\,}$ (275)
${\frac {m_{2}}{v}}=e^{\frac {H_{2}-c_{2}-a_{2}}{a_{2}}}t^{\frac {c_{2}}{a_{2}}}e^{\frac {\mu _{2}-E_{2}}{a_{2}t}},$
etc.

Equations (275) indicate that the relation between the temperature, the density of any component, and the potential for that component, is not affected by the presence of the other components. They may also be written

$\mu _{1}=E_{1}+t\left(c_{1}+a_{1}-H_{1}-c_{1}\log t+a_{1}\log {\frac {m_{1}}{v}}\right),$	$\scriptstyle {\left.{\begin{matrix}\ \\\\\ \\\ \ \end{matrix}}\right\}\,}$ (276)
etc.

Eliminating $\mu _{1},\mu _{2}$ etc. from (273) and (274) by means of (275) and (276), we obtain

$p=\sum _{1}{\frac {a_{1}m_{1}t}{v}},$

(277)

$\eta =\sum _{1}\left(m_{1}H_{1}+m_{1}c_{1}\log t+m_{1}a_{1}\log {\frac {v}{m_{1}}}\right)\cdot$

(278)

Equation (277) expresses the familiar principle that the pressure in a gas-mixture is equal to the sum of the pressures which the component gases would possess if existing separately with the same volume at the same temperature. Equation (278) expresses a similar principle in regard to the entropy of the gas-mixture.

From (276) and (277) we may easily obtain the fundamental equation between $\psi ,t,v,m_{1},m_{2},$ etc. For by substituting in (94) the values of $p,\mu _{1},\mu _{2},$ etc. taken from these equations, we obtain

$\psi =\sum _{1}\left(E_{1}m_{1}+m_{1}t\left(c_{1}-H_{1}-c_{1}\log t+a_{1}\log {\frac {m_{1}}{v}}\right)\right)\cdot$

(279)

If we regard the proportion of the various components as constant, this equation may be simplified by writing

$m$ for $\sum _{1}m_{1},$

$cm$ for $\sum _{1}(c_{1}m_{1}),$

$am$ for $\sum _{1}(a_{1}m_{1}),$

$Em$ for $\sum _{1}(c_{1}m_{1}),$

$m$ for $\sum _{1}(E_{1}m_{1}),$

and $Hm-am\log m$ for $\sum _{1}(H_{1}m_{1}-a_{1}m_{1}\log m_{1}).$