Page:Scientific Papers of Josiah Willard Gibbs.djvu/219

From this equation, by differentiation and comparison with (98), we obtain

${\frac {\eta }{v}}=L_{1}\left(c_{1}+a_{1}-{\frac {\mu -E_{1}}{t}}\right)t^{\frac {c_{1}}{a_{1}}}e^{\frac {\mu -E_{1}}{a_{1}t}}+L_{2}\left(c_{2}+a_{2}-{\frac {\mu -E_{2}}{t}}\right)t^{\frac {c_{2}}{a_{1}}}e^{\frac {\mu -E_{2}}{a_{2}t}},$

(344)

${\frac {m}{v}}=L_{1}t^{\frac {c_{1}}{a_{1}}}e^{\frac {\mu -E_{1}}{a_{1}t}}+L_{2}t^{\frac {c_{2}}{a_{2}}}e^{\frac {\mu -E_{2}}{a_{2}t}}.$

(345)

From the general equation (93) with the preceding equations the following is easily obtained,—

${\frac {\epsilon }{v}}=L_{1}(c_{1}t+E_{1})t^{\frac {c_{1}}{a_{1}}}e^{\frac {\mu -E_{1}}{a_{1}t}}+L_{2}(c_{2}t+E_{2})t^{\frac {c_{2}}{a_{2}}}e^{\frac {\mu -E_{2}}{a_{2}t}}.$

(346)

We may obtain the relation between $p,t,v,$, and $m$ by eliminating $\mu$ from (342) and (345). For this purpose we may proceed as follows. From (342) and (345) we obtain

$p-a_{2}t{\frac {m}{v}}=(a_{1}-a_{2})L_{1}t^{\frac {c_{1}+a_{1}}{a_{1}}}e^{\frac {\mu -E_{1}}{a_{1}t}},$

(347)

$a_{1}t{\frac {m}{v}}-p=(a_{1}-a_{2})L_{2}t^{\frac {c_{2}+a_{2}}{a_{2}}}e^{\frac {\mu -E_{2}}{a_{2}t}};$

(348)

and from these equations we obtain

$a_{1}\log(p-a_{2}t{\frac {m}{v}})-a_{2}\log(a_{1}t{\frac {m}{v}}-p)=(a_{1}-a_{2})\log(a_{1}-a_{2})+a_{1}\log L_{1}-a_{2}\log L_{2}+(c_{1}-c_{2}+a_{1}-a_{2})\log t-{\frac {E_{1}-E_{2}}{t}}\cdot$

(349)

(In the particular case when $a_{1}=2a_{2}$ this equation will be equivalent to (333).) By (347) and (348) we may easily eliminate $\mu$ from (346).

The reader will observe that the relations thus deduced from the fundamental equation (342) without any reference to the different components of the gaseous mass are equivalent to those which relate to the phases of dissipated energy of a binary gas-mixture with components which are equivalent in substance but not convertible, except that the equations derived from (342) do not give the quantities of the proximate components, but relate solely to those properties which are capable of direct experimental verification without the aid of any theory of the constitution of the gaseous mass.

The practical application of these equations is rendered more simple by the fact that the ratio $a_{1}:a_{2}$ will always bear a simple relation to unity. When $a_{1}$ and $a_{2}$ are equal, if we write a for their common value, we shall have by (342) and (345)

$pv=amt,$

(350)