Page:Scientific Papers of Josiah Willard Gibbs.djvu/460

Let us next consider the case in which the solutum appears with more than one molecular formula in the liquid or gas or both. Now there are two cases, that in which the quantities of the substance with the different molecular formulæ are independently variable, and that in which they are not. In the [first] case there is no question. If, for example, hydrogen appears with the molecular formula H₂O and also in molecules with the molecular formula H₂O, these are to be treated as separate substances, and we have the two equations

$\mu _{{\text{H}}_{2}}={\text{func}}(t)+{\frac {At}{M_{{\text{H}}_{2}}}}\log \gamma _{{\text{H}}_{2}},$

and

$\mu _{{\text{H}}_{2}{\text{O}}}={\text{func}}(t)+{\frac {At}{M_{{\text{H}}_{2}{\text{O}}}}}\log \gamma _{{\text{H}}_{2}{\text{O}}},$

and also if free oxygen is present

$\mu _{{\text{O}}_{2}}={\text{func}}(t)+{\frac {At}{M_{{\text{O}}_{2}}}}\log \gamma _{{\text{O}}_{2}}.$

But when the quantities of the substance associated in the different molecular combinations are not independently variable, then we have the equation

$M_{1}\mu _{1}+M_{2}\mu _{2}=M_{12}\mu _{12},$

[13]

which is exact and certain, and the considerations adduced on $p$. (^[1]), which are not limited to gases, seem to show that in this case the equations of the form (^[2]) all continue to subsist, but we have also the equation of form (^[3]).

It would therefore appear that we may regard the equation

$\mu ={\text{func}}(t)+{\frac {At}{M}}\log \gamma$

as expressing a general law of nature, where the letter $M$ is the molecular weight corresponding to any molecular combination in the liquid and $\gamma$ is the density of the matter which has that molecular formula, provided that the density $\gamma$ is so small that of the molecules which it represents only a negligible fraction at any time are within the spheres of each other's attraction. It goes without saying that the law is approximative, as the last condition can only be satisfied approximately for any finite value of $\gamma$. (Need of verification on account of the unknown $M$.)

[The author's manuscript for the proposed supplement ends, so far at least as a connected treatment is concerned, at this point. The following notes are appended.]

1. [Although left blank in the MS., this probably refers to p. 423.]
2. [Probably equation [12].]
3. [Probably equation [13].]