Page:Scientific Papers of Josiah Willard Gibbs.djvu/459

case the law of Henry and eq. [4] do not hold, it may be on account of a difference in the molecular weight in the gas and the liquid, and that the eq. [4] may still hold if we give the proper value to $M$ in that equation, viz., the molecular weight in the liquid.

But as these considerations, although natural, fall somewhat short of a rigorous demonstration, let us scrutinize the case more carefully. It is easy to give an a priori demonstration of Henry's law and equation [4] in cases in which there is only one molecular formula for the solutum in liquid and in gas, so long as the density both in liquid and in gas is so small that we may neglect the mutual action of the molecules of the solutum. In such a case the molecules of the solutum will be divided between the liquid and the gas in a (sensibly) constant ratio (the volume of the liquid and gas being kept constant), simply because every molecule, moving as if there were no others, would spend the same part of its time in the vapor and in the liquid as if the others were absent, and the number of the molecules being large, this would make the division sensibly constant. This proof will apply in cases in which the law of Henry can hardly be experimentally demonstrated, because the density of the solutum as gas is so small as to escape our power of measurement. Also in cases in which a semi-permeable diaphragm is necessary, an arrangement very convenient for theoretical demonstrations, but imperfectly realizable in practice. (Also in cases in [which a] difference of level is necessary, with or without diaphragm.) But in every case when the law of Henry is demonstrably untrue for dilute solutions, we may be sure that there is more than one value of the molecular weight of the solutum in the phases considered.

This theoretical proof will apply to cases in which experimental proof is impossible:

(1) When the density in gas is too small to measure.

(2) When the density in gas is too great, either the total density or the partial. (Diaphragm or vertical column.)

(3) When the liquid (or other phase) is sensitive to pressure and not in equilibrium with the gas.

Will the various theorems exist in these cases?

If one or both appear in a larger molecular form, the densities of $\gamma _{\text{M}}$ and $\gamma _{\text{M}}'$^[1] are proportional and

$\gamma _{2{\text{M}}}\propto \gamma _{\text{M}}^{2},$$\gamma _{\text{2M}}'\propto \gamma _{\text{M}}^{'2},$	$\scriptstyle {\left.{\begin{matrix}\ \\\\\ \\\ \ \end{matrix}}\right\}\,}$ [12]
$\gamma _{\text{2M}}=\gamma _{\text{M}}=\gamma _{\text{M}}'=\gamma _{\text{2M}}',$

hence one equation of form, $\mu _{\text{M}}={\frac {At}{M}}\log \gamma _{\text{M}}$ proves all.

1. [$\gamma$ refers to the liquid, and $\gamma '$ to the gaseous phase.]