Page:Scientific Papers of Josiah Willard Gibbs.djvu/451

quantities of the several kinds of molecules so regulated, that the pressures at all the diaphragms shall have the same two values.

It is evident that the vertical distance between successive connections must be everywhere the same, say $l$; also, that at all the diaphragms, on the side of the greater pressure, the proportion of molecules which can and which cannot pass the diaphragm must be the same. Let the ratio be $1\,:\,n$. If we write $\gamma _{\text{A}},\gamma _{\text{B}}$, etc., for the densities of the several kinds of molecules, and $\gamma$ for the total density, we have for the second cylinder

${\frac {\gamma _{\text{A}}+\gamma _{\text{B}}}{\gamma _{\text{A}}}}=1+n.$

For the third cylinder we have this equation, and also

${\frac {\gamma _{\text{A}}+\gamma _{\text{B}}+\gamma _{\text{C}}}{\gamma _{\text{A}}+\gamma _{\text{B}}}}=1+n,$

which gives

${\frac {\gamma _{\text{A}}+\gamma _{\text{B}}+\gamma _{\text{C}}}{\gamma _{\text{A}}}}=(1+n)^{2}.$

In this way, we have for the rth cylinder

${\frac {\gamma }{\gamma _{\text{A}}}}=(1+n)^{r-1}.$

Now the vertical distance between equal pressures in the first and rth cylinders, is

$(r-1)l.$

Now the equilibrium will not be destroyed if we connect all the cylinders with the first through diaphragms impermeable to all except A-molecules. And the last equation shows that as $\gamma /\gamma _{\text{A}}$ increases geometrically, the vertical distance between any pressure in the column when this ratio of densities is found, and the same pressure in the first cylinder increases arithmetically. This distance, therefore, may be represented by $\log(\gamma /\gamma _{\text{A}})$ multiplied by a constant. This is identical with our result for a volatile liquid, except that for that case we found the value of the constant to be $at/g$.

The following demonstration of van't Hoff's law, which is intended to apply to existing substances, requires only that the solutum, i.e., dissolved substance, should be capable of the ideal gaseous state, and that its molecules, as they occur in the gas, should not be broken up in the solution, nor united to one another in more complex molecules.

It will be convenient to use certain quantities which may be called the potentials of the solvent and of the solutum, the term being thus defined: In any sensibly homogeneous mass, the potential of any independently variable component substance is the differential coefficient of the thermodynamic energy of the mass taken with respect