Page:Scientific Papers of Josiah Willard Gibbs.djvu/449

VIII.

SEMI-PERMEABLE FILMS AND OSMOTIC PRESSURE.

[Nature, vol. lv, pp. 461, 462, Mar. 18, 1897.]

Lord Kelvin's very interesting problem concerning molecules which differ only in their power of passing a diaphragm (see Nature for January 21, p. 272), seems only to require for its solution the relation between density and pressure for the fluid at the temperature of the experiment, when this relation for small densities becomes that of an ideal gas; in other cases, a single numerical constant in addition to the relation between density and pressure is sufficient.

This will, perhaps, appear most readily if we imagine each of the vessels ${\text{A}}$ and ${\text{B}}$ connected with a vertical column of the fluid which it contains, these columns extending upwards until the state of an ideal gas is reached. The equilibrium which we suppose to subsist will not be disturbed by communications between the columns at as many levels as we choose, if these communications are always made through the same kind of semi-permeable diaphragm as that which separates the vessels ${\text{A}}$ and ${\text{B}}$. It will be observed that the difference of level at which any same pressure is found in the two columns is a constant quantity, easily determined in the upper parts (where the fluids are in the ideal gaseous state) as a function of the composition of the fluid in the A-column, and giving at once the height above the vessel ${\text{A}}$, where in the A-column we find a pressure equal to that in the vessel ${\text{B}}$.

In fact, we have in either column

$dp=-g\gamma dz,$

where the letters denote respectively pressure, force of gravity, density, and vertical elevation. If we set

${\frac {1}{\gamma }}={\text{F}}'(p)$

we have

${\text{F}}'(p)dp=-gdz.$

Integrating, with a different constant for each column, we get

${\text{F}}(p_{\text{A}})=-g(z-{\text{C}}_{\text{A}}),$

${\text{F}}(p_{\text{B}})=-g(z-{\text{C}}_{\text{B}}).$

${\text{F}}(p_{\text{A}})-{\text{F}}(p_{\text{B}})=g({\text{C}}_{\text{A}}-{\text{C}}_{\text{B}}).$