Page:Scientific Papers of Josiah Willard Gibbs.djvu/456

water-vapor, $\gamma _{\text{gas}}$, to be measured in a space containing only water-vapor and separated from the liquid by a diaphragm permeable to water and not to alcohol, then the above equation would probably be applicable, since then the water-vapor might probably be treated as an ideal gas. The same would be true (mutatis mutandis) of the potential for alcohol in a mixture of alcohol and water containing not more than ${\tfrac {1}{10}}$ of one per cent, of alcohol.^[1]

This law, however, which makes the potential in a liquid depend upon the density of the substance in some other phase is manifestly not convenient for use. We may get over this difficulty most simply by the law of Henry according to which the ratio of the densities of a substance in coexistent liquid and gaseous phases is (in cases to which the law applies) constant. If $\gamma$ be the density in the liquid phase and $\gamma _{\text{gas}}$ in the gas, we have

$\gamma _{\text{gas}}=c\gamma ,$

[3]

and by substitution in equation [2] we have

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

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

[4]

where the function of the temperature has been increased by ${\frac {At}{M}}\log c$.

With this value of the potential, which is manifestly demonstrated only to be used so far as the law of Henry applies, in connection with the general equation (98), ["Equilib. Het. Subs."] viz.,

$dp={\frac {\eta }{v}}dt+{\frac {m_{1}}{v}}d\mu _{1}+{\frac {m_{2}}{v}}d\mu _{2}...+{\frac {m_{n}}{v}}d\mu _{n},$

we may calculate the osmotic pressure, etc., etc., as we shall see more particularly hereafter.

I. Osmotic pressure.

II. Lowering freezing point.

III. Diminishing pressure of other gas.

IIIa. Effect on total pressure.

IIII. Raising boiling point with one pressure.

IIIIa. Raising boiling point with two pressures.

V. Interpolation formula for mixtures of liquids.

In fact, when $\gamma _{\text{D}}$^[2] is small, we have approximately

$\gamma _{\text{D}}d\mu _{\text{D}}={\frac {At}{M_{\text{D}}}}d\gamma _{\text{D}}=Atd{\frac {\eta _{\text{D}}}{v}},$

[5]

1. Also the potentials of water and alcohol in a mixture may be measured in a vertical tube of sufficient height. [See p. 413.]
2. [In the following discussion, $D$ indicates the dissolved substance, or solutum, and $S$ the solvent.]