Page:Scientific Papers of Josiah Willard Gibbs.djvu/453

This equation, in which a single constant may evidently take the place of ${\text{B}}$ and ${\text{C}}$, may be regarded as expressing the property of the solution implied in van't Hoff's law. For we have the general thermodynamic relation (ibid. p. 143) [this vol., p. 88].

$vdp=\eta dt+m_{1}d\mu _{1}+m_{2}d\mu _{2},$

where $v$ and $\eta$ denote the volume and entropy of the mass considered, and $m_{1}$ and $m_{2}$ the quantities of its components. Applied to this case, since $t$ and $\mu _{1}$ are constant, this becomes

$dp''=\gamma _{2}''d\mu _{2}''.$

Substituting the value of $d\mu _{2}''$, derived from the last finite equation, we have

$dp''=a_{2}td\gamma _{2}'',$

whence, integrating from $\gamma _{2}''=0$ and $p''=p'$, we get

$p''-p'=a_{2}t\gamma _{2}'',$

which evidently expresses van't Hoff's law.

We may extend this proof to cases in which the solutum is not volatile by supposing that we give to its molecules mutually repulsive molecular forces, which, however, are entirely inoperative with respect to any other kind of molecules. In this way we may make the solutum capable of the ideal gaseous state. But the relations pertaining to the contents of ${\text{R}}''$ will not be affected by these new forces, since we suppose that only a negligible part of the molecules of the solutum are within the range of such forces. Therefore these relations cannot depend on the new forces, and must exist without them.

To give up the condition that the molecules of the solutum shall not be broken up in the solution, nor united to one another in more complex molecules, would involve the consideration of a good many cases, which it would be difficult to unite in a brief demonstration. The result, however, seems to be that the increase of pressure is to be estimated by Avogadro's law from the number of molecules in the solution which contain any part of the solutum, without reference to the quantity in each.

J. Willard Gibbs.

New Haven, Connecticut, February 18.