Page:Scientific Papers of Josiah Willard Gibbs.djvu/452

to that component, the entropy and volume of the mass and the quantities of its other components remaining constant. The advantage of using such potentials in the theory of semi-permeable diaphragms consists pirtly in the convenient form of the conditions of equilibrium, the potential for any substance to which a diaphragm is freely permeable having the same value on both sides of the diaphragm, and partly in our ability to express van't Hoff's law as a relation between the quantities characterising the state of the solution, without reference to any experimental arrangement (see Transactions of the Connecticut Academy, vol. iii, pp. 116, 138, 148, 194) [this vol., pp. 63, 83, 92, 135].

Let there be three reservoirs, ${\text{R}}',{\text{R}}'',{\text{R}}'''$, of which the first contains the solvent alone, maintained in a constant state of temperature and pressure, the second the solution, and the third the solutum alone. Let ${\text{R}}'$ and ${\text{R}}''$ be connected through a diaphragm freely permeable to the solvent, but impermeable to the solutum, and let ${\text{R}}''$ and ${\text{R}}'''$ be connected through a diaphragm impermeable to the solvent, but freely permeable to the solutum. We have then, if we write $\mu _{1}$ and $\mu _{2}$ for the potentials of the solvent and the solutum, and distinguish by accents quantities relating to the several reservoirs,

$\mu _{1}''=\mu _{1}'={\text{const.,}}$⁠$\mu _{2}''=\mu _{2}'''.$

Now if the quantity of the solutum in the apparatus be varied, the ratio in which it is divided in equilibrium between the reservoirs ${\text{R}}''$ and ${\text{R}}'''$ will be constant, so long as its densities in the two reservoirs, $\gamma _{2}'',\gamma _{2}'''$, are small. For let us suppose that there is only a single molecule of the solutum. It will wander through ${\text{R}}''$ and ${\text{R}}'''$, and in a time sufficiently long the parts of the time spent respectively in ${\text{R}}''$ and ${\text{R}}'''$, which for convenience we may suppose of equal volume, will approach a constant ratio, say $1:{\text{B}}$. Now if we put in the apparatus a considerable number of molecules, they will divide themselves between ${\text{R}}'$ and ${\text{R}}''$ sensibly in the ratio $1:{\text{B}}$, so long as they do not sensibly interfere with one another, i.e., so long as the number of molecules of the solutum which are within the spheres of action of other molecules of the solutum is a negligible part of the whole, both in ${\text{R}}''$ and ${\text{R}}'''$. With this limitation we have, therefore,

$\gamma _{2}'''={\text{B}}\gamma _{2}''.$

Now in ${\text{R}}'''$ let the solutum have the properties of an ideal gas, which give for any constant temperature (ibid. p. 212) [this vol., p. 152]

$\mu _{2}'''=a_{2}t\log \gamma _{2}'''+{\text{C}},$

where $a_{2}$ is the constant of the law of Boyle and Charles, and ${\text{C}}$ another constant. Therefore,

$\mu _{2}''=a_{2}t\log({\text{B}}\gamma _{2}'')+{\text{C}}.$