Page:A history of the theories of aether and electricity. Whittacker E.T. (1910).pdf/408

the system would be decreased if the piston were to move outwards so as to admit more solvent into the solution; and therefore this movement of the piston would be assisted by a force—the "osmotic pressure of the solution," as it is called.^[1]

Consider, then, the case of a single electrolyte supposed to be perfectly dissociated; its state will be supposed to be the same at all points of any plane at right angles to the axis of x. Let ν denote the valency of the ions, and V the electric potential at any point. Since^[2] the available energy of a given quantity of a substance in very dilute solution depends on the concentration in exactly the same way as the available energy of a given quantity of a perfect gas depends on its density, it follows that the osmotic pressure p for each ion is determined in terms of the concentration and temperature by the equation of state of perfect gases

$Mp=RTc$,

where M denotes the molecular weight of the salt, and c the mass of salt per unit volume.

Consider the cations contained in a parallelepiped at the place x, whose cross-section is of unit area and whose length is dx. The mechanical force acting on them due to the electric field is -(vc/M)dV/dx.dx, and the mechanical force on them due to the osmotic pressure is -dp/dx.dx. If u denote the velocity of drift of the cations in a field of unit electric force, the total amount of charge which would be transferred by cations across unit area in unit time under the influence of the electric forces alone would be -(uνe/M)dV/dx; so, under the influence of both forces, it is

$-{\frac {uvc}{M}}\left({\frac {dV}{dx}}+{\frac {RT}{c\nu }}{\frac {dc}{dx}}\right)$.

Similarly, if v denote the velocity of drift of the anions in a

1. Cf. van't Hoff, Svenska Vet.-Ak. Handlingar xxi (1886), No. 17; Zeitschrift für Phys. Chem. i (1887), p. 481.
2. As follows from the expression obtained, supra, p. 383.