Page:Scientific Papers of Josiah Willard Gibbs.djvu/466

the cation (₁) has the charge $c_{1}$, the force necessary to prevent its migration would be

${\frac {d\mu _{1}}{dx}}+c_{1}{\frac {dV}{dx}}\cdot$

For an anion (₂) the force would be

${\frac {d\mu _{2}}{dx}}-c_{2}{\frac {dV}{dx}}\cdot$[1]

Now we may suppose that the same ion in different parts of a dilute solution will have velocities proportional to the forces which would be required to prevent its motion. We may therefore write for the velocity of the cation (₁),

$-{\frac {k_{1}}{c_{1}}}\left({\frac {d\mu _{1}}{dx}}+c_{1}{\frac {dV}{dx}}\right),$

and for the flux of the cation (₁),

$\phi _{1}=-{\frac {k_{1}}{c_{1}}}\gamma _{1}\left({\frac {d\mu _{1}}{dx}}+c_{1}{\frac {dV}{dx}}\right)=-At{\frac {k_{1}}{c_{1}M_{1}}}{\frac {d\gamma _{1}}{dx}}-k_{1}{\frac {dV}{dx}}\gamma _{1};$

(3)

for the flux of the anion (₂),

$\phi _{2}=-{\frac {k_{2}}{c_{2}}}\gamma _{2}\left({\frac {d\mu _{2}}{dx}}+c_{2}{\frac {dV}{dx}}\right)=-At{\frac {k_{2}}{c_{2}M_{2}}}{\frac {d\gamma _{2}}{dx}}+k_{2}{\frac {dV}{dx}}\gamma _{2},$

(4)

where $k_{1},k_{2}$ are constants ('migration velocities') depending on the solvent, the temperature, and the ion.^[2] Now whatever the number of ions the flux of electricity is given by the equation

$\phi =\textstyle \sum \displaystyle \pm c_{1}\phi _{1},$[3]

where the upper sign is for cations and the lower for anions, and the summation for all ions. This gives

$\phi =At\textstyle \sum \displaystyle \mp {\frac {k_{1}}{M_{1}}}{\frac {d\gamma _{1}}{dx}}-{\frac {dV}{dx}}\textstyle \sum \displaystyle c_{1}k_{1}\gamma _{1}.$

That is,

$\phi {\frac {dx}{\textstyle \sum \displaystyle c_{1}k_{1}\gamma _{1}}}=At{\frac {\textstyle \sum \displaystyle \mp {\frac {k_{1}}{M_{1}}}d\gamma _{1}}{\textstyle \sum \displaystyle c_{1}k_{1}\gamma _{1}}}-dV.$

The form of this equation shows that since $\phi$ is the current, ${\frac {dx}{\textstyle \sum \displaystyle c_{1}k_{1}\gamma _{1}}}$ is the "resistance" of an elementary slice of the cell, and the next term the (internal) electromotive force of that slice.

1. [$c_{2}$ is a positive number equal numerically to the negative charge on unit mass of the anion.]
2. [The positive direction for both these fluxes is the direction of increasing $x$.]
3. [The sign of the charge is not included in $c$. Honce the double sign is necessary.]