Electrical mobility

Electrical mobility is the ability of charged particles (such as electrons or protons) to move through a medium in response to an electric field that is pulling them. The separation of ions according to their mobility in gas phase is called Ion mobility spectrometry, in liquid phase it is called electrophoresis.

When a charged particle in a gas or liquid is acted upon by a uniform electric field, it will be accelerated until it reaches a constant drift velocity according to the formula:

where

In other words, the electrical mobility of the particle is defined as the ratio of the drift velocity to the magnitude of the electric field:

For example, the mobility of the sodium ion (Na⁺) in water at 25 °C is 5.19 × 10⁻⁸ m² V⁻¹ s⁻¹. This means that a sodium ion in an electric field of 1 V/m would have an average drift velocity of 5.19 × 10⁻⁸ m/s. Such values can be obtained from measurements of ionic conductivity in solution.

Electrical mobility is proportional to the net charge of the particle. This was the basis for Robert Millikan's demonstration that electrical charges occur in discrete units, whose magnitude is the charge of the electron.

Electrical mobility is also inversely proportional to the Stokes radius ${\displaystyle a}$ of the ion, which is the effective radius of the moving ion including any molecules of water or other solvent which move with it. This is true because the solvated ion moving at a constant drift velocity ${\displaystyle s}$ is subject to two equal and opposite forces: an electrical force ${\displaystyle zeE}$ and a frictional force ${\displaystyle F_{drag}=fs=(6\pi \eta a)s}$, where ${\displaystyle f}$ is the frictional coefficient, ${\displaystyle \eta }$ is the solution viscosity. For different ions with the same charge such as Li⁺, Na⁺ and K⁺ the electrical forces are equal, so that the drift speed and the mobility are inversely proportional to the radius a. In fact, conductivity measurements show that ionic mobility increases from Li⁺ to Cs⁺ and therefore that Stokes radius decreases from Li⁺ to Cs⁺. This is the opposite of the order of ionic radii for crystals, and shows that in solution the smaller ions (Li⁺) are more extensively hydrated than the larger (Cs⁺).

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