Electrical mobility

Mobility in gas phase

Mobility is defined for any species in the gas phase, encountered mostly in plasma physics and is defined as $${\displaystyle \mu =\frac{q}{m{\nu }_{\text{m}}},}$$ where

• ${\displaystyle q}$ is the charge of the species,
• ${\displaystyle {\nu }_{\text{m}}}$ is the momentum-transfer collision frequency,
• ${\displaystyle m}$ is the mass.

Mobility is related to the species' diffusion coefficient ${\displaystyle D}$ through an exact (thermodynamically required) equation known as the Einstein relation: $${\displaystyle \mu =\frac{q}{kT}D,}$$ where

• ${\displaystyle k}$ is the Boltzmann constant,
• ${\displaystyle T}$ is the gas temperature,
• ${\displaystyle D}$ is the diffusion coefficient.

If one defines the mean free path in terms of momentum transfer, then one gets for the diffusion coefficient $${\displaystyle D=\frac{\pi }{8}{\lambda }^2{\nu }_{\text{m}}.}$$

But both the momentum-transfer mean free path and the momentum-transfer collision frequency are difficult to calculate. Many other mean free paths can be defined. In the gas phase, ${\displaystyle \lambda }$ is often defined as the diffusional mean free path, by assuming that a simple approximate relation is exact: $${\displaystyle D=\frac{1}{2}\lambda v,}$$ where ${\displaystyle v}$ is the root mean square speed of the gas molecules: $${\displaystyle v=\sqrt{\frac{3kT}{m}},}$$ where ${\displaystyle m}$ is the mass of the diffusing species. This approximate equation becomes exact when used to define the diffusional mean free path.

Applications

Electrical mobility is the basis for electrostatic precipitation, used to remove particles from exhaust gases on an industrial scale. The particles are given a charge by exposing them to ions from an electrical discharge in the presence of a strong field. The particles acquire an electrical mobility and are driven by the field to a collecting electrode.

Instruments exist which select particles with a narrow range of electrical mobility, or particles with electrical mobility larger than a predefined value. The former are generally referred to as "differential mobility analyzers". The selected mobility is often identified with the diameter of a singly charged spherical particle, thus the "electrical-mobility diameter" becomes a characteristic of the particle, regardless of whether it is actually spherical.

Passing particles of the selected mobility to a detector such as a condensation particle counter allows the number concentration of particles with the currently selected mobility to be measured. By varying the selected mobility over time, mobility vs concentration data may be obtained. This technique is applied in scanning mobility particle sizers.