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{{Other uses|Electric vehicle}}
{{For|the electrical mobility of an [[electron]] or [[electron hole|hole]] in [[solid-state physics]]|Electron mobility}}

'''Electrical mobility''' is the ability of charged particles (such as [[electron]]s or [[proton]]s) 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]].

==Theory==
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
<math display="block">v_\text{d} = \mu E,</math>
where 
* <math>v_\text{d}</math> is the drift velocity ([[SI units]]: m/s),
* <math>E</math> is the magnitude of the applied electric field (V/m),
* <math>\mu</math> is the mobility (m<sup>2</sup>/(V·s)).

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:
<math display="block">\mu = \frac{v_\text{d}}{E}.</math>

For example, the mobility of the sodium ion (Na<sup>+</sup>) in water at 25&nbsp;°C is {{val|5.19|e=-8|u=m<sup>2</sup>/(V·s)}}.<ref>[[Keith J. Laidler]] and John H. Meiser, ''Physical Chemistry'' (Benjamin/Cummings 1982), p. 274. {{ISBN|0-8053-5682-7}}.</ref> This means that a sodium ion in an electric field of 1&nbsp;V/m would have an average drift velocity of {{val|5.19|e=-8|u=m/s}}. Such values can be obtained from measurements of [[Molar conductivity#Molar ionic conductivity|ionic conductivity]] in solution.

Electrical mobility is proportional to the net [[electric charge|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]] <math>a</math> of the ion, which is the effective radius of the moving ion including any molecules of water or other solvent that move with it. This is true because the solvated ion moving at a constant [[drift velocity]] <math>s</math> is subject to two equal and opposite forces: an electrical force <math>zeE</math> and a frictional force <math>F_\text{drag} = fs = (6 \pi \eta a)s</math>, where <math>f</math> is the frictional coefficient, <math>\eta</math> is the solution viscosity. For different ions with the same charge such as Li<sup>+</sup>, Na<sup>+</sup> and K<sup>+</sup> the electrical forces are equal, so that the drift speed and the mobility are inversely proportional to the radius <math>a</math>.<ref name=Atk>{{cite book |last1=Atkins |first1=P. W. |authorlink1=Peter Atkins |last2=de Paula |first2=J. |date=2006 |title=Physical Chemistry |url=https://archive.org/details/atkinsphysicalch00atki |url-access=limited |edition=8th |isbn=0198700725 |publisher=[[Oxford University Press]] |pages=[https://archive.org/details/atkinsphysicalch00atki/page/n796 764]–6}}</ref> In fact, conductivity measurements show that ionic mobility ''increases'' from Li<sup>+</sup> to Cs<sup>+</sup>, and therefore that Stokes radius ''decreases'' from Li<sup>+</sup> to Cs<sup>+</sup>. This is the opposite of the order of [[Ionic radius|ionic radii]] for crystals and shows that in solution the smaller ions (Li<sup>+</sup>) are more extensively [[solvation shell|hydrated]] than the larger (Cs<sup>+</sup>).<ref name=Atk/>

==Mobility in gas phase==
Mobility is defined for any species in the gas phase, encountered mostly in [[Plasma (physics)|plasma]] physics and is defined as
<math display="block">\mu = \frac{q}{m \nu_\text{m}},</math>
where
* <math>q</math> is the charge of the species,
* <math>\nu_\text{m}</math> is the momentum-transfer collision frequency,
* <math>m</math> is the mass.

Mobility is related to the species' '''diffusion coefficient''' <math>D</math> through an exact (thermodynamically required) equation known as the [[Einstein relation (kinetic theory)|Einstein relation]]:
<math display="block">\mu = \frac{q}{kT} D,</math>
where
* <math>k</math> is the [[Boltzmann constant]], 
* <math>T</math> is the [[gas]] temperature,
* <math>D</math> is the diffusion coefficient.

If one defines the [[mean free path]] in terms of [[momentum transfer]], then one gets for the diffusion coefficient
<math display="block">D = \frac{\pi}{8} \lambda^2 \nu_\text{m}.</math>

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, <math>\lambda</math> is often defined as the diffusional mean free path, by assuming that a simple approximate relation is exact:
<math display="block">D = \frac{1}{2} \lambda v,</math>
where <math>v</math> is the [[root mean square]] speed of the gas molecules:
<math display="block">v = \sqrt{\frac{3kT}{m}},</math>
where <math>m</math> 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.<ref>{{cite journal | author=E. O. Knutson and K. T. Whitby | title=Aerosol classification by electric mobility: Apparatus, theory, and applications | journal=J. Aerosol Sci. | year=1975 | volume=6 | pages=443–451 | doi=10.1016/0021-8502(75)90060-9 | issue=6| bibcode=1975JAerS...6..443K }}</ref>  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 sizer]]s.

==References==
{{Reflist}}

{{Electrophoresis}}

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[[Category:Physical quantities]]
[[Category:Electrophoresis]]
[[Category:Mass spectrometry]]