For the electrical mobility of an electron

Electron

The electron is a subatomic particle with a negative elementary electric charge. It has no known components or substructure; in other words, it is generally thought to be an elementary particle. An electron has a mass that is approximately 1/1836 that of the proton...

 or hole

Electron hole

An electron hole is the conceptual and mathematical opposite of an electron, useful in the study of physics, chemistry, and electrical engineering. The concept describes the lack of an electron at a position where one could exist in an atom or atomic lattice...

 in solid-state physics

Solid-state physics

Solid-state physics is the study of rigid matter, or solids, through methods such as quantum mechanics, crystallography, electromagnetism, and metallurgy. It is the largest branch of condensed matter physics. Solid-state physics studies how the large-scale properties of solid materials result from...

, see Electron mobility

Electron mobility

In solid-state physics, the electron mobility characterizes how quickly an electron can move through a metal or semiconductor, when pulled by an electric field. In semiconductors, there is an analogous quantity for holes, called hole mobility...

.

Electrical mobility is the ability of charged particles (such as electron

Electron

The electron is a subatomic particle with a negative elementary electric charge. It has no known components or substructure; in other words, it is generally thought to be an elementary particle. An electron has a mass that is approximately 1/1836 that of the proton...

s or proton

Proton

The proton is a subatomic particle with the symbol or and a positive electric charge of 1 elementary charge. One or more protons are present in the nucleus of each atom, along with neutrons. The number of protons in each atom is its atomic number....

s) to move through a medium in response to an electric field

Electric field

In physics, an electric field surrounds electrically charged particles and time-varying magnetic fields. The electric field depicts the force exerted on other electrically charged objects by the electrically charged particle the field is surrounding...

 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

Electrophoresis

Electrophoresis, also called cataphoresis, is the motion of dispersed particles relative to a fluid under the influence of a spatially uniform electric field. This electrokinetic phenomenon was observed for the first time in 1807 by Reuss , who noticed that the application of a constant electric...

.

Theory

When a charged particle

Charged particle

In physics, a charged particle is a particle with an electric charge. It may be either a subatomic particle or an ion. A collection of charged particles, or even a gas containing a proportion of charged particles, is called a plasma, which is called the fourth state of matter because its...

 in a gas

Gas

Gas is one of the three classical states of matter . Near absolute zero, a substance exists as a solid. As heat is added to this substance it melts into a liquid at its melting point , boils into a gas at its boiling point, and if heated high enough would enter a plasma state in which the electrons...

 or liquid

Liquid

Liquid is one of the three classical states of matter . Like a gas, a liquid is able to flow and take the shape of a container. Some liquids resist compression, while others can be compressed. Unlike a gas, a liquid does not disperse to fill every space of a container, and maintains a fairly...

 is acted upon by a uniform electric field

Electric field

In physics, an electric field surrounds electrically charged particles and time-varying magnetic fields. The electric field depicts the force exerted on other electrically charged objects by the electrically charged particle the field is surrounding...

, it will be accelerated until it reaches a constant drift velocity

Drift velocity

The drift velocity is the average velocity that a particle, such as an electron, attains due to an electric field. It can also be referred to as Axial Drift Velocity since particles defined are assumed to be moving along a plane. In general, an electron will 'rattle around' in a conductor at the...

 according to the formula:

where

• is the drift velocity (m/s)
• is the magnitude of the applied electric field (V/m)
• is the mobility (m²/(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:

Electrical mobility is proportional to the net charge

Electric charge

Electric charge is a physical property of matter that causes it to experience a force when near other electrically charged matter. Electric charge comes in two types, called positive and negative. Two positively charged substances, or objects, experience a mutual repulsive force, as do two...

 of the particle. This was the basis for Robert Millikan

Robert Millikan

Robert A. Millikan was an American experimental physicist, and Nobel laureate in physics for his measurement of the charge on the electron and for his work on the photoelectric effect. He served as president of Caltech from 1921 to 1945...

's demonstration that electrical charges occur in discrete units, whose magnitude is the charge of the electron

Electron

The electron is a subatomic particle with a negative elementary electric charge. It has no known components or substructure; in other words, it is generally thought to be an elementary particle. An electron has a mass that is approximately 1/1836 that of the proton...

.

Electrical mobility of spherical particles much larger than the mean free path of the molecules of the medium is inversely proportional to the diameter

Diameter

In geometry, a diameter of a circle is any straight line segment that passes through the center of the circle and whose endpoints are on the circle. The diameters are the longest chords of the circle...

 of the particles; for spherical particles much smaller than the mean free path, the electrical mobility is inversely proportional to the square of the particle diameter.

Mobility in gas phase

Mobility is defined for any species in the gas phase, encountered mostly in plasma

Plasma (physics)

In physics and chemistry, plasma is a state of matter similar to gas in which a certain portion of the particles are ionized. Heating a gas may ionize its molecules or atoms , thus turning it into a plasma, which contains charged particles: positive ions and negative electrons or ions...

 physics and is defined as:


where

• is the charge of the species,
• is the momentum transfer collision frequency, and
• is the mass.

Mobility is related to the species' diffusion coefficient through an exact (thermodynamically required) equation known as the Einstein relation

Einstein relation

Einstein relation can refer to*Einstein relation , a kinetic relation found independently by Albert Einstein and Marian Smoluchowski *Mass–energy equivalence, sometimes called Einstein's mass-energy relation...

:
,

where

• is the Boltzmann constant,
• is the gas
  Gas
  Gas is one of the three classical states of matter . Near absolute zero, a substance exists as a solid. As heat is added to this substance it melts into a liquid at its melting point , boils into a gas at its boiling point, and if heated high enough would enter a plasma state in which the electrons...
  
   temperature, and
• is a measured quantity that can be estimated. If one defines the mean free path
  Mean free path
  In physics, the mean free path is the average distance covered by a moving particle between successive impacts which modify its direction or energy or other particle properties.-Derivation:...
  
   in terms of momentum transfer, then one gets:

.

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, is often defined as the diffusional mean free path, by assuming a simple approximate relation is exact:
,

when is the root mean square

Root mean square

In mathematics, the root mean square , also known as the quadratic mean, is a statistical measure of the magnitude of a varying quantity. It is especially useful when variates are positive and negative, e.g., sinusoids...

 speed of the gas molecules:


where 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.