User:Kehrli/IonMobility

The ion mobility $$K$$ is a physical quantity which describes the speed of an ion moving in a drift gas, $$v_d$$ when exposed to an electric field $$E$$.


 * $$\vec v_d = K \vec E$$

Some authors use the symbol $$\mu$$ or $$Z$$ for representing the ion mobility. The official IUPAC sympol is $$\mu$$. The ion mobility can be measured with an ion mobility spectrometer. It can also be derived mathematically.

Low Field Limit
In the so called low field limit, i.e. if the electric field strength is weak or the pressure is high, the ion mobility is


 * $$K = \frac{3}{16}\ \sqrt{\frac 1 m_i + \frac 1 m_g}\ \sqrt{ \frac{2\pi}{kT}}\ \frac {Q}{n\sigma}$$

where $$n$$ is the number density of the gas particles, $$m_i$$ and $$m_g$$ are the masses of the ions and the gas particles, $$T$$ is the absolute temperature, $$Q$$ is the ion charge and $$\sigma$$ is the average cross section of the ion and the gas particle.

The low field limit is defined by :
 * $$\frac E N \ll 3\pi kT \sqrt{\frac {m_i}{m_i + m_g}}\ \frac {d^2}{Q}$$

Since $$K$$ strongly depends on gas pressure and temperature, it is common to tabulate the reduced mobility $$K_0$$ which is the mobility at standard conditions:


 * $$K_0 = K\ \frac{p}{p_0}\ \frac {T_0}{T}$$

where $$p_0$$ = 1013.25 mbar and $$T_0$$ = 273.15 K are standard pressure and standard temperature. The number density $$n$$ can be turned into a pressure using the ideal gas law:
 * $$n = \frac{N}{V} = \frac {p}{kT}$$

Using this and the reduced mass, the ion mobility then becomes:
 * $$K = \frac{3}{16}\ \sqrt{ \frac{2\pi kT}{\mu}}\ \frac {Q}{p\sigma}$$

Einstein Relation
The ion mobility $$K$$ is related to the diffusion constant $$D$$ by the Einstein relation:
 * $$K = \frac{QD}{kT}$$

Note that Einstein used a slightly different definition of the ion mobility, which therefore is expressed by the symbol $$\mu$$:
 * $$\vec v_d = \mu \vec F = \mu Q\vec E$$

Thereby the Einstein relation looks even simpler:
 * $$\mu = \frac{D}{kT}$$

High Field Limit
In the high field limit, the ion mobility $$K$$ becomes dependent on $$v_d$$ and therefore its calculation more difficult. However, it is a good approximation to replace the temperature $$T$$ by $$T_{eff}$$ :
 * $$T_{eff} = T(1 + \frac {m_g v_d^2}{3kT})$$


 * $$K = \frac{3}{16}\ \sqrt{\frac 1 m_i + \frac 1 m_g}\ \sqrt{ \frac{2\pi}{kT_{eff}}}\ \frac {Q}{n\sigma}$$

Applications
The following instruments are based on the ion mobility:
 * Ion Mobility Spectrometer (IMS)
 * Differential Mobility Analyzer (DMA)
 * Differential Mobility Spectrometer (DMS)
 * High-Field Asymmetric Waveform Ion Mobility Spectrometry (FAIMS, a brand of DMS)