Limitations and application of the Einstein relation

Question

The Einstein relation says that the diffusion coefficient $D$ and it's mobility $\mu$ under a force are related to each other as follows:

$$ D = \mu k_B T $$

I don't completely understand the derivation to this equation and what are the assumptions that went into deriving this relationship? In the case of an ideal gas, the self diffusion coefficient is given by

$$ D = \frac{3}{8} \frac{1}{nd^2}(\frac{k_B T}{\pi m})^{1/2}$$

This is also applicable in the case of a charged weakly ionized gas and $D$ can be seen as the diffusion coefficient for the ions through the neutral molecules. Would it be correct to use the Einstein relation in this case to obtain the ion mobility through the gas such that:

$$ \mu = \frac{3}{8} \frac{1}{nd^2}(\frac{1}{\pi m k_B T})^{1/2}$$

giving us:

$$ v_d = \mu F = \mu qE = \frac{3}{8} \frac{qE}{nd^2}(\frac{1}{\pi m k_B T})^{1/2}$$

Where $q$ is the charge on the ion.

Answer 1

The Einstein equation, often called Stokes-Einstein, assumes that the thermodynamic force that moves the particle is opposed by the drag of the fluid, which is expressed by the Stokes law, hence Stokes-Einstein. This further implies that the diffusing particle is much larger than the mean free path of the medium, so that the medium may be treated as a continuum that can be described by the Stokes law.

In liquid media the mean free path is very short, therefore, even molecular species can be described by the SE equation. In fact, Einstein in his 1908 paper applied his formula to sugar and diisoamyl ammonium.

The situation is different in gasses. At room temperature and pressure the mean free path of gas molecules is about 70 nm. To apply Stokes-Einstein in a gas the particle size must be roughly 10 times the mean free path, about a micron. For smaller particles there are slip corrections that basically account for the breakdown of continuum as the diffusing particle becomes comparable in size to the empty space between the molecules of the medium.

Nikolai Fuchs has done pioneering work in this area, his book The Mechanics of Aerosols is a highly recommended resource.