# Thermal Conduction and Diffusion in Gases

### Elementary theory of diffusion coefficient in gases[edit]

Random collisions of particles in a gas.

The diffusion coefficient {\displaystyle D} is the coefficient in the Fick’s first law {\displaystyle J=-D\,\partial n/\partial x}, where *J* is the diffusion flux (amount of substance) per unit area per unit time, *n* (for ideal mixtures) is the concentration, *x* is the position [length].

Let us consider two gases with molecules of the same diameter *d* and mass *m* (self-diffusion). In this case, the elementary mean free path theory of diffusion gives for the diffusion coefficient{\displaystyle D={\frac {1}{3}}\ell v_{T}={\frac {2}{3}}{\sqrt {\frac {k_{\rm {B}}^{3}}{\pi ^{3}m}}}{\frac {T^{3/2}}{Pd^{2}}}\,,}

where *k*_{B} is the Boltzmann constant, *T* is the temperature, *P* is the pressure, {\displaystyle \ell } is the mean free path, and *v _{T}* is the mean thermal speed:{\displaystyle \ell ={\frac {k_{\rm {B}}T}{{\sqrt {2}}\pi d^{2}P}}\,,\;\;\;v_{T}={\sqrt {\frac {8k_{\rm {B}}T}{\pi m}}}\,.}

We can see that the diffusion coefficient in the mean free path approximation grows with *T* as *T*^{3/2} and decreases with *P* as 1/*P*. If we use for *P* the ideal gas law *P* = *RnT* with the total concentration *n*, then we can see that for given concentration *n* the diffusion coefficient grows with *T* as *T*^{1/2} and for given temperature it decreases with the total concentration as 1/*n*.

For two different gases, A and B, with molecular masses *m*_{A}, *m*_{B} and molecular diameters *d*_{A}, *d*_{B}, the mean free path estimate of the diffusion coefficient of A in B and B in A is:{\displaystyle D_{\rm {AB}}={\frac {2}{3}}{\sqrt {\frac {k_{\rm {B}}^{3}}{\pi ^{3}}}}{\sqrt {{\frac {1}{2m_{\rm {A}}}}+{\frac {1}{2m_{\rm {B}}}}}}{\frac {4T^{3/2}}{P(d_{\rm {A}}+d_{\rm {B}})^{2}}}\,,}