Editing Fick's laws of diffusion (section)

=== Derivation of Fick's first law for gases ===

Four versions of Fick's law for binary gas mixtures are given below. These assume: thermal diffusion is negligible; the body force per unit mass is the same on both species; and either pressure is constant or both species have the same molar mass. Under these conditions, Ref.<ref>{{cite book| vauthors = Williams FA |year=1985 |chapter=Appendix E |title=Combustion Theory |publisher=Benjamin/Cummings}}</ref> shows in detail how the diffusion equation from the [[kinetic theory of gases]] reduces to this version of Fick's law:
<math display="block"> \mathbf{V_i}=- D\nabla \ln{\left( y_i \right)} , </math>
where {{math|'''V<sub>i</sub>'''}} is the diffusion velocity of species {{mvar|i}}. In terms of species flux this is
<math display="block">\mathbf{J_i}=- \frac{\rho D}{M_i}\nabla y_i . </math>

If, additionally, <math> \nabla \rho = 0</math>, this reduces to the most common form of Fick's law,
<math display="block"> \mathbf{J_i}=- D\nabla \varphi .</math>

If (instead of or in addition to <math> \nabla \rho = 0</math>) both species have the same molar mass, Fick's law becomes
<math display="block">\mathbf{J_i}=- \frac{\rho D}{M_i}\nabla x_i, </math>
where <math> x_i </math> is the mole fraction of species {{mvar|i}}.