# Derivation of Fick's second law

Consider a cylinder of unit cross sectional area. We take two cross-sections, separated by δx, and note that the flux through the 1st section will not be the same as the flux through the second section.

From Fick’s first law we can deduce that:

\[{J_1} = - D{\left\{ {\frac{{\partial C}}{{\partial x}}} \right\}_1}\] \[{J_2} = - D{\left\{ {\frac{{\partial C}}{{\partial x}}} \right\}_2}\] \[{J_2} = - D{\left\{ {\frac{{\partial C}}{{\partial x}}} \right\}_1} + \partial x\left\{ {\frac{{{\partial ^2}C}}{{\partial {x^2}}}} \right\}\]

In an increment of time,δt, there is a corresponding increment in concentration, δC.

\[\partial C\partial x = ({J_1} - {J_2})\partial t\] \[\frac{{\partial C}}{{\partial t}} = D\left\{ {\frac{{{\partial ^2}C}}{{\partial {x^2}}}} \right\}\]

This is Fick’s second law