\(\textbf{Chemistry}\) \[\textit{Fick’s Law of Diffusion}\]
Consider diffusion. It is the movement of particles from higher concentrations to lower concentrations. Consider a one-dimensional movement of ions across a cell membrane from regions where the concentration is higher to regions where it is lower across the concentration gradient. In other words, the \(\textit{diffusive flux}\), \( J_{diff} \) is equivalent to some scale of the concentration gradient.
\[J_{diff} \propto \frac{d[C]}{dx} \]
Where \(x\) is the position along the concentration gradient and \([C]\) is the concentration at that point.
This is known as Fick’s first law of diffusion in the most ideal scenario. It is a relatively simple concept, taken from the idea that diffusion depends, and is proportional to, the degree of magnitude of the concentration gradient. \(J_{diff}\) is given in the amount of substance per unit area per unit time.
Of course, the other value which \( J_{diff}\) depends upon is the substance itself. The diffusivity of a substance is how easy or difficult it is for the substance to diffuse, given its chemical properties such as mass and viscosity. We thus term \(D\) the diffusion constant of a substance with dimensions of area per second.
Lastly, since the movement is in the negative direction (from high to low), a negative sign must be multiplied.
\[J_{diff}=-D\frac{d[C]}{dx}\]
Typically speaking, the derivatives in this equation would be partial ones due to the other spatial dimensions which may determine the concentration in a given space. In this case, we simply take partial derivatives along each of the axes.
However, when looking at diffusion across a cell membrane, we are only considering diffusion in one specific direction (perpendicular to the membrane in the path joining the inside and outside of the cell), \(x\), say, such that
\[J_{diff}=-D\frac{\partial [C]}{\partial x}\]