Fick's Law

Fick’s first law relates the diffusive flux to the gradient of the concentration. It postulates that the flux goes from regions of high concentration to regions of low concentration, with a magnitude that is proportional to the concentration gradient (spatial derivative), or in simplistic terms the concept that a solute will move from a region of high concentration to a region of low concentration across a concentration gradient.

$$j=-D\frac{\partial N}{\partial x}$$

• $j$ is the diffusion flux and measures the amount of substance that will flow through a unit area during a unit time interval. $[j]=\text{particles}/m^{2}s$
• $D$ is the diffusion coefficient or diffusivity. $[D]=m^2/s$
• $\partial N/\partial x$ is the concentration gradient $[N]=\text{particles}/m^3$ $[x]=m$

Broadly we can then model D across Temperature (T) as:

$$D=D_0\text{exp}(-E_a/k_{B}T)$$

• $D$ is the diffusion coefficient, as before. $[D]=m^2/s$
• $D_0$ is the pre-exponential factor / frequency factor — the theoretical maximum diffusivity at infinite temperature. $[D_0]=m^2/s$
• $E_a$ is the activation energy — the energy barrier a particle must overcome to make a diffusive jump. $[E_a]=eV$ or $J$
• $k_B$ is the Boltzmann constant. $k_B=1.381\times 10^{-23}J/K$ (or $8.617\times 10^{-5}eV/K$)
• $T$ is the absolute temperature. $[T]=K$

Typically we plot this relation for different chemicals as an Arrhenius Plot:

For a single rate-limited thermally activated process, an Arrhenius plot gives a straight line, from which the activation energy ($E_a$) and the pre-exponential factor $A$ can both be determined.