Warburg coefficient

The Warburg coefficient (or Warburg constant; denoted $AW$ or $σ$) is the diffusion coefficient of ions in solution, associated to the Warburg element, $ZW$. The Warburg coefficient has units of $${\Omega}/\sqrt{\text{seconds}}={\Omega} s^{-1/2}$$

The value of $AW$ can be obtained by the gradient of the Warburg plot, a linear plot of the real impedance ($R$) against the reciprocal of the square root of the frequency ($${1}/\sqrt{\omega}$$). This relation should always yield a straight line, as it is unique for a Warburg.

Alternatively, the value of $AW$ can be found by:

$$A_W={\frac{R T}{An^2F^2\sqrt2}}{\left(\frac{1}{C_\mathrm{O}^b\sqrt{D_\mathrm{O}}}+{\frac{1}{C_\mathrm{R}^b\sqrt{D_\mathrm{R}}}}\right)}=\frac{R T}{An^2F^2\Theta C\sqrt{2D}}$$

where
 * $R$ is the ideal gas constant;
 * $T$ is the thermodynamic temperature;
 * $F$ is the Faraday constant;
 * $n$ is the valency;
 * $D$ is the diffusion coefficient of the species, where subscripts $O$ and $R$ stand for the oxidized and reduced species respectively;
 * $Cb$ is the concentration of the $O$ and $R$ species in the bulk;
 * $C$ is the concentration of the electrolyte;
 * $A$ denotes the surface area;
 * $Θ$ denotes the fraction of the $O$ and $R$ species present.

The equation for $AW$ applies to both reversible and quasi-reversible reactions for which both halves of the couple are soluble.