Warburg coefficient

The Warburg coefficient (or Warburg constant), ${\displaystyle A_{W}}$, is the diffusion coefficient of ions in solution, associated to the Warburg element, ${\displaystyle Z_{W}}$. The Warburg coefficient, ${\displaystyle A_{W}}$, also written as, ${\displaystyle {\sigma }}$, has the units of ${\displaystyle {\Omega }/{\sqrt {seconds}}={\Omega }(s^{-1/2})}$

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

Alternatively, the value of ${\displaystyle A_{W}}$ can be found by:

${\displaystyle A_{W}={\frac {RT}{An^{2}F^{2}{\sqrt {2}}}}{\left({\frac {1}{D_{O}^{1/2}C_{O}^{b}}}+{\frac {1}{D_{R}^{1/2}C_{R}^{b}}}\right)}={\frac {RT}{An^{2}F^{2}\Theta C{\sqrt {2D}}}}}$

where ${\displaystyle R}$ is the ideal gas constant, ${\displaystyle T}$ is the thermodynamic temperature, ${\displaystyle F}$ is the Faraday constant, ${\displaystyle n}$ is the valency, ${\displaystyle D}$ is the diffusion coefficient of the species where subscripts ${\displaystyle O}$ and ${\displaystyle R}$ stand for the oxidized and reduced species respectively, ${\displaystyle C^{b}}$ is the concentration of the ${\displaystyle O}$ and ${\displaystyle R}$ species in the bulk, C is the concentration of the electrolyte, ${\displaystyle A}$ denotes the surface area and ${\displaystyle \Theta }$ denotes the fraction of the ${\displaystyle R}$ and ${\displaystyle O}$ species present.

The equation for ${\displaystyle A_{W}}$ applies to both reversible and quasi-reversible reactions for which both halves of the couple are soluble.