Warburg coefficient

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

The value of $A W$ 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 $A W$ can be found by:

A_{W}={\frac {RT}{An^{2}F^{2}{\sqrt {2}}}}{\left({\frac {1}{C_{\mathrm {O} }^{b}{\sqrt {D_{\mathrm {O} }}}}}+{\frac {1}{C_{\mathrm {R} }^{b}{\sqrt {D_{\mathrm {R} }}}}}\right)}={\frac {RT}{An^{2}F^{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;
$C b$ 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 $A W$ applies to both reversible and quasi-reversible reactions for which both halves of the couple are soluble.

References

Ottova-Leitmannova, A. (2006). Advances in Planar Lipid Bilayers and Liposomes. Academic Press.

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