Kontsevich quantization formula

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

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:

$A_W=\frac{RT}{A{n}^2{F}^2\sqrt{2}}(\frac{1}{D_O^{1/2}{C}_O^b}+\frac{1}{D_R^{1/2}{C}_R^b})=\frac{RT}{A{n}^2{F}^2\mathrm{\Theta }C\sqrt{2D}}$

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 and $\mathrm{\Theta }$ denotes the fraction of the $R$ and $O$ 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

• A. Ottova-Leitmannova, Advances in Planar Lipid Bilayers and Liposomes, Academic Press (2006)
• http://www.consultrsr.com/resources/eis/warburg2.htm#ref1