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 Ω / seconds = Ω s − 1 / 2 {\displaystyle {\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 angular frequency ( 1 / ω {\displaystyle {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 = R T A n 2 F 2 2 ( 1 C O b D O + 1 C R b D R ) = R T A n 2 F 2 Θ C 2 D {\displaystyle 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; 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.
References Ottova-Leitmannova, A. (2006). Advances in Planar Lipid Bilayers and Liposomes. Academic Press.