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 ${\displaystyle {\Omega }/{\sqrt {\text{seconds}}}={\Omega }s^{-1/2}}$

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 angular frequency (${\displaystyle {1}/{\sqrt {\omega }}}$). 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:

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

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;
• 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.

The Warburg coefficient is mainly used in engineering. Engineers can calculate how quickly ions are moving through a material by measuring the Warburg Coefficient: this enables them to characterise battery performance.

References

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