Warburg coefficient

The Warburg coefficient (or Warburg constant), A W {displaystyle A_{W}} , is the diffusion coefficient of ions in solution, associated to the Warburg element, Z W {displaystyle Z_{W}} . The Warburg coefficient, A W {displaystyle A_{W}} , also written as, σ {displaystyle {sigma }} , has the units of Ω / s e c o n d s = Ω ( s − 1 / 2 ) {displaystyle {Omega }/{sqrt {seconds}}={Omega }(s^{-1/2})} The Warburg coefficient (or Warburg constant), A W {displaystyle A_{W}} , is the diffusion coefficient of ions in solution, associated to the Warburg element, Z W {displaystyle Z_{W}} . The Warburg coefficient, A W {displaystyle A_{W}} , also written as, σ {displaystyle {sigma }} , has the units of Ω / s e c o n d s = Ω ( s − 1 / 2 ) {displaystyle {Omega }/{sqrt {seconds}}={Omega }(s^{-1/2})} The value of A W {displaystyle A_{W}} can be obtained by the gradient of the Warburg plot, a linear plot of the real impedance ( R {displaystyle R} ) against the reciprocal of the square root of the 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 A W {displaystyle A_{W}} can be found by: A W = R T A n 2 F 2 2 ( 1 D O 1 / 2 C O b + 1 D R 1 / 2 C R b ) = 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}{D_{O}^{1/2}C_{O}^{b}}}+{frac {1}{D_{R}^{1/2}C_{R}^{b}}} ight)}={frac {RT}{An^{2}F^{2}Theta C{sqrt {2D}}}}} where R {displaystyle R} is the ideal gas constant, T {displaystyle T} is the thermodynamic temperature, F {displaystyle F} is the Faraday constant, n {displaystyle n} is the valency, D {displaystyle D} is the diffusion coefficient of the species where subscripts O {displaystyle O} and R {displaystyle R} stand for the oxidized and reduced species respectively, C b {displaystyle C^{b}} is the concentration of the O {displaystyle O} and R {displaystyle R} species in the bulk, C is the concentration of the electrolyte, A {displaystyle A} denotes the surface area and Θ {displaystyle Theta } denotes the fraction of the R {displaystyle R} and O {displaystyle O} species present. The equation for A W {displaystyle A_{W}} applies to both reversible and quasi-reversible reactions for which both halves of the couple are soluble.