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4 EIS at carbon fiber cylindrical microelectrodes At the reversible potential, the fitted parameters were π = 5.28 Β± 0.07 π β1/2, ππ = 13.7 Β± 0.4ππ , π =6470Β±30Ξ©, π =56Β±2ππΉπ 1βπ, π=0.843Β±0.003 and π =790Β± ππ‘0 π’ 9 Ξ©. Using Eq. (3), the radius of the fiber was estimated from the values of ππ and the mean diffusivity (Section 2) to be 3.42 Β± 0.05 ΞΌm. The standard rate constant calculated from Eq. (4.9) is π0 = 7.70 Β± 0.11 Γ 10-3 cm s-1. The value of π ππ‘ at the reversible potential can then be estimatedfromEq.(4.10)tobe3.4Ξ©cm2.Theerrorsinπ0andπ aredominatedbythefitting 0 errors in π and ππ respectively, and not the error in the mean diffusivity. The quoted errors are fitting errors only, and a more extensive study of many fibers will be required to clarify systematic errors. As in impedance methods for other geometries, no measurement of the surface area of the fiber is required. If the measured area is available, then a consistency check may be carried out by comparing π ππ‘ above with the directly fitted π ππ‘ in Ohms multiplied by the area. In the present case, the area was estimated from the radius found above and the known length, which gives π ππ‘ = 4.7 Ξ© cm2. Instead of starting with a known mean diffusivity, an alternative approach to this analysis would be to use the measured radius to estimate the diffusivity, and in turn the rate constant. Literature standard rate constants for the Fe(CN)3β/4β 6 system vary widely depending on the solution, the type of carbon surface and the surface pretreatment, and span the range 0.005 - 0.5 cm s-1 [154]. For the fastest rate constants (> 0.1 cm s-1) an activation procedure is generally required [155]. Lower values similar to that obtained here are typically obtained for unactivated surfaces such as regularly polished glassy carbon 0.005 cm s-1 [154, 156] or carbon paste 0.001 - 0.007 cm s-1 [157]. The analysis here assumes smooth surfaces, or at least a roughness scale small compared to the radius, in which case the true rate constant is the quoted value divided by the roughness factor, and the radius determined will be an average value. The effective double-layer capacitance calculated from the Brug formula (Eq. (8.12), Ref. [150]) is small, 12 ΞΌF cm s-2, similar to literature values [140, 158] for untreated fibers. Much higher capacitances are found for activated and severely roughened surfaces, for which the effective radius may underestimate the true radius [140]. 47PDF Image | Electron Transfer Kinetics in Redox Flow Batteries
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