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4 EIS at carbon fiber cylindrical microelectrodes Aside from this substitution, the mass transport impedance is coupled to the reaction kinetics in the usual way. For the standard redox reaction and kinetics, Eqs. (4.4,4.5) (4.4) (4.5) (4.6) 𝐹 The derivation of the Randles circuit [150, 151] gives 𝜎′ as in Eq. (4.6). 𝑅⇌ 𝑃+𝑒− 𝑗 = 𝑘𝑓𝑐𝑅 − 𝑘𝑏𝑐𝑃 −1 −1 𝜎′ = √2𝜎 = 𝑅𝑐𝑡 (𝑘𝑓𝐷𝑅 2 + 𝑘𝑏𝐷𝑃 2) In coupling the mass transport, and for the rest of the paper, we make the common assumption that 𝐷𝑅 = 𝐷𝑃 = 𝐷, and 𝜎′ then reduces to Eq. (4.7), where 𝑘 = 𝑘𝑓 + 𝑘𝑏. It is convenient to fit directly to the composite parameter 𝜓, Eq. (4.8), which is independent of electrode area. ′ −1 𝜎 = 𝑅𝑐𝑡𝑘𝐷 2 (4.7) (4.8) If both species are present in equal concentrations 𝑐𝑏 and their activity coefficients are equal, then at the reversible potential the standard rate constant 𝑘0 = 𝑘𝑓 = 𝑘𝑏, and the simple relationships of Eqs. (4.9) – (4.11) apply. 𝜎′ 𝑘 𝜓=𝑅 = 𝑐𝑡 √𝐷 2𝑘0 𝜓 = √𝐷 𝑅𝑐𝑡 = 𝑅𝑇 𝐹2𝑐𝑏𝑘0 𝜎′ = 2𝑅𝑇 𝐹2𝑐𝑏𝐷2 (4.9) (4.10) (4.11) 43 1PDF Image | Electron Transfer Kinetics in Redox Flow Batteries
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