Page - 24 - in Advanced Chemical Kinetics

whereS andP represent thedimensionless concentrations of substrate andproduct, τ repre- sents thedimensionless time,andR is thedimensionless radial co-ordinateof theparticle.The first termontheright-handsideof theaboveequationaccounts foractivespecies (substrateor product) diffusion,whereas the second term f(R,τ,S,P) and g(R,τ,S,P) represents the homo- geneous reaction term(nonlinear term),generallypolynomial in theconcentrationsandtime. 4.Commongeometriesandnonlinear reaction Most commonlyused electrodes/microelectrodes consist of a conductingmetal/glassy carbon or semiconducting surface embedded inan insulatingwall.When the conducting surface is a rectangle ordisc of a fewmillimeters, this is knownas a“planar” electrode.Diffusion to this surface is effectively planar (the effects of the edges are negligible), hence the nonlinear one- dimensional reactiondiffusionequation isgivenby: ∂C½ � ∂t ¼D∂ 2C½ � ∂x2 þ f C½ �ð Þ (3) Twoother electrodegeometrieswherediffusion occurs in onlyone spatial dimension are the hemispherical andhemicylindrical electrodes. Thenonlinear two-dimensional (hemispherical or spherical) reactiondiffusionequation is: ∂C½ � ∂t ¼D ∂ 2C½ � ∂x2 þ2 r ∂C½ � ∂r þ f C½ �ð Þ (4) andfor the latter is: ∂C½ � ∂t ¼D ∂ 2C½ � ∂x2 þ1 r ∂C½ � ∂r þ f C½ �ð Þ (5) Thehemispherecanbeachievedexperimentallyviaasmalldropofmercurypositionedovera smaller conducting disc. A soft polymer, rubber, or other similar materials are usually employed to fabricate a hemicylinder. The electrodes are usually employed in theoretical studies due to the lowdimensionality of themass-transport equation.Additional terms such asdiffusionandnonlinear reactionallow the equation tobe solvedanalytically. Furthermore, theelectrodesarenotaccuratelyoreasily fabricated forpracticalgeometries. The corresponding nonlinear reaction-diffusion issues in enzymekinetics are focused on the mathematical resolution. Table 1 shows the response of particular electrodes with special emphasisonearlier theoreticalworks in the field. Example1:Michaelis-Mentenkineticsandmicrocylinderelectrodes The model is written for an enzyme reaction to generate an electro-active product (e.g., hydrogen peroxide froman oxidase enzyme) that reacts at an immobilizationmatrix,which Advanced Chemical Kinetics24