Page - 28 - in Advanced Chemical Kinetics

is metallically conducting sites/particles. The reaction within the film under theMichaelis- Mentenkineticsmaybewrittenas follows: SþE1⇔ k1 k�1 E1S½ �����!kcat PþE2 (6) The consumption rate of S is given by k1cScE�k�1cES, where cidenotes the concentration of species i.Therate isequivalent to (kcat/KM) cScE,whereKM is theMichaelis constant,definedas KM=(k�1+kcat)/k1. The consumption rate of S in the film is compensated by diffusion. If the solution is stirreduniformly, so thatS is constantlysupplied to the film, themassbalance forS canbewritten incylindrical coordinates: DS r d dr r dcS dr � � � kcatcEcS cSþKM¼0 (7) where cS is the concentrationprofileof substrate, cE is the concentrationprofileof enzyme,DS is its diffusion coefficient, andKM is theMichaelis constant. The rate of consumptionwill be v(r)=k cH, where k is the rate constant for the hydrogen peroxide reaction and cH is the peroxide concentration. Then, the equation of continuum for hydrogenperoxide is generally expressed in thesteady-stateby DH r d dr r dcH dr � � þ kcatcEcS cSþKM�v rð Þ¼0 (8) At theelectrodesurface (r0) andat the filmsurface (r1), theboundaryconditionsare [29]: r¼ r0 : dcSdr ¼0, cH¼0 r¼ r1 : cS¼ c∗S, cH¼0 (9) where c∗S is the bulk concentration of S scaled by the partition coefficient of the film. The current isprovidedbytheconsumptionrateateachsite.Thus, the total currentatanelectrode of lengthL is expressedby[29] I=nF¼2πL ðr1 r0 rvdr (10) Theanalytical resultsof theproblemarediscussedbyEswariandRajendran [30]. Example2:enzymecatalysis reaction The reactions without spatial dependence on various concentrations have modeled the enzymekinetics inbiochemical systems.Nonlinear systemsofordinarydifferential equations are solely based on that. Michaelis and Menten were pioneers in explaining the enzyme reactionmodel. Inaddition, theyalso reported the freeenzymebinding to the reactant,which Advanced Chemical Kinetics28