Page - 168 - in Advanced Chemical Kinetics

microstructures, and thus properties, one has to precisely control the synthesis conditions during SHS. These conditions are primarily defined by the kinetics of the chemical reactions takingplace in thecombustionwave. In order to study and understand the kinetics of SHS reactions, it is important to examine the fundamentals of kinetics and how they relate to SHS itself. Let us start with general definitions. Thus, assuming that the concentrations of any initial reagent, ci, and the product are uniformly distributed throughout the entire volume (homogeneous or quasi homogeneous cases), the chemical reaction rate can be expressed by the following equa- tion: Wi¼dCidt orWi¼ dηi dt (1) whereWi is the reaction rate for the i th reagent or product, ηi = (c 0 i� ci)/c0 is the degree of conversion for the ith reagent, c0i is the initial concentrationof the i th reagent, and t is time. In the nineteenth century, C.M. Guldberg with P.Waage andN.N. Beketov independently formulated the lawofmass action. This essentially states that the chemical reaction rates at a given point are proportional to the concentration (mass) of the reactants raised to a propor- tional exponent. Thus, for an elementary chemical reactionbetween two reagentsAandBof the following form: υ1Aþυ2B!C, (2) whereυ1,υ2 are stoichiometric coefficients. For this reaction, the lawofmassactioncanbewritten in the formofakinetic equation: W¼ kcυ1Acυ2B , (3) where k is the reactionrateconstant. Alongwith thereactantconcentration, the temperatureaffects therateof thechemical reaction in a noncatalytic homogeneous reaction. However, the mechanisms of these processes are oftenunknownor too complicated.This is because the reactionsoccur inmultiple steps, each of which has unique reaction rates. In order to describe the chemical kinetics, a single-step approximationis typicallyused.Thisstates that therateof theprocesses inthecondensedstate isgenerallya functionof the temperatureanddegreeof conversion: dη dt ¼F T;η (4) The single-step approximation employs the assumption that the function in Eq. (4) can be expressedasaproductof twoseparable functions thatare independentofeachother; the first, K(T),dependssolelyonthetemperature,T, andthesecond,Φ(η),dependssolelyonthedegree of conversion,η: Advanced Chemical Kinetics168