1.3 Rate laws of overall and elementary reactions In general the equation describing the dependency of the reaction rate r on the concentrations ci of reaction components i is called the kinetic equation of the reaction: r = f(c A ,c B ,c D ,c E ,...,c N ) (1.16) Depending on the reaction the function f can be quite simple but sometimes it may as well be rather complicated. It usually depends on the concentrations of the components participating in the main reaction but it can also depend on the concentrations of catalysts and inhibitors. For the time being we will neglect this possibility. For many reactions in ideal systems (gaseous reactions and reactions in liquid phase) r is found to have the general form of rate law given in Eq. (1.17). r = k[ A ] [ B] ....[ L] = kcαAcβB……cλL α β λ (1.17) In this rate equation (rate law) k is the reaction rate constant (rate coefficient) and α, β and λ are often integers or half-integers. The rate constant k is a function of temperature and pressure, but the pressure dependency of k is small and is usually ignored. The reaction is said to have order α in respect of the component A, order β in respect of the component B, etc. and parameters α, β and λ are called partial orders of the reaction. The sum α + β +…+ λ = n is called the overall order (or simply order) of the reaction. Since r has the units of concentration over time, k has the units (concentration)1-n time-1. Therefore the rate constant of a first order (n = 1) reaction has the units s-1 and is independent of the units used for concentration which is a very important general property of a first order reaction. For elementary reactions (unit steps) only we can obtain the rate law immediately from the chemical reaction equation. For example if the forward reaction given by Eq. (1.1) can be considered as an elementary reaction its rate is given by r = k[ A ] − νA [ B] − ν = k[ A ] a [ B] b B (1.18) where νA and νB are the general stoichiometric numbers of the reactants involved in the elementary reaction given by Eq. (1.1) i.e. -νA = a and -νB = b. Then the negation of the sum of the general stoichiometric numbers of the reactants − ( νA + ν B +... ) = − ∑ νi (1.19) i = reactant is called the molecularity of the elementary reaction. Some experimentally observed rate laws for homogeneous reactions are given below: H2 + Br2 → 2HBr 2N2O5 → 4NO2 + O2 H2 + I2 → 2HI r= 1 2 k[ H 2 ][ Br2 ] 1 + j[ HBr ][ Br2 ] (1.20) r = k[N2O5] (1.21) r = k[H2][I2] (1.22) In general the rate law must be determined from experimental measurements of reaction rate and cannot be deduced from the reaction stoichiometry (except for in the special case of an elementary reaction). The reaction equation, Eq. (1.1) gives the overall stoichiometry of the reaction but does not tell us very much about the mechanism by which the reaction occurs. Each step in the mechanism of the overall reaction is an independent reaction and is called an elementary reaction (unit step). The most simple reaction consists of one single elementary step. The classical example of such elementary reaction is the reaction described by Eq. (1.22). It is explained by the following molecular level reaction mechanism A + B → A---B → C + B where A---B denotes the so called activated complex first formed between the reactant molecules A and B before the final rearrangement to product molecules C and B. The formally very similar reaction given by Eq. (1.20) is much more complicated and is called a complex (composite) reaction. A complex chemical reaction consists of at least two (and usually more) elementary steps. A specific characteristic of the complex reactions is that the exponentials (partial orders) in the rate law and stoichiometric coefficients in the reaction equation do not fit. In Chap. 6 we shall give a more detailed analyze for the mechanism of this reaction. An important concept is that some reactions can be characterized by pseudo orders and pseudo constants. For example for the hydrolysis of sucrose: C12H22O11 + H2O → C6H12O6 + C6H12O6 Sucrose Glucose Fructose (1.23) one finds that the rate is given by the rate law, r = k[C12H22O11]. However, since the solvent, H2O participates in the reaction, one could expect that the rate law would preferably have the more general form, r = k' [C12H22O11] [H2O]v. Because H2O is always present in the reaction vessel in great excess, [H2O]v is essentially constant and the rate law has the apparent form r = k[C12H22O11], where k = k´[H2O]v. The reaction is said to be of pseudo first order although the kinetic data indicates for v ≈ 6. The reaction rate constant k is respectively called the pseudo reaction rate constant. The number of molecules which react in an elementary step is called the molecularity of the elementary reaction. Molecularity is defined only for elementary reactions and should not be used to describe overall reactions that consist of more than one elementary step. In fact the only elementary reaction which exists is the bimolecular one. However, all the types of reactions given below are considered to be elementary reactions. The unimolecular reaction and the trimolecular reactions, however, are in fact simple reactions from the kinetic point of view since they can be explained by trivial combinations of bimolecular reactions. Examples of each case are given below: A → products unimolecular reaction A + B → products 2A → products A + B + C → products 2A + B → products 3A → products bimolecular reactions trimolecular(termolecular) reactions. No elementary reactions involving more than three molecules have been suggested because of the very low probability of collision of more than three molecules at the same moment of time, which of course is the perquisite of a chemical reaction. The rates of chemical reactions are obtained from measurements of concentrations of reaction mixture components as a function of time. Analytical chemistry methods may be applied to samples taken from reaction mixture when the reaction can be stopped suddenly. This may be done e.g. by rapid cooling of the samples for high-temperature reactions. Physical methods are especially useful for determining the rate of a chemical reactions because they offer the possibility to measure continuously the advancement of the reaction. Spectroscopic methods are most generally used techniques. An important characteristic of any measurement method is its response time. The measuring device must obviously respond much more rapidly than the concentrations are changing. Pulsed lasers have opened up new opportunities for investigation of very fast reactions. Especially reactions occurring in the picosecond (10-12 s) range may be studied in this way. To study gas reactions at high temperatures it is necessary to heat the gas to the reaction temperature very quickly because the reaction then occurs rapidly. This may be accomplished by means of shock tube in which a shock wave is used to heat the gas with high speed. Photochemical reactions may be initiated suddenly by a light pulse from a flash lamp or laser. In the flash photolysis method, a reaction vessel is exposed to a very high intensity flash of visible or ultraviolet light. The flash dissociates the molecules in the sample and the concentrations of these species are then determined over a period of time using subsequent flashes at much lower intensities.