Derivation of Rate Laws

A complex reaction requires more than one chemical equation and rate law for its stoichiometric and kinetics description, respectively. It can be thought of as yielding more than one set of products. The mechanisms for their production may involve some of the same intermediate species. In these cases, their rates of formation are coupled, as reflected in the predicted rate laws. 

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Open-Sequence Mechanisms Derivation of Rate Law from Mechanism... 

In this chapter, we examine the rates of chemical reactions. The first section deals with empirical rate laws, equations that summarize experimental rate measurements. Rate laws can often be understood in terms of the mechanisms of the reactions, and the derivation of rate laws from a postulated mechanism is the subject of the second section. 

The irreversible formation of the tetrahedral intermediate allowing for the derivation of rate law 6.7 from Equation 6.5 may alternatively be regarded as a borderline case of a reversible formation of the tetrahedral intermediate (here rate law 6.10 applies)—namely in the case the rate constant kiam approaches zero. Accordingly, Equation 6.10 for kretro = 0 transforms into Equation 6.7. 

Since a portion of this chapter is devoted to the derivation of rate laws and various microscopic expressions for the rate coefficients of condensed-phase chemical reactions, it is useful to first write down the phenomenological rate law we expect to obtain, to define the various rate coefficients and relaxation times, and to present the different points of view that we shall adopt in describing the system. 

The method of King and Altman rendered an invaluable service to enzymology because, with its help, the rate laws for many major reaction mechanisms in enzyme kinetics were developed. It is not necessary to understand the theory of the King-Altman method in order to apply it in practice, and indeed the theory is considerably more difficult than the practice. Therefore, we shall describe in the following sections the derivation of rate laws for several simple mechanisms... 

The exponents n and m in Eq. VII-1 are expected to be 1 and 3, respectively, under some conditions. Assuming spheres and letting Vr = V/Vq, derive the rate law dVrjdt =f(Vr). Expound on the peculiar nature of this rate law. 

Mechanisms. Mechanism is a technical term, referring to a detailed, microscopic description of a chemical transformation. Although it falls far short of a complete dynamical description of a reaction at the atomic level, a mechanism has been the most information available. In particular, a mechanism for a reaction is sufficient to predict the macroscopic rate law of the reaction. This deductive process is vaUd only in one direction, ie, an unlimited number of mechanisms are consistent with any measured rate law. A successful kinetic study, therefore, postulates a mechanism, derives the rate law, and demonstrates that the rate law is sufficient to explain experimental data over some range of conditions. New data may be discovered later that prove inconsistent with the assumed rate law and require that a new mechanism be postulated. Mechanisms state, in particular, what molecules actually react in an elementary step and what products these produce. An overall chemical equation may involve a variety of intermediates, and the mechanism specifies those intermediates. For the overall equation... 

Derive a rate law for the formation of ethylene C2H4 assuming k3 k5. The free radieals are CH3, and C2H5. 

Derive the rate law, making the steady-state approximation for the concentration of the intermediate (signified with an asterisk), which is a rearranged structure of the parent. For... 

Bond energies. The net reaction CD + RH = RC1 + HC1 proceeds by a chain mechanism in which the propagators are Cl and R (but not H ), and chain-breaking occurs by dimerization of Cl. Write a scheme consistent with this and derive its rate law. Show how one can use E and AH for the bond dissociation of CP to calculate an activation energy for an elementary reaction. 

Acid catalysis. Consider the reverse of the scheme written in Eqs.(10-37)-(10-38). Derive the rate law for the reverse reaction, and discuss different limiting forms as to the type of acid catalysis demonstrated. 

Measurements of the rates of the elementary forward reactions show that the slow step is the second step, the attack of O on 03. The reverse reaction 02 + O, - O + O 3 is so slow that it can be ignored. Derive the rate law implied by the mechanism, and confirm that it matches the observed rate law. 

The pre-equilibrium and the steady-state approximations are two different approaches to deriving a rate law from a proposed mechanism, (a) For the following mechanism, determine the rate law by the steady-state approximation, (c) Under what conditions do the two methods give the same answer (d) What will the rate law become at high concentrations of Br ... 

C15-0076. Problem asks about a mechanism for the hypothetical reaction of AB with C. Assume that the first step described in Problem is fast and reversible and your second step is rate-determining. Derive the rate law under these assumptions. 

The order of the above reaction is, therefore, 1.5 + 0.5 = 2. This is typical of situations where the order of reaction and the molecularity of the reaction are the same. It may, however, be noted that the form of rate law, which determines the order of a reaction, can only be derived by actual experiment, and that may or may not be equal to the molecularity of the reaction as provided by the equation representing that reaction. Thus, a general reaction... 

From the mechanism given in problem 7-8 for the decomposition of acetaldehyde, derive a rate law or set of independent rate laws, as appropriate, if H2 and C2Hs are major products (in addition to CH4 and CO). 

FIGURE 1.18. Deriving the rate law of an electron transfer involving immobilized reactants from the cyclic voltammetric responses, a Voltammograms recorded at 1 ( ), 10 (A), 100 (O), 1000 (v), 10,000 (O) V/s. b Derivation of the surface concentrations from the current responses, c Potential-dependent rate constant from the combination of a and b. MHL kinetics with /., = 0.85 eV. Adapted from Figure 4 in reference 43, with permission from the American Chemical Society. 

King EL, Altman C. 1956. A schematic method of deriving the rate laws for enzyme-catalyzed reactions. J Phys Chem 60 1375. 

The rate of a reaction is usually measured in terms of the change of concentration, with time, of one of the reactants or products, - d [reactant]/clt or +r/ [products]/r/t, and is usually expressed as moles per liter per second, or M s . We have already seen how this information might be used to derive the rate law and mechanism of the reaction. Now we are concerned, as kineticists, with measuring experimentally the concentration change as a function of the time that has elapsed since the initiation of the reaction. In principle, any property of the reactants or products that is related to its concentration can be used. A large number of properties have been tried. 

These regions are particularly useful since few, if any, reactions of transition metal complexes are unaccompanied by spectral absorption changes in these regions. We first show how optical absorbances may be substitued for the concentration changes required in deriving the rate law. 

Derive the rate laws for (a) and (b) with Lj in excess, and with assumption of a steady-state concentration of the 5-coordinate species in (a) and a rds associated with in (b). Under what conditions will the rate laws for (a) and (b) be identical With these conditions in 1,2-dichloroethane AH " = 14.7 cm moU when M = Cr, L —L = 3,6-dithiaoctane and L, = PlOCjHjlj. Which mechanism is supported ... 

Derivation of rate equations is an integral part of the effective usage of kinetics as a tool. Novel mechanisms must be described by new equations, and famihar ones often need to be modified to account for minor deviations from the expected pattern. The mathematical manipulations involved in deriving initial velocity or isotope exchange-rate laws are in general quite straightforward, but can be tedious. It is the purpose of this entry, therefore, to present the currently available methods with emphasis on the more convenient ones. 

As discussed in Appendix A, symmetric tensors have properties that are important to the subsequent derivation of conservation laws. As illustrated in Fig. 2.9, there is always some orientation for the differential element in which all the shear strain rates vanish, leaving only dilatational strain rates. This behavior follows from the transformation laws... 

Morgan (11) derived a rate law that adequately describes the observed kinetics and was able to extract rate constants for both homogeneous and particle-catalyzed reactions. In laboratory experiments with sterile, filtered synthetic solutions, Mn2+ oxidation proceeds much more slowly than in natural waters. It may not occur at all at neutral or acidic pH, especially in the absence of catalytically active surfaces such as preformed oxidation products (12, 13). 

Derive die rate law diat would describe the rate of product formation for die following reaction assuming that the cationic intermediate is a steady-state intermediate ... 

Derive the rate law tiiat would describe die rate of product formation for the following reaction where no assumptions are made as to die relative magnitudes of k, k-, or kp. 

A reaction mechanism consists of a series of elementary reaction steps, the sum of which is the stoichiometric chemical equation. An elementary reaction is a precise statement of the molecules which react or are formed in a single step of a mechanism. When we write a mechanism with its elementary reaction steps, we are forming a hypothesis that we hope to test with experimental rate data. The first step in such a test is to derive the rate law expected from the postulated mechanism. 

The simple relationship between the rate law and stoichiometry in elementary reactions allows one to derive a rate law for any multistep mechanistic scheme. The agreement between the derived rate law and that determined experimentally provides support for the proposed mechanism, although it does not prove it. The lack of agreement, on the other hand, definitely rules out the proposed scheme. 

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