Consistent rate law

Appending Ft to the Monod equation, we write a thermodynamically consistent rate law,... 

Towards a consistent rate law glass corrosion kinetics near saturation... 

However, the postulated trimolecular mechanism is highly questionable. The third-order rate law would also be consistent with mechanisms arising from consecutive bimolecular elementary reactions, such as... 

The system of coupled differential equations that result from a compound reaction mechanism consists of several different (reversible) elementary steps. The kinetics are described by a system of coupled differential equations rather than a single rate law. This system can sometimes be decoupled by assuming that the concentrations of the intennediate species are small and quasi-stationary. The Lindemann mechanism of thermal unimolecular reactions [18,19] affords an instructive example for the application of such approximations. This mechanism is based on the idea that a molecule A has to pick up sufficient energy... 

For tliis model tire parameter set p consists of tire rate constants and tire constant pool chemical concentrations l A, 1 (Most chemical rate laws are constmcted phenomenologically and often have cubic or otlier nonlinearities and irreversible steps. Such rate laws are reductions of tire full underlying reaction mechanism.)... 

The deomposition of AIBN in xylene at 77°C was studiedt by measuring the volume of N2 evolved as a function of time. The volumes obtained at time t and t = 00, are and, respectively. Show that the manner of plotting used in Fig. 6.1 is consistent with the integrated first-order rate law and evaluate k j. 

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... 

This rate law could correspond to formation of a CI2-AICI3 complex that acts as the active halogenating agent but is also consistent with a rapid equiUbrium involving formation of C1+ ... 

Show that each of these mechanisms is consistent with the observed rate law rate = fc[NO]2 X [02]. 

Certain features of the reaction schemes manifest themselves in the rate law in a regular way. These features guide the investigator to one or more mechanisms consistent with the data. The same considerations allow one to reject certain alternatives. Listed here is a set of rules, or more properly clues, that are useful guides to the correct scheme. Each is accompanied by examples as to how they can be applied. 

Another example is found in the scheme shown in Eqs. (4-50) to (4-52). The rate law, Eq. (4-54), contains two terms, consistent with an intermediate that branches along two channels. The same scheme, but with the first step (formation of the intermediate) rate-controlling, would not reveal this detail. The kinetics tells about what happens in the rate-controlling step, and sometimes prior to it, but not afterward. 

A second scheme is equally consistent with the kinetic data. The steps and the steady-state rate law are as follows ... 

Reaction scheme. Propose reaction steps consistent with the rate law for the hydrolysis of benzhydryl chloride relate a and /3 to the rate constants. 

Rate law to reaction scheme. Write reactions consistent with the rate equation given. Then compare the result to that given in Problem 4-5 and reconcile any differences. 

To explore this premise further, imagine that the second initiation step could be neglected. That is, [H2POj ] would be chosen such that k fc2[H2P03 ]. We then have a scheme consisting of Eqs. (8-33), (8-35)—(8-37), and (8-47). With the usual approximations, the rate law is... 

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. 

The reader can show that a third scheme also gives the same answer. In it the two cations first associate (however unlikely), and this dinuclear complex reacts with Cl-. To summarize any reaction scheme consistent with the rate law is characterized by the same ionic strength effects. In other words, it is useless to study salt effects in the hopes of resolving one kinetically indistinguishable mechanism from another. 

Kinetically indistinguishable chain mechanisms can be characterized by different ionic strength profiles, as was apparently first demonstrated in a study this author conducted with D. A. Ryan on the reaction of (aqua)-2-propylchromium cation with oxygen.17 This reaction was presented in Chapter 7. Two schemes that are consistent with the rate law are as follows ... 

How does one know when the complete roster of reaction schemes that are consistent with the rate law has been obtained One method is based on an analogy between electrical circuits and reaction mechanisms.13 One constructs an electrical circuit analogous to the reaction scheme. Resistors correspond to transition states, junctions to intermediates, and terminals to reactants and products. The precepts are these (1) any other electrical circuit with the same conductance corresponds to a different but kinetically equivalent reaction scheme, and (2) these circuits correspond to all of the fundamentally different schemes. 

To verify that a proposed reaction mechanism agrees with experimental data, we construct the overall rate law implied by the mechanism and check to see whether it is consistent with the experimentally determined rate law. However, although the constructed rate law and the experimental rate law may be the same, the proposed mechanism may still he incorrect because some other mechanism may also lead to the same rate law. Kinetic information can only support a proposed mechanism it can never prove that a mechanism is correct. The acceptance of a suggested mechanism is more like the process of proof in an ideal court of law than a proof in mathematics, with evidence being assembled to give a convincing, consistent picture. 

The rate law of an elementary reaction is written from the equation for the reaction. A rate law is often derived from a proposed mechanism by imposing the steady-state approximation or assuming that there is a pre-equilibrium. To be plausible, a mechanism must be consistent with the experimental rate law. 

A full development of the rate law for the bimolecular reaction of MDI to yield carbodiimide and CO indicates that the reaction should truly be 2nd-order in MDI. This would be observed experimentally under conditions in which MDI is at limiting concentrations. This is not the case for these experimements MDI is present in considerable excess (usually 5.5-6 g of MDI (4.7-5.1 ml) are used in an 8.8 ml vessel). So at least at the early stages of reaction, the carbon dioxide evolution would be expected to display pseudo-zero order kinetics. As the amount of MDI is depleted, then 2nd-order kinetics should be observed. In fact, the asymptotic portion of the 225 C Isotherm can be fitted to a 2nd-order rate law. This kinetic analysis is consistent with a more detailed mechanism for the decomposition, in which 2 molecules of MDI form a cyclic intermediate through a thermally allowed [2+2] cycloaddition, which is formed at steady state concentrations and may then decompose to carbodiimide and carbon dioxide. Isocyanates and other related compounds have been reported to participate in [2 + 2] and [4 + 2] cycloaddition reactions (8.91. 

When a reaction proceeds in a single elementary step, its rate law will mirror its stoichiometry. An example is the rate law for O3 reacting with NO. Experiments show that this reaction is first order in each of the starting materials and second order overall NO + 03- NO2 + O2 Experimental rate = i [N0][03 J This rate law is fully consistent with the molecular view of the mechanism shown in Figure 15-7. If the concentration of either O3 or NO is doubled, the number of collisions between starting material molecules doubles too, and so does the rate of reaction. If the concentrations of both starting materials are doubled, the collision rate and the reaction rate increase by a factor of four. 

The two proposed mechanisms for this reaction predict different rate laws. Whereas Mechanism I predicts that the rate is proportional to NO2 concentration. Mechanism II predicts that the rate is proportional to the square of NO2 concentration. Experiments agree with the prediction of Mechanism II, so Mechanism II is consistent with the experimental behavior of the NO2 decomposition reaction. Mechanism I predicts rate behavior contrary to what is observed experimentally, so Mechanism I cannot be correct. 

A rate law lhat is first order in each reactant is consistent with a mechanism whose first step is a collision between reaclant molecules lhat results in a slow reaction. Both reactants are stable substances, so it makes sense that although they can collide and react, the reaction is slow. 

The reaction mechanism must be consistent with the experimental rate law. 

Experiments show that this reaction is second order in NO2, as predicted by the second proposed mechanism. The one-step mechanism can be ruled out because it is not consistent with the experimental rate law. Agreement with the rate law does not prove that the second mechanism is the correct one, however, because other mechanisms may predict the same rate law. It is one strong piece of evidence that supports this particular two-step process. 

Nitrogen oxide converts ozone into molecular oxygen, as follows O3 + NO O2 + NO2 The experimental rate law is rate = "[03][N0 j. Which of the following mechanisms are consistent with the experimental rate law ... 

Mechanism I is a three-step process in which the first step is rate-determining. When the first step of a mechanism is rate-determining, the predicted rate law is the same as the rate expression for that first step. Here, the rate-determining step is a bimolecular collision. The rate expression for a bimolecular collision is first order in each collision partner Rate = j i[03 ][N0 j Mechanism I is consistent with the experimental rate law. If we add the elementary reactions, we find that it also gives the correct overall stoichiometry, so this mechanism meets all the requirements for a satisfactory one. 

Co (NH3)5 H2 O] [Co (NH3)5 ] + H2 O (a) Propose a slow second step that completes the mechanism and gives the correct overall stoichiomehy. (b) Derive the rate law that this mechanism predicts, (c) When the rate is studied in 1 M aqueous HCl solution that is 1 mM in [Co (NH3)5 H2 O], first-order experimental kinetics are observed. Is this observation consistent with the proposed mechanism State your reasoning clearly and in detail. 

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