Rate laws, approximate

Betts obtained evidence for the exchange in sulphate media being catalysed by light of wavelengths 340 m/t, the region of absorption of U(VI). In the presence of light a rate law, approximately... 

Figure 15-1 Total pressure dependence of the best pseudo-first-order kinetic rate constant when a first-order rate law approximates a Hougen-Watson model for dissociative adsorption of diatomic A2 on active catalytic sites. Irreversible triple-site chemical reaction between atomic A and reactant B (i.e., 2Acr - - Bcr -> products) on the catalytic surface is the rate-limiting step. The adsorption/desorption equilibrium constant for each adsorbed species is 0.25 atm. ...

In methanol solution the dioxygen complex, IX, reacted with added PMe2Ph to form Me2PhPO and regenerate [Co(CN)2(PMe2Ph)3] according to equation (87) with a rate law approximating equation (88), where k 4 x 10" M" sec". ... 

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... 

The order of the rate law with respect to the three reactants can be determined by comparing the rates of two experiments in which the concentration of only one of the reactants is changed. For example, in experiment 2 the [H+] and the rate are approximately twice as large as in experiment 1, indicating that the reaction is first-order in [H+]. Working in the same manner, experiments 6 and 7 show that the reaction is also first-order with respect to [CaHeO], and experiments 6 and 8 show that the rate of the reaction is independent of the [I2]. Thus, the rate law is... 

The result of the steady-state condition is that the overall rate of initiation must equal the total rate of termination. The application of the steady-state approximation and the resulting equality of the initiation and termination rates permits formulation of a rate law for the reaction mechanism above. The overall stoichiometry of a free-radical chain reaction is independent of the initiating and termination steps because the reactants are consumed and products formed almost entirely in the propagation steps. 

Steady-state. An erroneous rate law is shown below for the reaction scheme believed to represent the reaction between Fe3+ and I-, in that an extraneous denominator term appears. In the scheme shown, I2 and Fel2+ obey the steady-state approximation. Show what the incorrect part of the expression is. Suggest a simple derivation of the correct equation that avoids extensive algebraic manipulations. 

Making the steady-state approximation for [PFe], derive the rate law. Next, repeat the derivation including the reverse step with k-2. If [CO] and [02] are s> [PFe(O2)]0, what is the expression for ke, as defined in Chapter 3 ... 

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... 

With the steady-state approximation for [V(OH)Cr4+], the rate law becomes... 

In this scheme, CHO appears irrelevant we return to it later. The rate law can be derived by making the steady-state approximation for each of the chain-carrying radical intermediates ... 

The new pathway, too, is a chain reaction Note that the first term of Eq. (8-31) does not give a meaningful transition state composition. Since the scheme in Eqs. (8-20M8-23) seems valid for the Cu2+-free reaction, we can seek to modify it to accommodate the new result. This approach is surely more logical than inventing an entirely new sequence. To arrive at the needed modification, we simply replace Eq. (8-23) by a new termination step, Eq. (8-30). With that, and the steady-state approximation, the rate law is... 

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... 

Also, the rates of the propagation steps are equal to one another (see Problem 8-4). This observation is no surprise The rates of all the steps are the same in any ordinary reaction sequence to which the steady-state approximation applies, since each is governed by the same rate-controlling step. The form of the rate law for chain reactions is greatly influenced by the initiation and termination reactions. But the chemistry that converts reactant to product, and is presumably the matter of greatest importance, resides in the propagation reactions. Sensitivity to trace impurities, deliberate or adventitious, is one signal that a chain mechanism is operative. 

The rate law for each, with the steady-state and long-chain approximations, is... 

As before, the mechanism gives rise to an overall third-order rate law, in agreement with experiment. Although this procedure is much simpler than the steady-state approach, it is less flexible it is more difficult to extend to more complex mechanisms and it is not so easy to establish the conditions under which the approximation is valid. 

STRATEGY Construct the rate laws for the elementary reactions and combine them into the overall rate law for the decomposition of the reactant. If necessary, use the steady-state approximation for any intermediates and simplify it by using arguments based on rapid pre-equilibria and the existence of a rate-determining step. 

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. 

Models of population growth are analogous to chemical reaction rate equations. In the model developed by Malthus in 1798, the rate of change of the population N of Earth is dN/dt = births — deaths. The numbers of births and deaths are proportional to the population, with proportionality constants b and d. Derive the integrated rate law for population change. How well does it fit the approximate data for the population of Earth over time given below ... 

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 ... 

M, has observed a slow exchange in media 3 M with respect to HOO4. The experimental results suggested a complicated rate law of approximately the form... 

Keenan has made an investigation of the exchange reaction between Pu(IV) and Pu(in) in perchlorate media. The isotopic method was used with an a energy analyser to separate the tracer activity ( Pu) from that normally present from the major constituent ( Pu). Tributylphosphate extraction of the Pu(IV) formed the basis of the separation method. It was shown that the rate law has the approximate form... 

The authors propose that AgO oxidises water to H2O2 which is rapidly oxidised in turn by Ag(II) to H02 , itself oxidised by further Ag(II) to O2. Application of the steady-state approximation to [Ag(nr)], [AgO ], [H2O2] and [HO2 ] produces the observed rate law (A). 

Applying a steady-state approximation to [H02 ] the mechanism leads to the rate law... 

Reaction (61) implies that the transition state contains 3 SCN ions. A steady state approximation for [ (SCN)2 ] leads to the observed rate law. A temperature variation study of /fj and is included in this paper. It was also concluded that simple redox breakdown of FeSCN is of negligible importance. 

An enormous number of phase transitions are known to occur in common solid compounds. For example, silver nitrate undergoes a displacive phase transition from an orthorhombic form to a hexagonal form at a temperature of approximately 162°C that has a enthalpy of 1.85 kj/mol. In many cases, the nature of these transitions are known, but in other cases there is some uncertainty. Moreover, there is frequently disagreement among the values reported for the transition temperatures and enthalpies. Even fewer phase transitions have been studied from the standpoint of kinetics, although it is known that a large number of these transformations follow an Avrami rate law. There is another complicating feature of phase transitions that we will now consider. 

The most that can be said for relative activities at this stage is that to a good approximation in each period d > d > d Q, and in each group 4d>3d>5d. Only for cobalt has sufficient rate law data been obtained (12,13) to allow future comparisons of absolute activities with respect to CO hydrogenation to oxygenates. 

Figure 17.1 shows the calculation results. The mass of CrVI decreases at a rate mirroring the increase in Cr111 mass, which is twice the rate at which Reaction 17.28 proceeds. Dissolved sulfide in the simulation is divided approximately evenly between HS- and H2S(aq), since pH is held to 7. The reaction consumes H2S(aq) as well as Cr()/, causing the concentration of each to decline. Since the two concentrations appear as first order terms in the rate law, reaction rate also decreases with time. 

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