Rate law steady-state approximation

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... 

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. 

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. 

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 ... 

The most common states of a pure substance are solid, liquid, or gas (vapor), state property See state function. state symbol A symbol (abbreviation) denoting the state of a species. Examples s (solid) I (liquid) g (gas) aq (aqueous solution), statistical entropy The entropy calculated from statistical thermodynamics S = k In W. statistical thermodynamics The interpretation of the laws of thermodynamics in terms of the behavior of large numbers of atoms and molecules, steady-state approximation The assumption that the net rate of formation of reaction intermediates is 0. Stefan-Boltzmann law The total intensity of radiation emitted by a heated black body is proportional to the fourth power of the absolute temperature, stereoisomers Isomers in which atoms have the same partners arranged differently in space, stereoregular polymer A polymer in which each unit or pair of repeating units has the same relative orientation, steric factor (P) An empirical factor that takes into account the steric requirement of a reaction, steric requirement A constraint on an elementary reaction in which the successful collision of two molecules depends on their relative orientation. 

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. 

Equilibrium studies under anaerobic conditions confirmed that [Cu(HA)]+ is the major species in the Cu(II)-ascorbic acid system. However, the existence of minor polymeric, presumably dimeric, species could also be proven. This lends support to the above kinetic model. Provided that the catalytically active complex is the dimer produced in reaction (26), the chain reaction is initiated by the formation and subsequent decomposition of [Cu2(HA)2(02)]2+ into [CuA(02H)] and A -. The chain carrier is the semi-quinone radical which is consumed and regenerated in the propagation steps, Eqs. (29) and (30). The chain is terminated in Eq. (31). Applying the steady-state approximation to the concentrations of the radicals, yields a rate law which is fully consistent with the experimental observations ... 

What is meant by steady state approximation Determine the rate law, i.e. in the following reaction, using the steady state approximation ... 

In dealing with a complex reaction scheme as the one indicated above, one frequently introduces a so-called steady state approximation for reactive intermediates in order to find simplified rate laws (see, for example, Section 11.2). This approximation is usually sufficiently valid to give rise to useful results most physical chemistry texts discuss and use this application. In the steady state approximation for A, one writes... 

Using the steady-state approximation (s.s.a.) the rate law for the formation of the product, derived,... 

As indicated in Scheme 3.7, the first step of an ElcB mechanism can be reversible and therefore deprotonation at the 3-carbon does not always lead to product formation. By applying a steady-state approximation to the carbanion concentration, the following rate law is obtained for an ElcB reaction ... 

Consider a straight tubular runner of length L. A melt following the power-law model is injected at constant pressure into the runner. The melt front progresses along the runner until it reaches the gate located at its end. Calculate the melt front position, Z(f), and the instantaneous flow rate, Q t), as a function of time. Assume an incompressible fluid and an isothermal and fully developed flow, and make use of the pseudo-steady-state approximation. For a polymer melt with K = 2.18 x 10 N s"/m and n = 0.39, calculate Z(t) and Q(t)... 

The pre-equilibrium and the steady-state approximations are two different approaches to deriving a rate law from a proposed mechanism. 

The steady-state approximation assumes that since I is very reactive, its concentration will be very low at any time during the reaction and it will not change appreciably. Therefore dl/dr = 0. Solving the above expression for the concentration of I and substitution into the rate law for the formation product gives... 

Complicated rate-law focus on rate determining step. The intermediate formed at this step can be modeled using transition-state-theory. The steady-state approximation works for reactions with unstable intermediates. 

Different rate laws have been observed experimentally, depending on working conditions, i.e. first order in acetaldehyde and general base catalysed, or second order in acetaldehyde and specific base catalysed [ 17]. We shall use the steady-state approximation to deduce rate laws based upon the mechanism of Scheme 4.7, and see how different mechanistic assumptions lead to different rate law predictions. 

By applying the steady-state approximation to the concentration of radicals R in the reaction scheme in Equations 8.142-8.143 where (Co) represents the cobalt center and nonparticipating ligands, one obtains the rate law in Equation 8.144. At sufficiently high [TEMPO], so that /ct[TEMPO] /cr[(Con)], the rate constant kobs reduces to kf. 

The rate law for the halogenation reaction shown above is derived step by step in Equations 1.4-1.8. We will learn to set up derivations of this type in Section 2.4.1. There we will use a much simpler example. We will not discuss Bodenstein s steady-state approximation used in Equations 1.6 and 1.7 in more detail until later (Section 2.5.1). What will be explained there and in the derivation of additional rate laws in this book is sufficient to enable you to follow the derivation of Equations 1.4—1.8 in detail in a second pass through this book (and you should make several passes through the book to ensure you understand the concepts). 

Energy Profile and Rate Law of SN1 Reactions Steady State Approximation... 

The concentration of an intermediate in a multistep reaction is always very low when it reacts faster than it is produced. If this concentration is set equal to zero in the derivation of the rate law, unreasonable results may be obtained. In such a case, one resorts to a different approximation. The change of the concentration of this intermediate as afunction of time is set equal to zero. This is equivalent to saying that the concentration of the intermediate during the reaction takes a value slightly different from zero. This value can be considered to be invariant with time, i.e., steady. Consequently this approximation is called the steady state approximation. 

Gao et al. [71, 72] developed a mathematical model to describe the effect of formulation composition on the drug release rate for hydroxypropyl methylcellulose-based tablets. An effective drug diffusion coefficient T>, was found to control the rate of release as derived from a steady-state approximation of Fick s law in one dimension ... 

The rates of the overall reactions can be related to the rate law expressions of the individual steps by using the steady state approximation. However simple kinetic data alone may not distinguish a mechanism where, for example, a metal and an olefin form a small amount of complex at equilibrium that then goes on to react, from one in which the initial complex undergoes dissociation of a ligand and then reacts with the olefin. As a reaction scheme becomes more complex such steady state approximations become more complicated, but numerical methods are now available which can simulate these even for complex mixtures of reactants. 

It should be pointed out that all of the rate expressions were derived using the steady state approximation on an appropriate intermediate and represent limiting cases of the complete rate laws. These approximations are always used in the theoretical calculations, but the reaction order approach does not require such approximations since experimental data are treated directly. 

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