Reaction rates steady-state approximation

In formulating the kinetic expression for a heterogeneous as well as a homogeneous reaction, this steady state approximation is widely used with regard to the rate of change of extremely reactive intermediates, which are the surface complexes. Mathematical justification of this assumption has been demonstrated for certain simple situations [70, 71]. Thus, on applying the steady state approximation, Eqs. (2.4.30) and (2.4.31) become... 

For a sequenee of reaetion steps two more eoneepts will be used in kinetics, besides the previous rules for single reaetions. One is the steady-state approximation and the seeond is the rate limiting step eoneept. These two are in strict sense incompatible, yet assumption of both causes little error. Both were explained on Figure 6.1.1 Boudart (1968) credits Kenzi Tamaru with the graphical representation of reaction sequences. Here this will be used quantitatively on a logarithmic scale. 

The overall rate of a chain process is determined by the rates of initiation, propagation, and termination reactions. Analysis of the kinetics of chain reactions normally depends on application of the steady-state approximation (see Section 4.2) to the radical intermediates. Such intermediates are highly reactive, and their concentrations are low and nearly constant throughout the course of the reaction ... 

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. 

Consider further Scheme XIV and rate equations (3-139) to (3-141). Evidently Cb will be small relative to (Ca + Cc) if ( -i + 2) i- Then B plays the role of a reactive intermediate in the overall reaction A— C. This is the usual condition that is taken as a warrant for the application of the steady-state approximation. If Cb is small, it is reasonable that Cb will be small throughout most the reaction, so it is set equal to zero. As Wong (53) has pointed out, however, the condition Cb = 0 is a sufficient but unnecessary condition for Eq. (3-142) to hold. Erom Eq. (3-140) we obtain... 

If a reaction system consists of more than one elementary reversible reaction, there will be more than one relaxation time in general, the number of relaxation times is equal to the number of states of the system minus one. (However, even for multistep reactions, only a single relaxation time will be observed if all intermediates are present at vanishingly low concentrations, that is, if the steady-state approximation is valid.) The relaxation times are coupled, in that each relaxation time includes contributions from all of the system rate constants. A system of more than... 

Either step could be rate determining. Study of many reactions has shown that most occur with a significant isotopic effect, but for some reactions the isotope effect is absent.If we apply the steady state approximation to the intermediate, this reaction scheme leads to... 

Because the cationic intermediate is unstable, it will be permissible to apply the steady-state approximation, leading to Eq. (8-65) for the reaction rate. 

Given the postulated reaction scheme, the net rate of reaction often takes a simple form when it is expressed in terms of the concentration of the intermediate. Such an expression is algebraically correct, and is the form one needs so as to propose and interpret the mechanism. This form is, however, usually not useful for the analysis of the concentration-time curves. In such an expression the reaction rate is given in terms of the concentration of the intermediate, which is generally unknown at the outset. To eliminate the concentration term for the intermediate, one may enlist certain approximations, such as the steady-state approximation. This particular method is applicable when the intermediate remains at trace levels. 

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. 

Reactant fluxes. Calculate values of , for the combination of rate constants in Tables 4-1 and 4-2 for those systems for which the steady-state approximation holds. Construct a diagram of the fluxes at the start of the reaction when [A]o = 1. 

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 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 reaction rate of CO2 under the condition of steady-state approximation to formation of Cl is presented as follows ... 

Assuming that the catalytic reaction takes place in a flow reactor under stationary conditions, we may use the steady state approximation to eliminate the fraction of adsorbed intermediate from the rate expressions to yield ... 

It is important to realize that the assumption of a rate-determining step limits the scope of our description. As with the steady state approximation, it is not possible to describe transients in the quasi-equilibrium model. In addition, the rate-determining step in the mechanism might shift to a different step if the reaction conditions change, e.g. if the partial pressure of a gas changes markedly. For a surface science study of the reaction A -i- B in an ultrahigh vacuum chamber with a single crystal as the catalyst, the partial pressures of A and B may be so small that the rates of adsorption become smaller than the rate of the surface reaction. 

Note that it is assumed that reaction (1) is reversible, while reactions (2) and (3) are irreversible. Show, using steady-state approximations for the intermediates, that the overall reaction rate can be written as... 

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. 

In principle there is a competition for the HO2 radical between peroxydisulphate and hydrogen peroxide [reactions (63) and (86)] however, when the stoichiometry is 1 1 reaction (86) can be neglected. Assuming that the chain length is large, with the usual steady-state approximation, we obtain the following rate equation ... 

Here, we see that the rate of the reaction depends on the square root of the concentration of the initiator and linearly on the concentration of the monomer. The steady state approximation fails when the concentration of the monomer is so low that the initiation reaction cannot occur at the same rate as the termination reaction. Under these conditions, the termination reaction dominates the observed kinetics. 

The intermediate reaction complexes (after formation with rate constant, fc,), can undergo unimolecular dissociation ( , ) back to the original reactants, collisional stabilization (ks) via a third body, and intermolecular reaction (kT) to form stable products HC0j(H20)m with the concomitant displacement of water molecules. The experimentally measured rate constant, kexp, can be related to the rate constants of the elementary steps by the following equation, through the use of a steady-state approximation on 0H (H20)nC02 ... 

Instead of using the steady-state approximation in the manipulation of the individual rate expressions, the same result may be reached by assuming that a pseudo equilibrium condition is established with respect to reaction B and that reaction C continues to be the rate limiting... 

In 1919 Christiansen (25), Herzfeld (26), and Polanyi (27) all suggested the same mechanism for this reaction. The key factor leading to their success was recognition that hydrogen atoms and bromine atoms could alternately serve as chain carriers and thus propagate the reaction. By using a steady-state approximation for the concentrations of these species, these individuals were able to derive rate expressions that were consistent with that observed experimentally. 

ILLUSTRATION 4.3 USE OF THE BODENSTEIN STEADY-STATE APPROXIMATION TO DERIVE A RATE EXPRESSION FROM A CHAIN REACTION MECHANISM... 

Since the branching parameter a is greater than unity (usually it is 2), it is conceivable that under certain circumstances the denominator of the overall rate expression could become zero. In principle this would lead to an infinite reaction rate (i.e., an explosion). In reality it becomes very large rather than infinite, since the steady-state approximation will break down when the radical concentration becomes quite large. Nonetheless, we will consider the condition that Mol - 1) is equal to (fst T fgt) to be a valid criterion for an explosion limit. 

Reaction rate expressions for enzymatic reactions are usually derived by making the Bo-denstein steady-state approximation for the intermediate enzyme-substrate complexes. This is an appropriate assumption when the substrate concentration greatly exceeds that of the enzyme (the usual laboratory situation) or when there is both a continuous supply of reactant and a continuous removal of products (the usual cellular situation). 

The steady-state approximation allows the concentrations of each species to be determined by assuming that nothing is changing significantly with time. Placing H2+ formed in the reaction network defined by Equation 5.8 into steady state requires that the processes leading to the formation of H2+ should have a zero effect on the rate of change of H2+ with respect to time that is the derivative should be zero ... 

Steady-state approximation The balance of rates of formation and destruction of chemical reactions, which, when it equals zero, gives a steady-state concentration of species in the mixture. 

A standard kinetic analysis of the mechanism 4a-4e using the steady state approximation yields a rate equation consistent with the experimental observations. Thus since equations 4a to 4e form a catalytic cycle their reaction rates must be equal for the catalytic system to be balanced. The rate of H2 production... 

In this equation, kjy is the rate constant for the diffusion-limited formation of the encounter complex, d is the rate constant for diffusion apart, and ka is that for the activation step, i.e. M-L bond formation. Based on the steady-state approximation for the encounter complex concentration, the apparent rate constant for the on reaction is kon = k kj (k - ,+ka), and the activation volume is defined as... 

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 ... 

King et a/.54,138,155 applied the steady state approximation to both systems. In the case of Fe(CO)5, as shown in Scheme 7b, both C02 and H2 production rates should be the same, and so 2[Fe(CO)5][OH-] = fc4[H2Fe(CO)4]. Reactant concentrations are far from equilibrium, and the reactions are assumed to be driven to the right. 

Insofar as the intermediate B obeys the steady-state approximation, as is usually the case in practice, there are two limiting situations as to the nature of the rate-limiting step according to the value of the parameter A e/lc = k-rCp/kc, which measures the competition between the followup reaction and the backward electron transfer (see Section 6.2.7). 

Since the enzymatic reactions are generally rapid, it may be assumed that the steady-state approximation applies. Note, however, that although true is most systems, this is not always the case, as exemplified in Section 5.2.5. Each half-reaction is characterized by three rate constants, defined in Scheme 5.1. They may alternatively be characterized by the following... 

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