Reactants steady-state approximation

Kinetics of a two component system, PET and ABA in which phase separation occurs has been investigated. To retain simplicity of the analysis few assumptions were made. A generalized scheme in which acetic acid is produced through two channels is considered valid for PET rich and poor phase. Kinetically both these reactions were assumed and shown to be of second order with respect to reactants. Steady state approximation has been considered. Parameters were chosen such that the least squares deviation between moles of... 

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. 

Table 4-1 lists six combinations of rate constants for which an RCS can be defined and two others lacking one. A method has been presented for exploring the concept of the RCS by means of reactant fluxes.11 Consider the case k < (k- + k2), such that the steady-state approximation is valid. One defines an excess rate , for each step i as the difference between the forward rate of that step and the net forward rate v/. Thus, for Step 1,... 

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. 

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. 

In industry, as well as in a test reactor in the laboratory, we are most often interested in the situation where a constant flow of reactants enters the reactor, leading to a constant output of products. In this case all transient behavior due to start up phenomena have died out and coverages and rates have reached a constant value. Hence, we can apply the steady state approximation, and set all differentials in Eqs. (142)-(145) equal to zero ... 

It will be noted that conversion of the intermediate V(OH) Cr " to products involves a different number of H ions than its conversion back to reactants. It is considered likely that the binuclear intermediate has an inner-sphere structure. On applying the steady-state approximation to the concentration of this intermediate, it follows that... 

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 ... 

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). 

We now proceed to derive the rate expression by creating an intermediate X, as shown in Fig. 2 X is a species that is equally likely to form products or to return to the reactants. Applying the steady state approximation to X we obtain (5) where R is the reactant(s). 

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. 

Restricting ourselves to the rapid equilibrium approximation (as opposed to the steady-state approximation) and adopting the notation of Cleland [158 160], the most common enzyme-kinetic mechanisms are shown in Fig. 8. In multisubstrate reactions, the number of participating reactants in either direction is designated by the prefixes Uni, Bi, or Ter. As an example, consider the Random Bi Bi Mechanism, depicted in Fig. 8a. Following the derivation in Ref. [161], we assume that the overall reaction is described by vrbb = k+ [EAB — k EPQ. Using the conservation of total enzyme... 

In general, the effective overall rate constant associated with loss of reactants can be expressed in terms of the individual rate constants a-e in eq 3 by use of the steady state approximation. Simpler expressions can be obtained if the species related by diffusion (a,b) and activation (c,d) processes are assumed to be in thermal equilibrium. In such a case one finds straightforwardly that the effective first order rate constant, k (r), for electron transfer at separation r can be written as... 

In order to better understand the detailed dynamics of this system, an investigation of the unimolecular dissociation of the proton-bound methoxide dimer was undertaken. The data are readily obtained from high-pressure mass spectrometric determinations of the temperature dependence of the association equilibrium constant, coupled with measurements of the temperature dependence of the bimolecular rate constant for formation of the association adduct. These latter measurements have been shown previously to be an excellent method for elucidating the details of potential energy surfaces that have intermediate barriers near the energy of separated reactants. The interpretation of the bimolecular rate data in terms of reaction scheme (3) is most revealing. Application of the steady-state approximation to the chemically activated intermediate, [(CH30)2lT"], shows that. 

Therefore, we need to find approximate methods for simultaneous reaction systems that will permit finding analytical solutions for reactants and products in simple and usable form. We use two approximations that were developed by chemists to simplify simultaneous reaction systems (1) the equilibrium step approximation and (2) the pseudo-steady-state approximation... 

Radical chain reactions are complicated because multiple reactions occur, but the overall velocity of the sequence can be given in simplified form by applying steady-state approximations. An important feature of any chain reaction is that the velocities of all propagation steps must be identical because the radicals formed as products in each elementary reaction are the reactants in another elementary... 

This type of reaction for which the rate equation can be written according to the stoichiometry is called an elementary reaction. Rate equations for such cases can easily be derived. Many reactions, however, are non-elementary, and consist of a series of elementary reactions. In such cases, we must assume all possible combinations of elementary reactions in order to determine one mechanism that is consistent with the experimental kinetic data. Usually, we can measure only the concentrations ofthe initial reactants and final products, since measurements of the concentrations of intermediate reactions in series are difficult. Thus, rate equations can be derived under assumptions that rates of change in the concentrations of those intermediates are approximately zero (steady-state approximation). An example of such treatment applied to an enzymatic reaction is shown in Section 3.2.2. 

The general rate expressions for the hydrolysis and formation of an ester by this mechanism are complicated, and even the mechanism as written is a simplification, since ROH and ROH2+ will act as general base and general acid, respectively, just as H20 and H30+ do. In a solution containing only the reactants the initial rate of disappearance of ester, E, can be shown, by the use of the steady-state approximation for the tetrahedral intermediate, to be130... 

It remains now to solve Eq. (2.3). Here, there are various approaches, depending on the conditions. When a non-steady-state solution is required, one can introduce the decoupling approximation of Sumi and Marcus, if there is the difference in time scales mentioned earlier. Or one can integrate Eq. (2.3) numerically. For the steady-state approximation either Eq. (2.3) can again be solved numerically or some additional analytical approximation can be introduced. For example, one introduced elsewhere [44] is to consider the case that most of the reacting systems cross the transition state in some narrow window (X, X i jA), narrow compared with the X region of the reactant [e.g., the interval (O,Xc) in Fig. 2]. In that case the k(X) can be replaced by a delta function, fc(Xi)A5(X-Xi). Equation (2.3) is then readily integrated and the point X is obtained as the X that maximizes the rate expression. The A is obtained from the width of the distribution of rates in that system [44]. 

For example, the kinetics may be different within cells, where molar concentrations of enzymes often exceed those of substrate, than in the laboratory. In most laboratory experiments the enzyme is present at an extremely low concentration (e.g., 10 8 M) while the substrate is present in large excess. Under these circumstances the steady-state approximation can be used. For this approximation the rate of formation ofES from free enzyme and substrate is assumed to be exactly balanced by the rate of conversion ofES on to P. That is, for a relatively short time during the duration of the experimental measurement of velocity, the concentration of ES remains essentially constant. To be more precise, the steady-state criterion is met if the absolute rate of change of a concentration of a transient intermediate is very small compared to that of the reactants and products.19... 

This represents the case where there is a fast preequilibrium preceding the rate-determining step. It is again not necessary to work through the steady-state approximation if this situation is known to exist. If the intermediate I is in equilibrium with the reactant A, then... 

The rate of a reaction involving a high-energy intermediate appears to depend on an observed first-order rate constant associated with the formation of the product (or disappearance of reactant), which can be expressed in a simplified manner in most cases by applying the steady-state approximation as obs = i 2/( -i +k2). The overall forward reaction (that includes steps associated with ki and k2) is significantly suppressed to the extent that k is comparable in magnitude to k2- In the case of decarboxylation, we propose that the reaction can be accelerated by a catalyst that is capable of effectively... 

The solution obtained using this system of equations is plotted as dashed lines in Figure 3.4. The solution based on this quasi-steady state approximation closely matches the solution obtained by solving the full kinetic system of Equations (3.27). The major difference between the two solutions is that the quasi-steady approximation does not account explicitly for enzyme binding. Therefore a + b remains constant in this case, while in the full kinetic system a + b + c remains constant. Since the fraction of reactant A that is bound to the enzyme is small (c/a << 1), the quasi-steady approximation is relatively accurate. 

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. 

In cases where a specific rate-determining step cannot be chosen, an analysis called the steady-state approximation is often used. The central feature of this method is the assumption that the concentration of any intermediate remains constant as the reaction proceeds. An intermediate is neither a reactant nor a product but something that is formed and then consumed as the reaction proceeds. 

This is the analytical expression of the steady-state approximation the time derivatives of the concentrations of reactive intermediates are equal to zero. Equation (4.2.9) must not be integrated since the result that y = constant is false [see Equation (4.2.7)]. What is important is that B varies with time implicitly through A and thus with the changes in A (a stable reactant). Another way to state the steady-state approximation is [Equation (4.2.4) with dy/dt = 0] ... 

An important feature of these electrical analogs is that when the steady-state approximation is valid the reciprocals of the indicated resistances are the analogs of fc[A] [B] [C]. . . , where A, B, C,. . . are the initial reactants, and k is the effective rate constant for the formation of the activated complex directly from the initial reactants, even if intermediates are involved. This means that the over-all rate law is found by combining the individual rate terms according to the rules for combining the analogous reciprocal resistances. 

If the reaction is such that the conversion from reactants to products takes place with no hesitation at the transition point as in Figure 12-1 (a), the structure at that state is called the transition state. If there is a structure that lasts a bit longer as in Figure 12-1(b), and particularly if it is detectable by some experimental means, it is called an intermediate. Frequently, the kinetic equations include intermediates, even if they remain undetected. Their presence allows treatment by a steady-state approximation, in which the concentration of the intermediate is assumed to be small and essentially unchanging during much of the reaction. Details of this approach are described later. 

The quasi-steady-state approximation works by replacing the differential equations for the rates of change of the intermediate species by algebraic conditions obtained by setting d[H]/dt = 0 etc. (see Section 4.8.5). In some cases, the resulting equations can be solved and manipulated algebraically allowing substitution into the overall rate equation to obtain a form that only involves explicitly the concentrations of the reactants and (perhaps) products. Such rate equations can then be compared with the empirical rate equations determined from experiment to test the validity of the assumed mechanism and to obtain quantitative values for the rate coefficients involved. 

Assuming that the steady-state approximation holds for [M A ], and that reactant and helium are both in large excess, integrating over the residence time t in the reactor yields an expression for the reactivity R ... 

Using values of k s for the E2AdE3 model in Equation 8 with the assumption k2/k3 - 0, we obtain the steady state (SS) first order rate constant k j gg = 36 s (mol/1) . This value is consistent with the first order rate constant k j = 31 obtained in earlier work (Narayan, R. Antal, Jr., M.J., submitted to J. Am. Chem. Soc.) from experimental data at low initial concentrations of 1-propanol reactant. Apparently the steady state approximation is valid at low reactant concentrations. Nevertheless, Figure 5... 

The steady-state approximation is a more general method for solving reaction mechanisms. The net rate of formation of any intermediate in the reaction mechanism is set equal to 0. An intermediate is assumed to attain its steady-state concentration instantaneously, decaying slowly as reactants are consumed. An expression is obtained for the steady-state concentration of each intermediate in terms of the rate constants of elementary reactions and the concentrations of reactants and products. The rate law for an elementary step that leads directly to product formation is usually chosen. The concentrations of all intermediates are removed from the chosen rate law, and a final rate law for the formation of product that reflects the concentrations of reactants and products is obtained. 

Note that the steady-state approximation does not imply that [X] is even approximately constant, only that its absolute rate of change is very much smaller than that of [A] and [D], Since according to the reaction scheme d[D]/dt = 2[X][C], the assumption that [X] is constant would lead—for the case in which C is in large excess—to the absurd conclusion that formation of the product D will continue at a constant rate even after the reactant A has been consumed. (2) In a stirred-flow reactor, a steady state implies a regime so that all concentrations are independent of time. 

Because the electronic factors become more favorable with decreasing separation of the two reactants, the most favorable configuration for electron transfer is one in which the two reactants are in contact. As a consequence, the first step in bimolecular electron-transfer reactions is the formation of a close-contact (or bridged) precursor complex from the separated reactants. The actual electron transfer occurs within the precursor complex to form a successor complex. This is followed by the dissociation of the successor complex to give the separated products. Provided that the formation of the precursor complex is not rate determining, the observed (second-order) rate constant for the electron transfer is equal to K k, where is the equilibrium constant for the formation of the precursor complex and k, is the first-order rate constant for electron transfer within the precursor complex. If the formation of the precursor complex is rate determining, then it is necessary to use a steady-state approximation for its concentration. 

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