Steady-state approximation derivation

Assuming for simplicity that only a particular base B is an effective catalyst in equation 1, application of the steady-state approximation derives in equation 2 the expression of the second-order rate constant, k, at a given concentration of B. 

Using the steady-state approximation, derive the rate law for this mechanistic scenario. 

Using the steady-state approximation, derive the copolymer equation for the free radical synthesis of monomer Mt with monomer M,. Express your answer in tenns of the mole fraction of monomer 1 in the copolymer (Fj) and the mole fraction of monomer 1 in the feed (/j). 

Using the steady-state approximation, derive a rate equation for the disappearance of ethane. Simplify the rate expression as much as possible by neglecting the rates of any steps that are comparatively insignificant... 

It is assumed that irreversible aggregation occurs on contact. The rate of coagulation is expressed as the aggregation flux J of particles towards a central particle. Using a steady-state approximation, the diffusive flux is derived to be... 

A second common approximation is the steady-state condition. That arises in the example if /fy is fast compared with kj in which case [i] remains very small at all times. If [i] is small then d[I] /dt is likely to be approximately zero at all times, and this condition is commonly invoked as a mnemonic in deriving the differential rate equations. The necessary condition is actually somewhat weaker (9). Eor equations 22a and b, the steady-state approximation leads, despite its different origin, to the same simplification in the differential equations as the pre-equihbrium condition, namely, equations 24a and b. 

Assume that the steady-state approximation can be applied to the intermediate TI. Derive the kinetic expression for hydrolysis of the imine. How many variables must be determined to construct the pH-rate profile What simplifying assumptions are justified at very high and very low pH values What are the kinetic expressions that result from these assumptions ... 

Using the Bodenstein steady state approximation for the intermediate enzyme substrate eomplexes derives reaetion rate expressions for enzymatie reaetions. A possible meehanism of a elosed sequenee reaetion is ... 

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... 

Derive the expression shown for the rate constant for product buildup, making the steady-state approximation for the intermediates. It is best to use inventive shortcuts rather than tedious algebra. 

Steady-state mechanism. Derive the expression for -d[hR]/dt in this scheme, making the steady-state approximation for [A] and [B]. The answer must contain no concentration other than [AB],... 

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 ... 

Derive the rate expression. Make the steady-state approximation for the radical intermediates, and assume that the chains are long. 

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 ... 

There are several assumptions inherent in the HMD multicomponent derivation that should be understood. In the model derivation, the authors made a steady-state approximation, which means that the solution is applicable for Cases I and II only after the surface films have formed. The model is not applicable during the initial early dissolution phase. HMD also point out that the time re-... 

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. 

However, the complete solution of these three simultaneous differential equations is difficult to obtain and is no more instructive than the approximate solution that can be obtained by means of the steady-state approximation for intermediates. If one sets the time derivatives in equations 4.2.17 and 4.2.18 equal to zero and adds these equations, their sum is found to be... 

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

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). 

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 ... 

Eqs. (19) and (20) were derived applying the steady-state approximation to the oxidized Fe-TAML species and using the mass balance equation [Fe-TAML] = 1 + [oxidized Fe-TAML] ([Fe-TAML] is the total concentration of all iron species, which is significantly lower than the concentrations of H2O2 and ED). The oxidation of ruthenium dye 8 is a zeroth-order reaction in 8. This implies that n[ED] i+ [H202]( i+ m). Eq. (19) becomes very simple, i.e.,... 

The kinetic analysis of the mechanism 6a-6e,2 is more complicated than that of the mechanism 4a-4e because of the external reaction 6e but nevertheless is feasible using the steady state approximation. By a procedure similar to the derivation of equation 5 the following equation can be derived ... 

Similar to irreversible reactions, biochemical interconversions with only one substrate and product are mathematically simple to evaluate however, the majority of enzymes correspond to bi- or multisubstrate reactions. In this case, the overall rate equations can be derived using similar techniques as described above. However, there is a large variety of ways to bind and dissociate multiple substrates and products from an enzyme, resulting in a combinatorial number of possible rate equations, additionally complicated by a rather diverse notation employed within the literature. We also note that the derivation of explicit overall rate equation for multisubstrate reactions by means of the steady-state approximation is a tedious procedure, involving lengthy (and sometimes unintelligible) expressions in terms of elementary rate constants. See Ref. [139] for a more detailed discussion. Nonetheless, as the functional form of typical rate equations will be of importance for the parameterization of metabolic networks in Section VIII, we briefly touch upon the most common mechanisms. 

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... 

Traditionally, reaction mechanisms of the kind above have been analysed based on the steady-state approximation. The differential equations for this mechanism cannot be integrated analytically. Numerical integration was not readily available and thus approximations were the only options available to the researcher. The concentrations of the catalyst and of the intermediate, activated complex B are always only very low and even more so their derivatives [Cat] and [B]. In the steady-state approach these two derivatives are set to 0. 

Textbooks state that the pseudo-steady-state approximation will be valid if the concentration of a species is small. However, one then proceeds by setting its time derivative equal to zero (]/t/f = 0) in the batch reactor equation, not by setting the concentration (CH3CO ) equal to zero. This logic is not obvious from the batch reactor equations because setting the derivative of a concentration equal to zero is not the same as setting its concentration equal to zero. 

As for the quasi (pseudo)-steady-state case, the basic assumption in deriving kinetic equations is the well-known Bodenshtein hypothesis according to which the rates of formation and consumption of intermediates are equal. In fact. Chapman was first who proposed this hypothesis (see in more detail in the book by Yablonskii et al., 1991). The approach based on this idea, the Quasi-Steady-State Approximation (QSSA), is a common method for eliminating intermediates from the kinetic models of complex catalytic reactions and corresponding transformation of these models. As well known, in the literature on chemical problems, another name of this approach, the Pseudo-Steady-State Approximation (PSSA) is used. However, the term "Quasi-Steady-State Approximation" is more popular. According to the Internet, the number of references on the QSSA is more than 70,000 in comparison with about 22,000, number of references on PSSA. 

The quasi-steady-state approximation then allows the concentration time derivatives to be set equal to zero,... 

Existence and uniqueness of the particular solution of (5.1) for an initial value y° can be shown under very mild assumptions. For example, it is sufficient to assume that the function f is differentiable and its derivatives are bounded. Except for a few simple equations, however, the general solution cannot be obtained by analytical methods and we must seek numerical alternatives. Starting with the known point (tD,y°), all numerical methods generate a sequence (tj y1), (t2,y2),. .., (t. y1), approximating the points of the particular solution through (tQ,y°). The choice of the method is large and we shall be content to outline a few popular types. One of them will deal with stiff differential equations that are very difficult to solve by classical methods. Related topics we discuss are sensitivity analysis and quasi steady state approximation. 

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 mechanism depicted in Scheme 2 involves two main steps. Rupture of the first metal-nitrogen bond accompanied by coordination of a water ligand at the metal center is followed by reversible deprotonation and intramolecular reduction of the metal center. Under the experimental conditions wherein the concentration of base is much larger than the concentration of tris(diimine) complex, and, applying the steady-state approximation to the concentration of the intermediate species with the monodentate diimine ligand, Eq. (6) can be derived as... 

The following treatment applies to the case where the solids are stationary in a shallow packed bed, so that they can be considered to be in well-mixed conditions, and that the solute initially saturates the solid, as in the case of vegetable oil in crushed seeds. For the quasi--steady-state approximation, Brunner [51] derived a practical equation ... 

Moreover if we apply equation (2) to the deep interior of stars (r=0), eather the velocity or the density should become infinitely large at r=0. Therefore we cannot get any normal stellar structures. This means that equation (2) is inadequate to the interior part of stars. This difficulty comes from the steady-state approximation (1). The mass flux must reduce zero at the center of the stars or the surface of the degenerate stars. The interior flow therefore should be described by another steady states, not by equation (1). Therefore we will present a new steady-state approximation and derive mass-loss equations which is available also to the deep interior of stars. 

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

Steady-state approximation is based on the concept that the formation of [ES] complex by binding of substrate to free enzyme and breakdown of [ES] to form product plus free enzyme occur at equal rates. A graphical representation of the relative concentrations of free enzyme, substrate, enzyme-substrate complex, and product is shown in figure 7.8 in the text. Derivation of the Michaelis-Menten expression is based on the steady-state assumption. Steady-state approximation may be assumed until the substrate concentration is depleted, with a concomitant decrease in the concentration of [ES]. 

---