Chemistry
Steady State Approximation
The steady state approximation is a method used to simplify the analysis of chemical reactions by assuming that the concentration of certain reactive intermediates remains constant over a short period of time. This allows for the derivation of rate equations and the determination of reaction mechanisms. The approximation is particularly useful in understanding complex reaction networks.
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8 Key excerpts on "Steady State Approximation"
eBook - PDF- John H. Lawrence, John W. Gofman, John H. Lawrence, John W. Gofman(Authors)
- 2013(Publication Date)
- Academic Press(Publisher)
The use of the steady-state kinetic approach requires that concen-trations of all forms of the enzyme be constant during the time that measurements are being made. It is generally assumed that such is the case under normal conditions for enzyme measurements. Since this assumption is not always valid (19), care must be taken to show that the accumulation of product is a linear function of time. In addition, it must be recognized that the concentration of active enzyme, substrates, and products are known only at the very beginning of the reaction. Therefore, it is important that activity measurements emphasize the earliest possible part of the reaction course in order to obtain a velocity that approaches the initial velocity. The transient state is usually so short that it can be ignored. For the remainder of this chapter it will be assumed that steady-state conditions obtain in reaction mixtures or that some special case of the steady state such as quasi-equilibrium obtains. Furthermore, it is assumed that initial velocities are being measured and product accu-mulation is linear over the reaction times used to define the initial velocities. 1 6 8 JAMES R. FISHER AND VINCENT D. HOAGLAND, JR. II. DERIVATION OF STEADY-STATE RATE EXPRESSIONS The purpose of this section is to describe a method that has been devised for deriving rate expressions for model systems. These systems may be reversible, involve many sequential steps, and involve several alternate pathways. Furthermore, this method allows exact rate expres-sions to be formulated directly from inspection of the models. This capability is extremely important since kinetic studies bearing on enzyme mechanisms must involve consideration of many systems. The symbolism, terminology, and mathematical operations used in this approach are presented by considering a series of progressively more complex models. The final portion of this section demonstrates the formulation of exact rate expressions by inspection.
eBook - ePubEnzymes
A Practical Introduction to Structure, Mechanism, and Data Analysis
- Robert A. Copeland(Author)
- 2023(Publication Date)
- Wiley(Publisher)
Equation 5.12 , and despite the differences between the rapid equilibrium and steady‐state approaches, the final steady‐state equation is commonly referred to as the Henri–Michaelis–Menten equation.Steady state refers to a time period of the enzymatic reaction during which the rate of formation of the ES complex is exactly matched by its rate of decay to free enzymes and products. This kinetic phase can be attained when the concentration of substrate molecules is in great excess of the free enzyme concentration. To achieve a steady state, certain conditions must be met, and these conditions allow us to make some reasonable assumptions, which greatly simplify the mathematical treatment of kinetics. These assumptions are as follows:- During the initial phase of the reaction progress curve (i.e., conditions under which we are measuring the linear initial velocity), there is no appreciable buildup of any intermediates other than the ES complex. Hence, all the enzyme molecules can be accounted for by either the free enzyme or by the enzyme–substrate complex. The total enzyme concentration [E] is, therefore, given by:
(5.13)
- As in the rapid equilibrium treatment, we assume that the enzyme is acting catalytically, so that it is present in a very low concentration relative to the substrate, that is, [S] ≫ [E]. Hence, the formation of the ES complex does not significantly diminish the concentration of free substrate. We can, therefore, make the approximation: [S]f ∼ [S], where [S]f
eBook - ePubPhysical Chemistry
Kinetics
- Horia Metiu(Author)
- 2006(Publication Date)
- CRC Press(Publisher)
t) is the concentration of the product when the steady-state approximation is valid.§18. The Concentration of the Complex and of the Enzyme in the Steady-state Approximation.The calculation of C(t) and E(t) in the steady-state approximation uses Eqs 9.24 and 9.23. These express C(t) and E(t) in terms of R(t), which, in turn, is given by Eq. 9.34.The Michaelis–Menten Mechanism: How Good is the Steady-state Approximation?
§19. Introduction.
Approximations simplify a problem but add new burdens: if we plan to use the results to fit the experiments, we must ensure that the data used in the analysis were taken under conditions for which the errors made by the approximation are acceptable.In this section I examine, by using two examples, whether the steady-state approximation works well for the Miehaelis–Menten mechanism. We will find that this is an excellent approximation when the enzyme concentration is much smaller than that of the reactant. This is the case in many enzyme-catalyzed reactions.Fig. 9.4 shows a comparison of the numerically exact C(t) with the one obtained from the steady-state approximation. The calculations were made in Cell 2 of Workbook K9.4, for R(0) = E(0) = 1, k1 = 1.2, k−1 = 0.6, and k2 = 2.8. Note that the rate constants are of comparable magnitude and the initial concentration of the enzyme is the same as that of the reactant. In practice the concentration of the enzyme is much lower than that of the reactant, but we examine this case to see how well the steady-state approximation works under these conditions.Figure 9.4 The change in concentration of the complex with time in the case when R(0) = 1, E(0) = 1, P(0) = 0, k1 = 1.2, k−1 = 0.6, and k2 = 2.8. The solid line shows the result of the exact calculation, and the dotted line, that of the steady-state approximation.There are substantial differences between the exact and the steady-state values of C(t) at short time. This is not a surprise: the initial concentration C
eBook - PDFKinetics of Chemical Processes
Butterworth-Heinemann Series in Chemical Engineering
- Michel Boudart, Howard Brenner(Authors)
- 2014(Publication Date)
- Butterworth-Heinemann(Publisher)
This is an example where departure from the steady-state condition can be produced deliberately in order to reach a useful kinetic goal. But, in general, if there are reasons to believe that the steady-state approximation does not apply, the situation is usually very bad. Thus, in addition polymeri-zation, while the steady-state approximation normally applies to the total concentration of active centers, it is much more questionable to apply the steady-state condition to individual classes of active centers, i.e., active polymer chains of a given length. Yet, to obtain a distribution of molecular weights, the steady-state approximation, as applied to active polymer chains, is the only technique leading to tractable results. Without it, calculations become quite formidable. It may be noted that the distribution of molecular weights in the polymeric product can be shown to be sensitive to the mode of termination and even to the type of reaction system chosen. Thus different types of distribution are obtained in batch and in stirred-flow reactors. Since the molecular weight distribution affects the properties of the polymer, it is clear that a number of basic kinetic principles are of decisive importance in polymer chemistry — the steady-state approximation and the type of reactor in particular. Problem 3.6.1 Show that the chain transfer constant c s can also be determined by measuring the average degree of polymerization both in the absence [DP]o and in the presence [DP] of chain transfer agent: _!_ = _!_ + , SÊL [DP] [DP], (M) This expression shows the effect of S on the polymer chain length. BIBLIOGRAPHY 3.1 Although the term active center is frequently associated with reactive sites at a solid surface following the proposal of Hugh S. Taylor in 1925, the name has also been used extensively, especially by Semenov's school, to denote reactive 80 The Steady-State Approximation: Catalysis Chap. 3 intermediates in chain reactions.
eBook - ePub- Athel Cornish-Bowden(Author)
- 2013(Publication Date)
- Wiley-Blackwell(Publisher)
Chapter 14 Fast Reactions14.1 Limitations of steady-state measurements
14.1.1 The transient state
It should be obvious that experimental methods for investigating fast reactions, with half-times of much less than 1 s, must be different from those used for slower reactions, because in most of the usual methods it takes seconds or more to mix the reactants. Less obviously, the kinetic equations needed for the study of fast reactions are also different, because the steady state of an enzyme-catalyzed reaction is usually established fast enough to be considered to exist throughout the period of investigation, provided that this period does not include the first second after mixing (Section 2.5 ).§ 2.5, pages 43–45Consequently, most of the equations that have been discussed in this book are based on the steady-state assumption. By contrast, fast reactions are concerned, almost by definition, with the transient state (or transient phase ) of a reaction before the establishment of a steady state and cannot be described by steady-state rate equations. This chapter deals with experimental and analytical aspects of this phase.The differential equations that define simple chemical reactions, such as those considered in Chapter 1 , are linear and have solutions that consist of exponential terms of the form A exp(−λt ), where t is the time, A is a constant known as the amplitude, and λ is a constant that is called the frequency constant in this chapter (but see Section 14.1.2 ). For example, the second term of equation 1.3 is an exponential term with frequency constant k and amplitude a 0 . Such an exponential term is equal to the amplitude when t
eBook - ePub- Mark E. Davis, Robert J. Davis, Robert J. Davis(Authors)
- 2013(Publication Date)
- Dover Publications(Publisher)
CHAPTER 4The Steady-StateApproximation: Catalysis
4.1 | Single Reactions
One-step reactions between stable molecules are rare since a stable molecule is by definition a quite unreactive entity. Rather, complicated rearrangements of chemical bonds are usually required to go from reactants to products. This implies that most reactions do not proceed in a single elementary step as illustrated below for NO formation from N2 and O2 :Normally, a sequence of elementary steps is necessary to proceed from reactants to products through the formation and destruction of reactive intermediates (see Section 1.1 ).Reactive intermediates may be of numerous different chemical types (e.g., free radicals, free ions, solvated ions, complexes at solid surfaces, complexes in a homogeneous phase, complexes in enzymes). Although many reactive intermediates may be involved in a given reaction (see Scheme 1.1.1), the advancement of the reaction can still be described by a single parameter—the extent of reaction (see Section 1.2 ). If this is the case, the reaction is said to be single. Why an apparently complex reaction remains stoichiometrically simple or single, and how the kinetic treatment of such reactions can be enumerated are the two questions addressed in this chapter.There are two types of sequences leading from reactants to products through reactive intermediates. The first type of sequence is one where a reactive intermediate is not reproduced in any other step of the sequence. This type of sequence is denoted as an open sequence. The second type of sequence is one in which a reactive intermediate is reproduced so that a cyclic reaction pattern repeats itself and a large number of product molecules can be made from only one reactive intermediate. This type of sequence is closed and is denoted a catalytic or chain reaction cycle. This type of sequence is the best definition of catalysis .A few simple examples of sequences are listed in Table 4.1.1 . The reactive intermediates are printed in boldface and the stoichiometrically simple or single reaction is in each case obtained by summation of the elementary steps of the sequence. While all reactions that are closed sequences may be said to be catalytic, there is a distinct difference between those where the reactive intermediates are provided by a separate entity called the catalyst
- John Happel(Author)
- 2012(Publication Date)
- Academic Press(Publisher)
Chapter 5 Steady-State Tracing 5.1 Introduction As was noted in the previous chapter, the steady-state tracing equations are a special case of the general equations for superposition modeling in which the time derivatives become zero. In steady-state tracing we are not able to determine surface concentrations of intermediates because the concentrations of tracer in such species do not change in the course of an experiment. The parameters to be determined are the velocities of individual mechanistic steps. An earlier survey (Happel, 1972) discusses some of the general principles involved and gives a number of industrial catalysis examples. At about the same time Emmett (1972) published a general review of the use of isotopic tracers in catalysis that was devoted to steady-state tracing. His review for the most part is not devoted to modeling for the purpose of estimating rates of individual reaction steps. While less comprehensive than transient modeling, the steady-state method can be applied more readily in the case of plug-flow reactors because of the elimination of the time variable present in transient tracing. Steady-state tracing is also advantageous where exchange of tracer atoms can occur with the catalyst support or with a molten phase in the catalyst pores, because of possible difficulties in the interpretation of transient data. In an elegant statistical-mechanical development, Horiuti showed how transition-state theory could be applied to developing a relationship between forward and reverse velocities of individual reaction steps in a mechanism. These ideas were further developed and summarized by Horiuti and 100 5.1 Introduction 101 Nakamura (1967) and Horiuti (1973). Such relationships are especially useful in steady-state tracing because the information obtained by the tracing procedure itself is more limited than in the case of transient tracing. The ideas are detailed in the following sections of this chapter.
eBook - PDF- George W. Roberts(Author)
- 2015(Publication Date)
- Wiley(Publisher)
The steady-state approximation must now be applied to eliminate this unknown concentration from the rate expression. The steady-state approximation for O–S * is r OS ffi 0 ¼ k 1 ½S ½H 2 O k 2 ½CO½O---S (5-5) This expression contains the concentration of the second active center, S * . The steady-state approximation for S * is r S ffi 0 ¼k 1 ½S ½H 2 Oþ k 2 ½CO½O---S (5-6) Unfortunately, Eqn. (5-6) is just Eqn. (5-5) multiplied by 1. The two equations are not independent; they cannot be solved for both [S * ] and [O–S * ]. Why did the steady-state approximation ‘‘fail’’ in this case? The problem is as follows. The total concentration of sites on the catalyst is fixed. Sites are either empty (S * ) or they are occupied by a bound O atom (O–S * ). New sites are not created in the reaction sequence, nor are existing sites destroyed. The first SSA equation (Eqn. (5-5)) tells us that the rate of disappearance of S * is equal to the rate of formation of O–S * . The second SSA equation (Eqn. (5-6)) tells us that the rate of disappearance of O–S * is equal to the rate of formation of S * . This is not a new piece of information; it follows directly from the first SSA equation, plus the fact that the total number of sites, i.e., the total of S * and O–S * , is constant. The failure of the two equations for r S and r O---S to produce expressions for the concentrations of these two species can be looked at from another viewpoint. In earlier applications of the SSA, there was a net creation of active centers in some reactions (the initiation reactions) and a net destruction of active centers in other reactions (the termination reactions). This is not the case with the two reactions above. These reactions merely involve the transformation of one kind of active center into a different kind of active center. There is nothing in the given reaction mechanism that allows us to calculate the total concentration of active centers.
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