Steady State Approximation Example

4 Key excerpts on "Steady State Approximation Example"

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  eBook - ePub

  Fundamentals of Chemical Reaction Engineering

  • Mark E. Davis, Robert J. Davis, Robert J. Davis(Authors)
  • 2013(Publication Date)
  • Dover Publications

    (Publisher)

  CHAPTER 4

  The 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

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  eBook - PDF

  Kinetics 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.

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  eBook - ePub

  Fundamentals of Enzyme Kinetics

  • Athel Cornish-Bowden(Author)
  • 2013(Publication Date)
  • Wiley-Blackwell

    (Publisher)

  Chapter 14 Fast Reactions

  14.1 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 43–45

  Consequently, 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

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  eBook - PDF

  Chemical Reactions and Chemical Reactors

  • 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 OS  ffi 0 ¼ k 1 ½S  ½H 2 O k 2 ½CO½O---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 ¼k 1 ½S  ½H 2 Oþ k 2 ½CO½O---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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