8 Key excerpts on "First Order Reaction"

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  eBook - ePub

  General Chemistry for Engineers

  • Jeffrey Gaffney, Nancy Marley(Authors)
  • 2017(Publication Date)
  • Elsevier

    (Publisher)

  2 O on a hot platinum surface.

  2N 2

  O g +

  2N 2

  g →

  O 2

  g

  The N2 O molecules that can react under these conditions are limited to those that are attached to the surface of the solid platinum. Once all of the limited surface sites are occupied, the gas phase N2 O molecules cannot react until some of the adsorbed molecules decompose and free up a surface site. So, the rate is dependent on the number of sites on the hot platinum surface, but not on the gas phase concentration of N2 O. The overall order of the reaction is zero because it is not dependent on the concentration of the reactant. Another example of a zero order reaction is enzyme reactions in living organisms. These reactions begin with the attachment of the reactant to an active site on the enzyme, forming an enzyme-reactant complex. If the number of active enzyme sites is limited compared to the concentration of reactant molecules, the reaction is independent of reactant concentration and so is zero order.

  A first order chemical reaction is one with a rate law in which the sum of the exponents is equal to one. The reaction rate is then proportional to the concentration of one reactant. Even though other reactants may be present, their concentrations will not affect the reaction rate and so each of these other reactants will be zero order. The rate law for a First Order Reaction is given by;

  Reaction rate =

  − d A

  dt

  = k A

    (4)

  Since the reaction rate is equal to the rate constant multiplied by the concentration of one reactant, the units of the rate constant are in inverse time (s− 1 ).

  A second order

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  eBook - ePub

  BIOS Instant Notes in Chemistry for Biologists

  • J Fisher, J.R.P. Arnold, Julie Fisher, John Arnold(Authors)
  • 2020(Publication Date)
  • Taylor & Francis

    (Publisher)

  order of reaction. The order of a reaction should not be confused with stoichiometry, as illustrated by the following example;

  By experiment the rate of reaction is found to be;

  note that in this case x = 1, not 2 as stoichiometry would suggest. This and similar processes are therefore referred to as first order reactions.

  Using the more general reaction (1) the rate for this if it were a First Order Reaction would be written as; if k = k′a (a is the number of moles of A, and k′ the rate constant for conversion of A to products) then; Upon integration;

  where [A]0 is the initial concentration of A. Consequently a plot of ln [A]/[A]0 versus time will produce a straight line with slope −k.

  Second order reactions

  There are two possibilities for a second order reaction. Consider first reaction (1) under conditions where the rate is found to depend solely on [A], that is; and Integrating produces; Alternatively if the rate was found to depend on both [A] and [B], that is; Several integrations and substitutions lead to the following expression;

  where [A]0 and [B]0 are the initial concentrations of A and B.

  Rates for equilibrium and multistep processes

  Most chemical reactions are reversible to some degree. Clearly then, when considering the rate of a reversible reaction, both the forward and backward steps must be considered. Take the simple example of a reactant A being converted to B, in a reversible manner. There are two elementary steps in such a reaction; Thus; At equilibrium there is no net change in concentration, hence d[A]/dt = 0 Therefore; When a reaction involves more than one step, then each step of the reaction must be considered when determining a reaction rate. Consider the following reaction;

  [A] i.e the reaction is first order and,

  and,

  To simplify the determination of k2 a steady state approximation

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  eBook - ePub

  AP® Chemistry Crash Course Book + Online

  • Adrian Dingle, Derrick C. Wood(Authors)
  • 2014(Publication Date)
  • Research & Education Association

    (Publisher)

  1.   General formula for rate equation. For the generic reaction

  Rate of reaction = k [A ]x [B ]y [C ]z

  where k is the rate constant and x , y , z are the orders with respect to A , B , and C , respectively, but are not necessarily the stoichiometric coefficients of A , B , and C .

  2.   Orders

      i.      The order with respect to a reactant is the exponent of the concentration term in the rate equation (a.k.a. the rate law ).

  Rate of reaction = k [H2 O2 ]

  In this case, the reaction is first order with respect to the reactant H2 O2 since the concentration of H2 O2 is raised to the power of 1 in the rate equation.

      ii.     Zero order rate reaction

         The rate of the reaction is independent of the concentration of the reactant(s).

      iii.    First order rate reaction

         The rate of the reaction is directly proportional to the concentration of one of the reactants.

  (sometimes the “1” is omitted; i.e., Rate = k [A ])

         Radioactive decay reactions are a very common example of a first-order process. They have a constant half-life, meaning the time taken for half of the atoms to decay is constant and independent of the initial concentration.

  For these reactions, .

      iv.    Second order rate reaction

         The rate of the reaction is directly proportional to the square of the concentration of one of the reactants.

      v.     Determining orders

  Conduct multiple experiments with changing concentrations of each reactant and measure the rate in some way.

         Comparing Trial 1 and Trial 2—the concentration of NO is doubled, and the concentration of Cl2 remains the same. The reaction rate is increased 4 times, meaning the rate of reaction with respect to NO is second order (22 = 4).

         Comparing Trial 1 and Trial 3—the concentration of Cl2 is doubled, and the concentration of NO stays the same. The reaction rate is doubled, meaning the rate of reaction with respect to Cl2 is first order (21

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  eBook - ePub

  Fundamentals of Enzyme Kinetics

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

    (Publisher)

  second-order reaction it is proportional to the product of two concentrations or to the square of one concentration; and so on.

  For a simple reaction that consists of a single step, or for each step in a complex reaction, the order is usually the same as the molecularity (though this may not be apparent if one concentration, for example that of the solvent if it is also a reactant, is so large that it is effectively constant). However, many reactions consist of sequences of unimolecular and bimolecular steps, and the molecularity of the complete reaction need not be the same as its order. Indeed, a complex reaction often has no meaningful order, as the overall rate often cannot be expressed as a product of concentration terms. As we shall see in later chapters, this is almost universal in enzyme kinetics, where not even the simplest enzyme-catalyzed reactions have simple orders. Nonetheless, the individual steps in enzyme-catalyzed reactions nearly always do have simple orders, usually first or second order, and the concept of order is important for understanding enzyme kinetics. The binding of a substrate molecule to an enzyme molecule is a typical example of a second-order bimolecular reaction in enzyme kinetics, whereas conversion of an enzyme–substrate complex into products or into another intermediate is a typical example of a first-order unimolecular reaction.

  Figure 1.1. Order of reaction. When a reaction is of first order with respect to a reactant A the rate is proportional to its concentration a. If it is of second order the rate is proportional to a2 ; if it is of zero order it does not vary with a.

  1.2.2 First-order kinetics

  The rate v of a first-order reaction A → P can be expressed as

  (1.1)

  in which a and p are the concentrations of A and P respectively at any time t, k is a first-order rate constant and a0 is a constant. As we shall see throughout this book, the idea of a rate constant1 is fundamental in all varieties of chemical kinetics. The first two equality signs in the equation represent alternative definitions of the rate v: because every molecule of A that is consumed becomes a molecule of P, it makes no difference to the mathematics whether the rate is defined in terms of the appearance of product or disappearance of reactant. It may make a difference experimentally, however, because experiments are not done with perfect accuracy, and in the early stages of a reaction the relative changes in p are much larger than those in a (Figure 1.2 ). For this reason it will usually be more accurate to measure increases in p than decreases in a.

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  eBook - ePub

  Physical Chemistry of Foods

  • Pieter Walstra(Author)
  • 2002(Publication Date)
  • CRC Press

    (Publisher)

  -1 , etc.

  For a zero-order reaction, the rate remains constant: see Table 4.1 . Approximately zero-order reactions occur, for instance, if small quantities of a substance, say one causing an off—flavor, are slowly formed from a very large reservoir of a parent component.

  For a first-order reaction of the type , we have

  (4.2a)

  where [A] stands for the molar concentration of A and k is the rate constant, which in this case is in s-1 ·k varies with temperature and pressure, but it is generally assumed to be constant otherwise, i.e., independent of concentration; this is often (nearly) true, but not always. Integrating the equation, and introducing the initial concentration of A, we obtain

  (4.2b)

  or

  (4.2c)*

  where τ is the relaxation time. If we plot the log of the concentration versus time, we thus obtain a straight line. The relations are illustrated in Figure 4.1 . If, for example, 10% of A is left after D s, this means that 1% is left after 2D s, 0.1% after 3D s, and so on; D, which equals 2.3/k, is called the decimal reduction time and is mostly used by microbiologists. The killing of microorganisms and the inactivation of enzymes at high temperature often follow first-order kinetics, at least approximately. Also bacterial growth in the so-called exponential phase follows first order kinetics, but now the sign in Eq. (4.2a ) is positive.

  The relaxation time is mostly used by physical chemists and is the time needed for a certain change to occur over (1- 1 /e

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  eBook - ePub

  BIOS Instant Notes in Physical Chemistry

  • Gavin Whittaker, Andy Mount, Matthew Heal(Authors)
  • 2000(Publication Date)
  • Taylor & Francis

    (Publisher)

  Thus a reaction which is second order overall must have a rate constant with dimensions of concentration −1.time −1 in order to provide the right hand side of the rate law with dimensions equal to the dimensions of concentrationtime 1 for rate of reaction. The exact units of k depend on the units of concentration and time used, but mol dm −3 and s, respectively, are common. A rate constant for a particular reaction has a fixed value at a particular temperature, although it usually varies with temperature and the temperature dependence is often conveniently described by an Arrhenius equation (see Topic F3). Rate constants of elementary reactions do not vary with pressure so the observation of a pressure dependence in the rate of reaction indicates a more complex multistep reaction mechanism (see Topics F4, F5 and F6). Order of reaction If the rate law for a reaction can be written in the form, rate ∝[A] α [B] β … then the reaction is classified as α- order in A, β- order in B,…and as (α+β+…)-order overall. Where the exponent, or sum of exponents, equals one the reaction is said to be first order with respect to that species, or first order overall, respectively. Where the exponent, or sum of exponents, equals two the reaction is described as second order with respect to that species, or second order overall, respectively, and so on. Both the rate laws: and are second order overall, but whereas the first rate law is second order in species A only, the second rate law is first order in each of species A and B. If a reactant species appears in the balanced chemical equation for the reaction but does not appear in the rate law then the reaction is zero order with respect to that species. Zero order terms are not usually written in rate law equations since the concentration of any species to the power zero is just unity

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  eBook - ePub

  How Enzymes Work

  From Structure to Function

  • Haruo Suzuki(Author)
  • 2019(Publication Date)
  • Jenny Stanford Publishing

    (Publisher)

  5 ]. Copyright (1975) American Chemical Society.

  5.2    Analysis of the First-Order Reaction

  This section describes how to determine the rate constant of the first-order reaction. In various cases in enzymology, reactions can be analyzed by the first-order reaction, like the pseudo-first-order treatment of a second-order reaction. The more complex and higher-order reactions are described in ref. [6 ] and in refs. [2 , 11 in Chapter 2 ].

  5.2.1    Order of Reaction

  The zero-order reaction is the one that the rate is independent of the concentration of reactant.

  S → k P

  (5.1)

  The rate (v) of formation of product is

  v = − d [ S ] d t = k

  (5.2)

  Integration of Eq. 5.2 from time 0 to t gives

  [ S ] t = − k t + [ S ] 0

  (5.3)

  where [S]

  t

  and [S]0 represent the concentration of S at time t and time 0, respectively. The substrate concentration decreases linearly with time (Fig. 5.4 ). The zero-order reaction is observed in the enzyme reaction with sufficiently high substrate concentration.

  Figure 5.4 The zero-order reaction. The plot of Eq. 5.3 is shown.

  The first-order reaction is the one that the rate of reaction is dependent on the concentration of reactant. In reaction 5.1, the rate (v) of formation of product is

  − d [ S ] d t = k [ S ]

  (5.4)

  Integration of Eq. 5.4 from time 0 to t gives

  ln[S ] t = − k t + ln [ S ] 0

  (5.5)

  Equation 5.5 is expressed by the exponential function

  [ S ] t = [ S ] 0 e − k t

  (5.6)

  The equation shows that the concentration of S decreases exponentially with time. Here, the useful concept is introduced, half-life (t½

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  eBook - ePub

  Fundamentals of Chemical Reaction Engineering

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

    (Publisher)

  CHAPTER 2

  Rate Constants ofElementary Reactions

  2.1 | Elementary Reactions

  Recall from the discussion of reaction networks in Chapter 1 that an elementary reaction must be written as it proceeds at the molecular level and represents an irreducible molecular event. An elementary reaction normally involves the breaking or making of a single chemical bond, although more rarely, two bonds are broken and two bonds are formed in what is denoted a four-center reaction. For example, the reaction:

  is a good candidate for possibly being an elementary reaction, while the reaction: is not. Whether or not a reaction is elementary must be determined by experimentation.

  As stated in Chapter 1 , an elementary reaction cannot be written arbitrarily and must be written the way it takes place. For example (see Table 1.4.3 ), the reaction:

  cannot be written as:

  since clearly there is no such entity as half a molecule of dioxygen. It is important to note the distinction between stoichiometric equations and elementary reactions (see Chapter 1 ) is that for the stoichiometric relation:

  one can write (although not preferred):

  The remainder of this chapter describes methods to determine the rate and temperature dependence of the rate of elementary reactions. This information is used to describe how reaction rates in general are appraised.

  2.2 | Arrhenius Temperature Dependence of the Rate Constant

  The rate constant normally depends on the absolute temperature, and the functional form of this relationship was first proposed by Arrhenius in 1889 (see Rule III in Chapter 1 ) to be:

  where the activation energy, E , and the pre-exponential factor, , both do not depend on the absolute temperature. The Arrhenius form of the reaction rate constant is an empirical relationship. However, transition-state theory provides a justification for the Arrhenius formulation, as will be shown below. Note that the Arrhenius law (Equation 2.2.1) gives a linear relationship between ln k and T −1

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