First Order Reaction

11 Key excerpts on "First Order Reaction"

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  Transport Modeling for Environmental Engineers and Scientists

  • Mark M. Clark(Author)
  • 2012(Publication Date)
  • Wiley

    (Publisher)

  These are critical tools for the environmental scientist and engineer. We point out here that kinetic studies are an important tool in understand-ing reaction mechanisms. An in-depth discussion of mechanism is outside the scope of this chapter, but the reader can follow up on this subject in some of the excellent chemistry and physical chemistry references at the end of the chapter. 9.2. FIRST-ORDER REACTIONS Consider that we find the following relationship between reactant A and product B, and that there are no other reactants or products: A -> B (9.2) Equation 9.2 represents the stoichiometry of an irreversible reaction: 1 mol of A is converted to 1 mol of B. The reaction rate is defined as the rate of change in the concentration of a reactant or product. For computational purposes, we will need to quantify the reaction rate as a differential equation. We make the assumption that the rate of change in the concentration of A and B is proportional to the concentration of A, rA = d[A] = _ k[A] ( 9 3 ) dt and r B = ^ = k[A] (9.4) dt Here r A and r B stand for the reaction rates of A and B, k is the reaction rate or kinetic constant, and [A] and [B] represent the appropriate concentrations of A and B. The concentration units could be molar, mass, or colloidal number concentration in water, or partial pressure or aerosol number concentration in air. In molar concentration units, the correct units for the r values are mol-L~ 3 -7 and the correct units for k are T~ x . From Eqs. 9.2 to 9.4, we con-clude that rA .m..m.^ <9 .5) dt dt In the present case, we have used the stoichiometry (Eq. 9.2) to guess at the kinetics law (Eqs. 9.3 and 9.4). Reactions in which the kinetics are suggested by the stoichiometry are called elementary reactions (see Section 9.6). FIRST-ORDER REACTIONS 497 Reactions 9.3 and 9.4 are called first-order reactions. First-order reactions are called linear because Eqs. 9.3 and 9.4 are linear differential equations. Integrating Eq.

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  Foundations of Chemistry

  An Introductory Course for Science Students

  • Philippa B. Cranwell, Elizabeth M. Page(Authors)
  • 2021(Publication Date)
  • Wiley

    (Publisher)

  8 Chemical kinetics – the rates of chemical reactions At the end of this chapter, students should be able to: • Define and understand the following terms: Rate of reaction Reaction mechanism Rate expression Rate constant Overall order of a reaction Order with respect to a reactant Half-life of a reaction Rate-determining step • Use collision theory to explain how changes in reaction conditions of tem-perature, concentration, and the presence of a catalyst affect the rate of a chemical reaction • Outline the difference in behaviour between heterogeneous and homoge-neous catalysts • Define the average rate of reaction and the instantaneous rate of reaction, and understand how the latter changes throughout a reaction • Describe how experiments can be used to determine the rate expression for a chemical reaction Foundations of Chemistry: An Introductory Course for Science Students , First Edition. Philippa B. Cranwell and Elizabeth M. Page. © 2021 John Wiley & Sons Ltd. Published 2021 by John Wiley & Sons Ltd. Companion website: www.wiley.com/go/Cranwell/Foundations • Be familiar with plots of concentration against time for zero-, first-, and second-order reactions • Be familiar with the appropriate plots used to obtain the rate constant from the integrated rate expressions for zero-, first-, and second-order reactions • Understand how the half-life of a reaction can be obtained from concen-tration against time plots, and how half-life depends upon the order of the reaction • Predict a rate expression consistent with the reaction mechanism for a sim-ple multi-step reaction • Use the Arrhenius equation to obtain the activation energy for a reaction from rate constant and temperature data 8.1 Introduction We saw in the previous chapter that the economic viability of industrial chemical processes depends upon a number of factors.

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  An Introduction to Chemical Metallurgy

  International Series on Materials Science and Technology

  • R. H. Parker, D. W. Hopkins(Authors)
  • 2016(Publication Date)
  • Pergamon

    (Publisher)

  For the moment we will assume that the temperature, which has an important effect on reaction rates, remains constant. We can define the order of the reaction as the sum of the powers to which the concentrations of the reacting atoms or molecules must be raised to determine the rate of reaction. It must be emphasized that the order of a reaction does not neces-sarily bear any relation to the molecularity of the reaction which is the number of atoms or molecules taking part in a reaction. For example, in the reaction 2HI = H 2 + I 2 two molecules of hydrogen iodide take part and therefore the reaction is bimolecular. Experiments have shown that the reaction rate is proportional to the square of the hydrogen iodide concentration— indicating a second order reaction, whose rate depends on the concentrations of two molecules. In this case, the molecularity is the same as the order of the reaction, but where one reactant is present in excess, its con-centration is not changed appreciably as the reaction proceeds, as in the reaction of hydrogen iodide with hydrogen peroxide in aqueous solution, HI + H 2 0 2 = HIO+H 2 0, 124 AN INTRODUCTION TO CHEMICAL METALLURGY where the hydrogen iodide is present in excess. The rate of this reaction depends only on the concentration of the hydro-gen peroxide because the concentration of HI is not altered significantly by the reaction, and it is therefore of first order even though it is bimolecular. In this particular case, the reaction is not completed after this first stage, and a further stage in the reaction takes place, HI + HIO = H 2 0 +1 2 , giving an overall reaction 2HI + H 2 0 2 = I 2 + 2H 2 0. Measurement of the rate of this overall reaction, which is often used as a convenient laboratory experiment in teaching courses, shows that it still depends only on the concentration of the hydrogen peroxide and is therefore of first order.

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  Basic Physical Chemistry

  The Route to Understanding

  • E Brian Smith(Author)
  • 2012(Publication Date)
  • ICP

    (Publisher)

  255 256 | Basic Physical Chemistry Thus, − d [ A ] d t = k r [ A ] n [ B ] m would be a typical expression of the rate of the reaction illustrated above. n and m are often simple integers (or, sometimes, half integers) and k r is a constant at any given temperature termed the rate constant . 1 The sum of the exponents, (n + m) , is defined as the overall order of the reaction . n and m are the orders with respect to each of the reactants and a knowledge of these orders, together with a knowledge of the rate constant, k r , enables us to calculate the rate of the reaction when different concentrations of the reactants are present. A simple example is H 2 + I 2 → 2HI , for which we find d [ HI ] d t = k r [ H 2 ][ I 2 ] . This corresponds to an overall order of 2 and is first-order with respect to both H 2 and I 2 . By contrast, for H 2 + Br 2 → 2HBr, the reaction rate shows a complex dependence on H 2 and Br 2 , even though the stoichiometric equation is the same as that for the formation of HI (Section 11.10). Another example is the reaction S 2 O 2 − 8 + 2I → I 2 + 2SO 2 − 4 , which might be expected to be a third-order reaction, but experiment shows it to be second-order. Reactions are sometimes defined in terms of molecularity . This is a theoretical concept which indicates the number of molecules participating in an elementary step in a reaction. The molecularity and the order of a reaction are often different. 11.2 First-order reactions The simplest type of kinetic behaviour is that represented by first-order rate equations. An example is radioactive decay of an unstable isotope, X → Y + α or β radiation. The carbon 14 C nucleus decays in this way, with a half-life , the time for the initial concentration to be reduced to half its value, of some 5200 years. An example of a chemical reaction which occurs with first-order kinetics is the decomposition of N 2 O 5 , N 2 O 5 ( g ) → 2 NO 2 ( g ) + 1 2 O 2 ( g ).

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  Geochemical Kinetics

  • Youxue Zhang(Author)
  • 2021(Publication Date)
  • Princeton University Press

    (Publisher)

  A good example of a first-order (pseudo-first-order) chemical reaction is the hydration of CO 2 to form carbonic acid, Reaction 1-7f, CO 2 (aq) þ H 2 O(aq) ? H 2 CO 3 (aq). Because this is a reversible reaction, the concentration evolution is considered in Chapter 2. 1.3 KINETICS OF HOMOGENEOUS REACTIONS 21 1.3.5.3 Second-order reactions Most elementary reactions are second-order reactions. There are two types of second-order reactions: 2A ? C and A þ B ? C. The first type (special case) of second-order reactions is 2A ! C : (1-50) The reaction rate law is d x = d t ¼ k [A] 2 ¼ k ([A] 0 2 x ) 2 : (1-51) The solution can be found as follows: d x = ([A] 0 2 x ) 2 ¼ k d t : (1-51a) Then d([A] 0 2 x ) = ([A] 0 2 x ) 2 ¼ 2 k d t : (1-51b) Then 1 = ([A] 0 2 x ) 1 = [A] 0 ¼ 2 kt : (1-51c) That is 1 = [A] 1 = [A] 0 ¼ 2 kt : (1-52) Or [A] ¼ [A] 0 = (1 þ 2 k [A] 0 t ) : (1-53) The concentration of the reactant varies with time hyperbolically. The second type (general case) of second-order reactions is A þ B ! C : (1-54) The reaction rate law is d x = d t ¼ k [A][B] ¼ k ([A] 0 x )([B] 0 x ) : (1-55) If [A] 0 ¼ [B] 0 , The solution is the same as Equation 1-53. For [A] 0 = [B] 0 , the so-lution can be found as follows: d x = {([A] 0 x )([B] 0 x )} ¼ k d t : (1-56) u d x = ([A] 0 x ) u d x = ([B] 0 x ) ¼ k d t , where u ¼ 1 = ([B] 0 [A] 0 ) : u ln {([A] 0 x ) = [A] 0 } u ln {([B] 0 x ) = [B] 0 } ¼ kt : ln{([A] 0 x ) = [A] 0 } ln{([B] 0 x ) = [B] 0 } ¼ k ([B] 0 [A] 0 ) t x ¼ [A] 0 [B] 0 ( q 1) = ( q [A] 0 [B] 0 ), where q ¼ exp{ k ([B] 0 [A] 0 ) t } : (1-57) 22 1 INTRODUCTION Figure 1-1 compares the concentration evolution with time for zeroth-, first-, and the first type of second-order reactions. Table 1-2 lists the solutions for concen-tration evolution of most elementary reactions.

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  Kinetics of Geochemical Processes

  • Anthonio C. Lasaga, James Kirkpatrick(Authors)
  • 2018(Publication Date)
  • De Gruyter

    (Publisher)

  Suppose that the true rate obeys the equation dC A - dT = k C A C B ' < 25 > i.e. , a second-order rate law. If we now choose the initial concentra- tion of I (25) as: tion of B, C°, to be much greater than C°, then we can rewrite equation Β A - ÏÏT B k C l C A • k ' C A < 26 > where k' Ξ k C°. Equation (26) is a first-order equation; our choice of Β initial conditions has thus reduced the experimental order of our system. The reaction is now referred to as a pseudo-first-order reaction. With suitable constraints, the kinetic behavior of geochemical reactions can be reduced to that exhibited by simpler reactions. The simplest possible case is that of a zero-order reaction. Such a reac- tion obeys the equation - g - k (27) 18 which integrates to C = C Q - kt (28) where C is the initial concentration. Zero-order kinetics yield o straight lines, if concentration is plotted versus time. Equation (28) can only hold for t such that C >_ 0. The reduction of the sulfate ion by the bacteria Desulfovibrio in marine environments seems to obey a zero-order rate law with respect to sulfate if the concentration of 2- SO^ is greater than 2 mM (Berner, 1971; Goldhaber and Kaplan, 1974). Thus if the amount of decomposable organic matter, the temperature, and 2- other variables are kept constant (except SO^ ), the concentration of sulfate in the interstitial waters of sediments would obey equation (28). The rate of leaching of alkalis from single crystals of alkali feldspar also follows a zero-order rate law (Tole and Lasaga, pers. comm.; Fung and Sanipelli, 1980). (Pseudo) First-Order and Second-Order Reactions A first-order reaction obeys the equation dC A - dT = k C A < 29 > Rate constants for first-order reactions have units of time ^.

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  Chemistry

  Structure and Dynamics

  • James N. Spencer, George M. Bodner, Lyman H. Rickard(Authors)
  • 2011(Publication Date)
  • Wiley

    (Publisher)

  Assume, for the moment, that the reaction between reagents X and Y is stud- ied under conditions for which there is a large excess of Y. If this is true, the con- centration of Y will remain essentially constant during the reaction. As a result, the rate of the reaction will be zero-order for the excess reagent. Instead, it will rate = k(X)(Y) appear to be first-order in the other reactant, X. A plot of ln(X) versus time will therefore give a straight line with a slope of k . If we now run the reaction in the presence of a large excess of X, the reaction will appear to be first-order in Y. Under these conditions, a plot of ln(Y) versus time will be linear with a slope of k . The value of the rate constant obtained from either of these equations won’t be the actual rate constant for the reaction. It will be the product of the rate constant for the reaction times the concentration of the reagent that is present in excess. The reaction between phenolphthalein and excess OH  ion described in Sec- tion 14.3 is an example of a peudo-first-order reaction because the initial concentra- tion of the OH  ion was roughly 120 times the initial concentration of phenolph- thalein. There was so much OH  ion in the solution that the concentration of this ion remained virtually constant throughout the course of the data collection. The reaction therefore appeared to be first-order in phenolphthalein when the data were analyzed. 14.15 The Activation Energy of Chemical Reactions Only a small fraction of the collisions between reactant molecules convert the reactants into the products of the reaction. This can be understood by turning, once again, to the reaction between ClNO 2 and NO. In the course of this reaction, a chlorine atom is transferred from one nitrogen atom to another. For the reaction to occur, the nitrogen atom in NO must collide with the chlorine atom in ClNO 2 . The reaction won’t occur if the oxygen from the NO molecule collides with the chlorine atom on ClNO 2 .

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  Physical Chemistry for Engineering and Applied Sciences

  • Frank R. Foulkes(Author)
  • 2012(Publication Date)
  • CRC Press

    (Publisher)

  Thus, for a First Order Reaction, we substitute c = 1 2 c o into Eqn [5] as follows: ln c c o £ ¤ ² ¥ ¦ ´ = – kt . . . [5] giving ln 0.5c c o o £ ¤ ² ¥ ¦ ´ = ln 1 2 = – ln 2 = – kt 1/2 which rearranges to t 1/2 = ln 2 k Half-life for a First Order Reaction . . . [9] Eqn [9] shows that for a First Order Reaction the half-life is independent of the initial concentration of reactant. During each successive duration of t 1/2 , the concentration of the reactant in a First Order Reaction decays to half its value at the start of that period. After n such periods, the reactant con-centration will be 1 2 n ( ) of its initial concentration. 26-8 CHEMICAL REACTION KINETICS Example 26-4 There is a constant flux of particles falling on the earth as a result of nuclear processes occurring in the sun and other parts of the universe; these particles, which are mostly protons, are called cosmic rays . When cosmic rays collide with N 2 molecules in the earth’s atmosphere they produce carbon-14, an unstable radioactive isotope of carbon that has a half-life of 5720 years. Accordingly, when a living tree takes in CO 2 from the air, a certain amount of C 14 becomes incorporated into its wood. When the tree dies, it no longer ingests CO 2 , and the radioactivity present in the wood gradually disappears through the first-order process of radioactive disintegration. Thus, by measuring the residual amount of C 14 present in an ancient sample of wood and comparing it with the amount found in living wood, we are able to estimate how long the tree has been dead, and therefore the age of the sample. 4 This is the principle behind the technique of carbon-14 dating . An archeological sample contained wood that had only 72% of the C 14 found in living trees. Estimate the age of the sample. Solution The process of radioactive decay is a first order process, so we can write d dt m = < km where k is the first order rate constant and m is the mass of C 14 present in the sample.

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  Modeling of Chemical Kinetics and Reactor Design

  • A. Kayode Coker(Author)
  • 2001(Publication Date)
  • Gulf Professional Publishing

    (Publisher)

  Instead of the velocity constant, a quantity referred to as the half-life t 1/2 is often used. The half-life is the time required for the concentration of the reactant to drop to one-half of its initial value. Substitution of the appropriate numerical values into Equation 3-33 gives k t t = = 1 2 0 693 1 2 1 2 . ln . (3-39) Figure 3-5. First Order Reaction. Reaction Rate Expression 121 Equation 3-39 shows that in the First Order Reactions, the half-life is independent of the concentration of the reactant. This basis can be used to test whether a reaction obeys first order kinetics by measur-ing half-lives of the reaction at various initial concentrations of the reactant. SECOND ORDER REACTIONS A second order reaction occurs when two reactants A and B interact in such a way that the rate of reaction is proportional to the first power of the product of their concentrations. Another type of a second order reaction includes systems involving a single reactant. The rate at any instant is proportional to the square of the concentration of a single reacting species. Case 1 Consider the reaction A B products k + ⎯ → ⎯ . The rate equation for a constant volume batch system (i.e., constant density) is: − ( ) = − = − = r dC dt dC dt kC C A A B A B (3-40) The amount of A and B that have reacted at any time t can be described by the following mechanism and set of equations. From stoichiometry: A B Amount at time t = 0 C AO C BO Amount at time t = t C A C B Amounts that have reacted C AO – C A C BO – C B C B = C BO – (C AO – C A ) (3-41) Substituting Equation 3-41 into Equation 3-40, rearranging and integrating between the limits gives − − − ( ) [ ] = ∫ ∫ dC C C C C k dt A A BO AO A C C t AO A 0 (3-42)

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  Elementary Chemical Reactor Analysis

  Butterworths Series in Chemical Engineering

  • Rutherford Aris(Author)
  • 2013(Publication Date)
  • Butterworth-Heinemann

    (Publisher)

  Treat the equation subject to 1 dy 2 P dx 2 -%+*■ ¥£ + y=n « dx : 1 at x = = 0, x = 0 1, as a special case of Eqs. (5.1.9) and (5.1.12) and hence solve it. (We shall meet this equation in Chap. 9.) Exercise 5.1.3. The solution of the equation %=q~py, X0) = y 0 , with constant p and q, is worth having at your finger tips. Solve it and grasp it—not just by learning it by heart, but by really seeing what each term means. 5.2 The First Order Reaction (i) Irreversible First Order Reaction. For the typical first order irreversible dissociation we may write A — > products, for, since it is irreversible, its rate is unaffected by the number or concentration of its products. If c is the concentration of A, the statement that the reaction is first order is just a way of saying that the rate of decrease of c is proportional to c itself, or Sec. 5.2 The First Order Reaction # = -kc. dt 89 (5.2.1) If c 0 is the value of c at / = 0 we have, by method I of Sec. 5.1, £f: = ,„f)=-„ (5.2.2, or c(t) = c 0 e-kt . (5.2.3) The concentration of A thus decreases exponentially to zero and the con-centrations of the product or products increase proportionally to their stoichiometric coefficients and the amount of A that has been used up, namely, c 0 — c = c 0 (1 — e~ kt ). The time course of c and c 0 — c = ξ are seen in Fig. 5.1. ► Time Fig. 5.1 Variation of concentrations and extent with time during the course of a first order reversible reaction. (ii) Reversible First Order Reaction. Consider the reaction A x — A 2 = 0 in which A 2 reacts to form Αχ ; it will be first order if r = kc 2 - k'c x . (5.2.4) This reaction provides the simplest nontrivial way of comparing three ways of treating the problem. First, we use the notion of extent: here Ci = c l0 + ξ, c 2 = c 20 — £ (5.2.5) so that r = (kc 20 - k'c lQ ) - {k + k')£ = r 0 -(k + *')f ; 90 The Progress of the Reaction in Time Chap. 5 and g = r = r 0 -(k + k*)£ 9 (5.2.6) where £ = 0 when t = 0.

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  Kinetics of Chemical Processes

  Butterworth-Heinemann Series in Chemical Engineering

  • Michel Boudart, Howard Brenner(Authors)
  • 2014(Publication Date)
  • Butterworth-Heinemann

    (Publisher)

  As the extent of reaction is a thermodynamic variable, the general defini-tion of the rate of reaction as dX/dt is also a thermodynamic quantity and indeed it plays a central role in the thermodynamics of irreversible processes. Being a thermodynamic quantity, it is totally unrelated to any molecular interpretation as to how the chemical reaction actually occurs. In particular, the definition applies to any single reaction, i.e., one the advancement of which Sec. 1.4 The Rate Function for a Single Reaction 13 can be described by a single variable. How this is possible in spite of the fact that the reaction normally takes place through reactive intermediates that do not appear in the equation for reaction, will be discussed in Chapter 3. 1.4 General Properties of the Rate Function for a Single Reaction The rate of reaction is generally a function of temperature, composition and pressure. To find the form of the rate function r is a central problem of applied chemical kinetics. Once r is known, information can frequently be in-ferred on rates of individual steps for theoretical studies. Or a system, i.e., a reactor, can be designed for carrying out the reaction under optimum conditions. In this paragraph, we present five general rules on the form of the rate function and much of the book will be devoted to the understanding and elucidation of these rules, of their range of validity, and of their exceptions. The rules are of an approximate nature but are sufficiently general that ex-ceptions to them usually reveal something of interest. It must be stressed that the utility of these rules is their applicability to many single reactions. Appli-cation to elementary steps only would be far too restrictive and can be dis-cussed readily in theoretical terms (see Chapter 2). Rule I The rate Junction r at constant temperature generally decreases in monotonie fashion with time or extent of reaction (Fig. 1.4.1).

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