Chemistry
Second Order Reactions
Second order reactions in chemistry refer to chemical reactions where the rate of the reaction is directly proportional to the square of the concentration of one reactant or the product of the concentrations of two reactants. These reactions are characterized by their rate laws, which typically follow the form rate = k[A]^2 or rate = k[A][B]. The overall order of the reaction is 2.
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12 Key excerpts on "Second Order Reactions"
eBook - PDF- Praveen Bhai Patel(Author)
- 2023(Publication Date)
- Arcler Press(Publisher)
2.2.6. Zero Order Reaction In these reactions, the reactants’ concentration has no impact on the reaction rate. The oxidation of CH 3 CH 2 OH (ethanol) to CH 3 CHO is an example of this kind of reaction (acetaldehyde) (Varshney et al., 1996). 2.2.7. First Order Reaction Because the rates of these reactions are determined only by the concentration of a single reactant, the reaction order is 1. Although these reactions may include a large number of reactants, only one will have a first-order concentration, while the others will have a zero- order concentration (Ball, 1998): 2H 2 O 2 + 2H 2 O + O 2 is a first-order reaction. Basics of Catalysis Reactions and Chemical Kinetics 31 2.2.8. Pseudo-First Order Reaction Because the concentration of one reactant stays constant in a pseudo-first- order reaction, it is included in the rate constant in the rate statement. Because it is plentiful in relation to other reactants’ concentrations, or because it acts as a catalyst, the reactant concentration may remain constant. A pseudo-first-order reaction is as follows: CH 3 COOCH 3 + H 2 O →CH 3 COOH + CH 3 OH 2.2.9. Second Order Reaction When the reaction order is two, it is referred to as a second-order reaction. The rate of these reactions may be determined by squaring either the one reactant’s concentration or the concentrations of two independent reactants. r = k[A]2 or r = k[A][B] may be used as the rate equation. NO 2 + CO = NO + CO 2 is a second-order reaction. 2.2.10. Complex Order Reaction The number of atoms, molecules, or ions that must collide in a brief period of time in order for a chemical reaction to occur is referred to as the molecularity of the reaction. The following table outlines the critical properties of molecularity and reaction order (Mok and Chau, 1996). When a reaction happens in a fractional-order, it is often the result of a complex process or some chain reaction. Acetaldehyde pyrolysis is an example of a fractional reaction order in chemistry.
- 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.
eBook - ePub- Michael C. Flickinger(Author)
- 2013(Publication Date)
- Wiley(Publisher)
We need to introduce some basic principles of kinetic analysis of chemical and enzymatic reactions. Quantitative description and understanding of microbial growth dynamics and kinetics are impossible without some elementary knowledge in underlying scientific disciplines. Enzymatic and chemical reactions play an essential role in biotechnology, which is one of the most important fields in industrial development.17.3.1 Order and Molecularity of Chemical Reactions
The molecularity of any chemical reaction is defined by the number of molecules that are altered in the reaction (Table 17.3 ). The order is a description of the number of concentration terms multiplied together in the rate equation (Table 17.4 ). Hence, in a first-order reaction, the rate is proportional to one concentration of reactant; in a second-order reaction, it is proportional to two concentrations or to the square of one concentration. For a simple single-step reaction, the order is generally the same as the molecularity. For a complex reaction involving a sequence of unimolecular and bimolecular steps, the molecularity is not the same as its order. Reactions of molecularity greater than 2 are common, but reactions of order greater than 2 are very rare. For instance, a trimolecular reaction, such as A + B + C → P, as a rule proceeds through two elementary steps, A + B → X and X + C → P, each of which are of the second or first order. Very often, bimolecular reactions between S1 and S2 occurs under the condition that their respective concentrations s2 s1 (e.g. if the second reactant S2 is solvent), then we have a pseudo first-order
- A. Kayode Coker(Author)
- 2001(Publication Date)
- Gulf Professional Publishing(Publisher)
Therefore, the second order reaction becomes a pseudo-first order reaction. Case 2 Case 2 involves a system with a single reactant. In this case, the rate at any instant is proportional to the square of the concentration of A. The reaction mechanism is 2A products k ⎯ → ⎯ . The rate expres-sion for a constant volume batch system (i.e., constant density) is − ( ) = − = = − ( ) r dC dt kC kC X A A A AO A 2 2 2 1 (3-50) Figure 3-6. Test for the reaction A + B → products. 124 Modeling of Chemical Kinetics and Reactor Design Rearranging Equation 3-50 and integrating between the limits yields − = ∫ ∫ dC C k dt A A C C t AO A 2 0 (3-51) 1 C kt A C C AO A ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ = (3-52) 1 1 C C kt A AO − = (3-53) or 1 1 C C kt A AO = + (3-54) In terms of the fractional conversion, X A , Equation 3-50 becomes C dX dt kC X AO A AO A = − ( ) 2 2 1 (3-55) Rearranging Equation 3-55 and integrating gives dX X kC dt A A X AO t A 1 2 0 0 − ( ) = ∫ ∫ (3-56) Equation 3-56 yields 0 1 1 X A AO A X kC t − ( ) ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ = (3-57) Equation 3-57 gives Reaction Rate Expression 125 X X kC t A A AO 1 − ( ) = (3-58) Figure 3-7 gives plots of Equations 3-54 and 3-58, respectively. Consider the second order reaction 2A B products k + ⎯ → ⎯ , which is first order with respect to both A and B, and therefore second order overall. The rate equation is: Figure 3-7. Test for the reaction 2A → products. 126 Modeling of Chemical Kinetics and Reactor Design − ( ) = − = r dC dt kC C A A A B (3-59) 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 and from stoichiometry (C AO – C A ) = 2(C BO – C B ).
eBook - ePub- John R. Whitaker(Author)
- 2018(Publication Date)
- Routledge(Publisher)
5). The individual rate constants are calculated by recourse to Eq. (51). C. Second-Order Rate Laws A second-order rate process is characterized by the rate being dependent on two molecules either of the same compound or of two different compounds. Consider the reaction described by Figure 5 First-order reversible rate process. The line is drawn according to Eq. (54). The slope of the line is the sum of the forward and reverse specific reaction rate constants. A + B → k P (55) The rate of the reaction may be written as − d A d t = − d B d t = d P d t = k (A) (B) (56) Experimentally, we can distinguish three types of second-order reactions and the forms of the integrated rate expressions will be dependent on the type. 1 Type I Type 1 second-order reactions will be designated as those in which the initial concentrations of A and B are identical and in which the two react in equimolar amounts. In this case Eq. (56) may be written as − d A d t = k (A) 2 (57) Integration between the limits of A 0 and A at times t 0 and t, − ∫ A 0 A d A (A) 2 = k ∫ t 0 t d t (58) gives − [ − 1 (A) + 1 (A 0) ] = k t − k t 0 = k t (when t 0 = 0) (59) which rearranges to the. form (A 0) − (A) (A) (A 0) = k t = (P) (A 0) [ (A 0) − (P) ] (60) A plot of data according to Eq. (60) gives a straight line with intercept of zero and slope of k provided that the reaction is type I second order (Fig. 6). The half-life, t 1/2, of the reaction is calculated by using the relation (A) = 0.5(A 0), and thus Eq. (60) becomes t 1 / 2 = 1 k (A 0) (61) The half-life of a second-order reaction (type I) depends on the initial concentration of the reactant, but the half-life of a first-order reaction does not. This provides a simple experimental method of distinguishing between the two. 2 Type II Type II second-order reactions will be designated as those in which the initial concentrations of A and B are not equal but are similar. In actual practice this is when (B 0)/(A 0) > 1 < 20
eBook - PDF- Chung K. Law(Author)
- 2010(Publication Date)
- Cambridge University Press(Publisher)
A prominent example is the unimolecular reaction to be studied in Section 2.3, for which the reaction order is 1 at high pressures but becomes 2 at low pressures. The measured reaction order is sometimes called the pseudo- molecularity of the reaction. This potential loss of sensitivity at large concentrations also forms the basis of the isolation method in determining the reaction orders of individual components of a complex reaction scheme. That is, by keeping all but one of the reactants in high concentrations, the reaction order of the lean component can be approximately identified as the apparent overall reaction order, which can be easily measured. This concept is also useful in modeling premixed combustion phenomena because the concentration of the stoichiometrically abundant reactant can be assumed to be constant during the course of reaction. Since reaction order describes both elementary and global reactions, it will be used from now on in specifying all reactions. 2.2. THEORIES OF REACTION RATES: BASIC CONCEPTS 2.2.1. The Arrhenius Law The specific reaction rate constant k(T) gives the functional dependence of the re- action rate on temperature. For an elementary reaction the Arrhenius law states that d ln k(T) dT = E a R o T 2 , (2.2.1) where E a is called the activation energy of the reaction, having the unit of cal/mole or joule/mole. If E a is a constant with respect to temperature, integrating Eq. (2.2.1) yields k(T) = Ae −E a / R o T , (2.2.2) where A is called the frequency factor or the preexponential factor. We note that since R o is a constant, it is convenient to define a new quantity, T a = E a R o , and call it the activation temperature of the reaction. Furthermore, following tradi- tion, we shall also use the uppercase letter to designate molar quantities. For constant values of A and E a , a plot of ln k(T) versus 1/ T exhibits a linear relationship, with A and E a respectively determined from the intercept and slope of such a plot.
eBook - PDF- Irvin Glassman(Author)
- 2012(Publication Date)
- Academic Press(Publisher)
In Eq. (2) £ v} is also given the symbol n, which is called the overall order of the reaction; v) itself would be the order of the reaction with respect to species j . In an actual reacting system, the rate of change of the Β. THE RATES OF REACTIONS AND THEIR TEMPERATURE DEPENDENCY 33 concentration of a given species i is given by = [ν'/ -v'JRR = [y'l -v'Jk Π W (3) at j = 1 since v moles of M, are formed for every v- moles of M t consumed. For the previous example, d(H)/dt = — 2 /c (H) 3 . The use of this complex representa-tion prevents error in sign and eliminates confusion when stoichiometric coefficients are different from one. In many systems Μ,· can be formed not only from a single-step reaction such as that represented by Eq. (3), but also from many different such steps leading to a rather complex formulation of the overall rate. However, for a single-step reaction such as Eq. (3), Σ v' f not only represents the overall order of the reaction, but also the molecularity, which is defined as the number of molecules that interact in the reaction step. Generally the molecularity of most reactions of interest will be 2 or 3. For a complex reaction scheme the concept of molecularity is not appropriate and the overall order can take various values including fractional ones. 1. The Arrhenius Rate Expression Most chemical reactions that take place have their rates dominated by collisions of two species that may have the capability to react. Thus, most simple reactions are second order. Other reactions are dominated by a loose bond breaking step and thus are first order. Most of these latter type reactions fall in the class of decomposition processes. Isomerization reactions are also found to be first order. According to Lindemann's theory [1,4] of first-order processes, first-order reactions occur as a result of a two-step process. This point will be discussed in a subsequent section.
eBook - PDF- Gordon Skinner(Author)
- 2012(Publication Date)
- Academic Press(Publisher)
2.3 REACTION ORDER The rate of a reaction is usually found to depend on the concentra-tion of some of the reactants and is often influenced by the presence of other substances deliberately or accidentally added to the reaction mixture. At a given temperature, and perhaps within a limited range of concentrations, one can write a rate law for a reaction of the form rate = k H [AJ^M* 3 ' (2.3) i where the A» are the reactants, the X 3 are substances that are not reactants but do influence the rate, the as and f3's are coefficients that are not neces-sarily related to the stoichiometric coefficients v, and & is a rate constant. For a long time it was thought that the a 's and p's would be integers, but it is now clear that they need not be. Two kinds of reaction order are commonly defined. The overall order of a reaction is the sum of all the a 's and p's in the rate law expression. The overall order tells us how the rate responds to changes in the absolute concentration at constant relative concentration, such as one would produce by changing the pressure in a gaseous system or diluting a liquid system with an inert solvent. The order with respect to A» is a iy and similarly the order with respect to Xy is 13j. These individual orders not only tell us how sensitive the system is to changes in the concentration of each species, but may also suggest the chemical mechanism of the reaction. 2.3 REACTION ORDER For many elementary reactions, the rate law coefficients are equal to the stoichiometric coefficients. This is true for some of the elementary reactions given above, provided the concentrations are moderate. That is, for O H + CH 3 Br CH3OH + Br the rate law is rate = /c [OH-][CH 3 Br] in dilute solutions. Since the reaction is elementary, the rate depends on the number of collisions between the reactants, which in turn depends on the product of the concentrations.
eBook - PDFElementary Chemical Reactor Analysis
Butterworths Series in Chemical Engineering
- Rutherford Aris(Author)
- 2013(Publication Date)
- Butterworth-Heinemann(Publisher)
(4.2.4) Such a reaction is said to be of the second order overall (since the sum of the powers of the concentrations is two) and of the first order with respect to A a or A 3 individually (since c 20 and c 30 are each raised to the first power). When there is no product present, the reaction can only go in one direc-tion, but as soon as some of the product is formed the reaction rate slows down. It does so for two reasons. In the first place, if the reaction is at all reversible the net reaction rate is the difference between the forward reaction rate (the rate at which the product is being formed) and the back reaction rate (the rate at which the product is dissociating). We may write r = r f -r b (4.2.5) and note that as soon as r b is greater than zero it will tend to slow down the reaction. Equilibrium will be attained when r f = r b . In the second place, r f will be a function of the concentration of the reactants and by analogy with Eq. (4.2.4) (where r 0 = r f9 since there can be no back reaction if no product is formed) we will suppose r f = kc 2 c z = k(c 20 — f)(c 30 — £). (4.2.6) Now as the reaction proceeds and ξ increases, these concentrations decrease and so does r f . It is important to emphasize that there is no a priori reason for writing r f = kc 2 c 3 ; it is a perfectly valid and a not unusual form of depen-dence, but for any particular reaction it must be tested experimentally. We would expect the rate of the back reaction to increase with an in-creasing concentration of the product, which presumably would make its dissociation easier. Let us assume that it has been found that r b = k' Cl . (4.2.7) Constants k and k' are determined experimentally at a given temperature and are themselves functions of temperature as we shall see shortly.
eBook - ePubChemical Kinetics
From Molecular Structure to Chemical Reactivity
- Luis Arnaut, Hugh Burrows(Authors)
- 2006(Publication Date)
- Elsevier Science(Publisher)
Complex reaction mechanisms can conveniently be grouped within the following classification: consecutive reactions, parallel reactions and reversible reactions. Parallel reactions are those in which the same species participates in two or more competitive steps. Consecutive reactions are characterised by the product of the first reaction being a reactant in a subsequent process, leading to formation of the final product. Reversible reactions are those in which the products of the initial reaction can recombine to regenerate the reactant.As complex reactions follow a reaction mechanism involving various elementary steps, the determination of the corresponding kinetic law involves the solution of a system of differential equations, and the complete analytical solution of these systems is only possible for the simplest cases. In slightly more complicated cases it may still be possible to resolve the system of corresponding differential equations using methods such as Laplace transforms or matrix methods. However, there are systems which cannot be resolved analytically, or whose analytical solution is so complex that it is not easily applied. In the absence of information on the orders of magnitude of the rate constants involved, the treatment of these kinetic systems is made using numerical methods. These methods allow us to obtain concentrations of the reactants for discrete time intervals, which can then be represented graphically and compared with experimental data to provide an insight into the changes occurring. Further, although the results obtained numerically are inherently approximations, we can estimate and allow for the errors involved, which can be expressed as error limits when compared with experimental data. In some cases, it may also be possible to introduce some changes of a chemical nature in the system to help simplify and analyse these complex systems. For example, if we know the relative magnitudes of some of the rate constants, it may be possible to simplify the system of complex differential equations involved by modifying the initial concentrations of reactants and obtaining a rate equation that can be solved analytically. Finally, when these procedures cannot be used to resolve complex systems of differential equations, it is still possible to use stochastic treatments such as Markov chains or the Monte Carlo method.
eBook - PDFBasic 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 ).
eBook - PDF- George W. Roberts(Author)
- 2015(Publication Date)
- Wiley(Publisher)
Chapter 2 Reaction Rates—Some Generalizations LEARNING OBJECTIVES After completing this chapter, you should be able to 1. use the Arrhenius relationship to calculate how reaction rate depends on temperature; 2. use the concept of reaction order to express the dependence of reaction rate on the individual species concentrations; 3. calculate the frequency of bimolecular and trimolecular collisions; 4. determine whether the rate equations for the forward and reverse rates of a reversible reaction are thermodynamically consistent; 5. calculate heats of reaction and equilibrium constants at various temperatures (review of thermodynamics). In order to design a new reactor, or analyze the behavior of an existing one, we need to know the rates of all the reactions that take place. In particular, we must know how the rates vary with temperature, and how they depend on the concentrations of the various species in the reactor. This is the field of chemical kinetics. This chapter presents an overview of chemical kinetics and introduces some of the molecular phenomena that provide a foundation for the field. The relationship between kinetics and chemical thermodynamics is also treated. The information in this chapter is sufficient to allow us to solve some problems in reactor design and analysis, which is the subject of Chapters 3 and 4. In Chapter 5, we will return to the subject of chemical kinetics and treat it more fundamentally and in greater depth. 2.1 RATE EQUATIONS A ‘‘rate equation’’ is used to describe the rate of a reaction quantitatively, and to express the functional dependence of the rate on temperature and on the species concentrations. In symbolic form, r A ¼ r A ðT , all C i Þ where T is the temperature. The term ‘‘all C i ’’ is present to remind us that the reaction rate can be affected by the concentrations of the reactant(s), the product(s), and any other compounds that are present, even if they do not participate in the reaction.
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