Chemical Engineering - Chemical Reaction Engineering and Applied Chemical Kinetics

Chapter 2

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1 Notes of Applied Chemical Kinetics course Carlo Cavallotti Chapter 1. Introductory aspects and classification of chemical reactions Chapter 2. Kinetic schemes and reaction mechanisms Chapter 3. Kinetic theory of Gases Chapter 4. Fundamentals of Statistical Thermodynamics and Molecular Quantum Mechanics Chapter 5. Transition State Theory and further developments 2 Chapter 2. Kinetic schemes and reaction mechanisms 2.1 Introduction The schematization of reaction mechanisms introduced in the previous chapter seems to indicate a well-defined path to predict the kinetics of any reagent system. Given any reagent system, it seems possible to study its reactivity in a systematic way by applying a protocol that consists in the definition of all the chemical species that can be reasonably formed during the reaction, and in the construction of a kinetic scheme that includes all the reactions that can occur between them. Thus, the problem is reduced to the estimation of the kinetic constants of every single elementary act. This approach is endearing for many reasons. At least because it seems to suggest the existence of a universal method to study any reagent system. Actually it is a path followed by some research groups in the kinetics field, whose aim is the creation of an ‘automatic generator of kinetic schemes’. Basically it would be a software capable of generating automatically a kinetic scheme given any reactive system, and of estimating the kinetic constants of every reaction that appears automatically (e.g. using the methods described in chapter 5 of these notes). However, the few examples reported in literature are unconvincing. The problems are essentially two. The first is the a priori identification of the chemical intermediates that can be formed during the reaction. In fact, very often, extremely unstable chemical species are produced during a reaction to follow the paths with the minimum energy (and, as we will see, with the maximum entropy). These unstable species are difficult to observe, but are very reactive. They are species whose formation is difficult to be predicted a priori and it can be argued for decades about their contribution to a chemical process, e.g. radical species in gas phase reactions, and ions in liquid phase reactions, or plasmas. The second problem that anyone who wants to study a reagent system has to face, is the estimation of the kinetic constants of the elementary acts. The possible paths to follow are two: the experimental evaluation, and the theoretical estimation. Anyhow it is a complex problem, both for the difficulty of setting up an experimental apparatus capable of measuring the kinetic constant of a single reaction, thus extrapolating it from the set of reactions that occur together in any reacting context, and for the complexity of determining the parameters necessary to apply adequately the most advanced kinetic theories. 3 Although these considerations indicate that the possibility of automatically constructing a kinetic scheme without having detailed knowledge about it are now remote, it is however interesting to observe that the level of understanding of many kinetic mechanisms, even if significantly complex, is very high, e.g. pyrolysis and oxidation mechanisms of a large number of hydrocarbons, or even many catalytic mechanisms and some aspects of atmospheric and biologic chemistry. The reason of the intelligibility of many reagent systems lies in the fact that often only few, if not only one, elementary acts condition the rate of the entire process, as the number of the reactions and the complexity of the chemical species produced may be high. The analysis and study of the mechanism and logic of evolution of a kinetic scheme often allow extrapolating from an apparently incoherent system of reactions, a reaction law that provides a level of understanding sufficient to interpret the global reactivity of the system. Thus the subject of this chapter is the definition and application of the necessary criteria to obtain this result, and also the explanation of why, in some situations, a particular chemical reaction acquires a role of such importance. 2.2 Analysis of kinetic mechanisms: reactions in series and in parallel At a first level of complexity, it is possible to interpret the set of reactions that occur in a reagent system in terms of: - parallel reactions from different reactants; - parallel reactions from the same reactant; - reactions in series. Though simple, this classification has multiple applications. For example it helps to interpret experimental kinetic data, by allowing to understand if during a reaction an intermediate is formed and providing a scheme for the estimation through regression of the kinetic data. Furthermore, it allows interpreting if preferential reaction mechanisms exist inside complex kinetic schemes, which lead to the production of a specific reaction product by proceeding in series or in parallel. As mentioned in section 1.3, the study of the chemical evolution of a reagent system requires the integration of the mass balance for the chemical species involved in the reaction. A particular simplification of the equations involved is obtained if a system homogeneous in space maintained at constant temperature and pressure is assumed. These are called microkinetic conditions. For a closed system, the form of the resulting mass balance is equivalent to that of a batch reactor. For an open steady 4 system it is equivalent to that of a CSTR. Both these typologies of reactor are in fact experimentally used to determine kinetic data. In the following analysis we will refer to a batch reactor, as we are interested in the study of the evolution in time of the considered systems. To interpret in an intuitive way the nature of the reactivity of a reacting system, it is more effective to study the variation in time of some of its characteristic parameters than to report directly the laws of the evolution the chemical species involved. Once defined nA, nB, and nA0 as the number of moles of A and B present at time t, and the number of moles present at the instant 0, for a generic reaction A ! B, the most used parameters for this aim are: conversion (ξ) : number of moles reacted / number of initial moles = (nA0-nA)/nA0; yield (η) : number of moles reacted to give the desired product / number of initial moles = (nB/νB)/nA0; where νB is the stoichiometric coefficient of the species B. Also, if multiple products of reaction are possible (e.g. if a parallel reaction A ! C exists), the following parameter is introduced: selectivity of B with regard to C (SB/C): number of moles reacted to give B / number of moles reacted to give C ) (nB/νB)/ (nc/νc). It is important to stress here the conceptual difference between conversion and yield, where the latter represents, from a process optimization point of view, the parameter to focus on. In fact, it is not always true that the desired product is obtained at high conversions. Thus, the analysis of the three reagent systems is done as follows: A) parallel reactions from different reactants The kinetic scheme has the following form: 1) A ! B 5 2) C ! D Under the hypothesis of irreversible reactions, the mass balances associated are (and under the hypothesis of negligible variations of volume): € dC A dt = − k 1 C A (2.1) € dC C dt = − k 2 C C (2.2) The mass balances of the products are redundant, because they can simply be derived from the reactants imposing the conservations of the mass. Equations 2.1 and 2.2 can be integrated directly by parts. The resulting kinetic laws, with the boundary conditions of concentration of compounds A and C present at time 0 set to € C A 0 e € C C 0 and that B and D to sero, are: € C A = C A 0 exp − k 1 t ( ) (2.3) € C C = C C 0 exp − k 2 t ( ) (2.4) € C B = C A 0 − C A (2.5) € C D = C D 0 − C D (2.6) where the concentrations of B and D are obtained by imposing that the concentration of A and C at time 0 are equal to the sum of reactants and products of reaction at a generic instant. On a Cartesian graph, the variation in time of A and C assumes the form schematized in Figure 2.1, where the initial concentration of A is assumed to be equal to 1 mol/m3 and that the kinetic constant k1 is 10 cm-1. 6 Figure 2.1. Variation in time of the concentrations of compounds A and C. From the analysis of the graphs reported in figure 2.1, it can be noted that a particular time exist when the concentration of the reactant reach a value that is half the initial one. Such time is called half-life period of compound A and is often used in the analysis of characteristic times to define the period in which a determined compound, consumed by a chemical reaction, is present in a significant concentration. In the case of irreversible unimolecular reactions, it is simple to calculate, by means of equation 2.3, that the half-life period is: € t 1 / 2 = ln 2 ( ) k 1 (2.7) As the natural logarithm of 2 is approximately equal to 1, it is often assumed that the characteristic time is simply equal to the inverse of the kinetic constant. It is now possible to calculate conversion, yield and selectivity from the concentrations (the equations are similar if the volume does not change during the reaction). Referring to the compounds A and B, we get: Conversion: 0 0.2 0.4 0.6 0.8 1 1.2 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 t (s) C (mol/m3) CA CC 7 € ξ = 1 − exp − k 1 t ( ) Yield: € η = 1 − exp − k 1 t ( ) It can be noted that yield and conversion are equal. That is due to the fact that there are not reactions in competition (i.e. reactions that can lead to different products starting from the same reactant). For the same reason, the selectivity analysis is meaningless for this reagent system. B) parallel reactions from the same reactant The kinetic scheme has the following form: 1) A ! B 2) A ! C Proceeding in a way similar to that followed for the previous set of reactions, the mass balances assume the following form: € dC A dt = − k 1 C A − k 2 C A (2.8) € dC B dt = k 1 C A (2.9) € dC C dt = k 2 C A (2.10) The integration of equations 2.8-2.10 can be done by parts, similarly to what seen in the previous paragraph, substituting in 2.9 and 2.10 the result of 2.8. Assuming that at time 0 only A is present, we get: € C A = C A 0 exp − k 1 + k 2 ( ) t ( ) (2.11) € C B = k 1 k 1 + k 2 C A 0 1 − exp − k 1 + k 2 ( ) t ( ) [ ] (2.12) € C C = k 2 k 1 + k 2 C A 0 1 − exp − k 1 + k 2 ( ) t ( ) [ ] (2.13) 8 The evolution in time of the reacting system, calculated assuming that at time 0 the concentration of A is 1 mol/m3 and that kinetic constants k1 and k2 are 10 s-1 and 5 s-1 respectively, assumes the form reported in Figure 2.2. Figure 2.2. Variation in time of concentrations of compounds A, B, and C for a parallel reaction mechanism. It can be noted that the rate of consumption of the reactant is given by the sum of the rates of the parallel reactions, which thus compete for the consumption of the precursor. The kinetic law for the disappearance of the reactant and its half-life period are similar to those calculated for a single reaction with an effective constant given by the sum of the two parallel processes. A further analysis of the properties of the reagent system can be done by studying conversion, yield and selectivity. The parameters calculated from 2.11-2.13 are: Conversion: € ξ = 1 − exp − k 1 + k 2 ( ) t ( ) 0 0.2 0.4 0.6 0.8 1 1.2 0 0.05 0.1 0.15 0.2 0.25 t (s) C (mol/m3) CA CB CC 9 Yield in B: € η = 1 − k 1 k 1 + k 2 exp − k 1 + k 2 ( ) t ( ) Selectivity of B with respect to C: € S B / C = k 1 k 2 Compared to the previous system, in this case conversion and yield differ. This is due to the fact that not all the molecules converted give the desired product, as measured by yield, because of the competition with the process of production of C. The selectivity of B with regard to C measures the ratio between the two products in this reagent system. As it appears evident from graphs reported in Figure 2.2, such ratio is constant and it is equal to the ratio of the kinetic constants of the two processes that occur in parallel. C) series or consecutive reactions In this case, the kinetic scheme expects the formation of a product B, that is an intermediate of the reaction. In its simplest form, the kinetic scheme can be formulated as: 1) A ! B 2) B ! C The correspondent equations of mass balance are: € dC A dt = − k 1 C A (2.14) € dC B dt = k 1 C A − k 2 C B (2.15) € dC C dt = k 2 C B (2.16) The integration of these equations is a little more complex than what seen until now. As it is a class of equation often encountered in the field of chemical kinetics that has 10 the quality of admitting an analytical solution, we will discuss in detail the solution procedure. This initially consists in the integration by parts of equation 2.14. If at time zero it is assumed that the concentrations of compounds B and C are zero and that the concentration of A is equal to € C A 0 , the solution of the integration of equation 2.14 is equation 2.3. Inserting it in 2.15 we get: € dC B dt = k 1 C A 0 exp − k 1 t ( ) − k 2 C B (2.17) It is a first-order non-homogeneous ordinary differential equation. The solution to equation 2.17 can be found by summing one particular solution to the solution of the associated homogeneous form (obtained by eliminating the non-homogeneous term, i.d. the one dependent on t). The solution obtained has the following form: € C B = A exp − k 2 t ( ) + B exp − k 1 t ( ) (2.18) where A and B are two constants of integration. The term B is thus calculated by substituting 2.18 in 2.17, while A may be calculated imposing the boundary conditions (in this case CB(0) = 0). We get: € B = k 1 C A 0 k 2 − k 1 € A = − k 1 C A 0 k 2 − k 1 From which, substituting in 2.18, we obtain the desired solution: € C B = − k 1 C A 0 k 2 − k 1 exp − k 2 t ( ) + k 1 C A 0 k 2 − k 1 exp − k 1 t ( ) (2.19) The kinetic law for CC can be obtained both by substituting 2.19 into 2.16 and integrating by parts, and deriving it from the mass conservation law: € C A 0 = C A + C B + C C . 11 € C C = C A 0 1 − k 2 k 2 − k 1 exp − k 1 t ( ) + k 1 k 2 − k 1 exp − k 2 t ( ) # $ % & ' ( (2.20) The evolution in time of the reagent system, calculated in a way similar to the previous case, by assuming that at time 0 the concentration of A is 1 mol/m3, and that the kinetic constants k1 and k2 are 10 s-1 and 15 s-1 respectively, assumes the form reported in Figure 2.3. Figure 2.3. Variation in time of concentrations of compounds A, B, and C for a mechanism consisting of reactions in series. The behavior of a system with reaction in series appears immediately different from a parallel one. In fact the trend in time of the concentrations of the two products of reaction, B and C, is different. As it can be noted, compound B shows a maximum in time, while an inflection point characterizes the trend of the concentration of compound C in time. Depending on whether the compound C or the compound B is the desired product of the reaction, we say that B is an intermediate of the reaction or also that the reaction that from B leads to C is a secondary or parasitic reaction, as it involves the loss of the desired product of the reaction. Conversion, selectivity and yield can be calculated in a similar way to that used to derive relations 2.11-2.13: 0 0.2 0.4 0.6 0.8 1 1.2 0 0.05 0.1 0.15 0.2 0.25 t (s) C (mol/m3) CA CB CC 12 Conversion: € ξ = 1 − exp − k 1 t ( ) Yield in C: € η C = 1 − k 2 k 2 − k 1 exp − k 1 t ( ) + k 1 k 2 − k 1 exp − k 2 t ( ) Selectivity with respect to B: € S C / B = 1 − k 2 k 2 − k 1 exp − k 1 t ( ) + k 1 k 2 − k 1 exp − k 2 t ( ) # $ % & ' ( / − k 1 k 2 − k 1 exp − k 2 t ( ) + k 1 k 2 − k 1 exp − k 1 t ( ) # $ % & ' ( The characteristic of systems with reactions in series is to have maximum values both for concentrations and for yields of intermediate compounds. Instead selectivities show increasing or decreasing trends depending on whether the desired compound is C or B. A particular case of a system with reactions in series is when an intermediate compound is extremely more reactive than the reactant, thus once it has been formed, it will disappear rapidly. When this occurs, some approximations that allow simplifying considerably the mathematical problem associated to the integration of the system of differential equations of the mass balances become acceptable. 2.3 Analysis and simplification of kinetic schemes One of the most important results that can be obtained from the analysis of a kinetic scheme is the kinetic law that determines the rate of production or consumption of every desired compounds. It is, as mentioned in the first chapter of these notes, a mathematical relation, where the rate of formation of the desired compound (Ri) is expressed as a function of the concentrations of reactants and products of the reaction. This usually requires the analytical solution of the set of equations that express the mass balance of every component. Unfortunately, this is a type of problem that rarely admits an analytical solution. However, methods of analysis of kinetic schemes exist that in many cases allow obtaining analytical solutions that, on the basis of some basic 13 assumptions, allow simplifying considerably the reacting system. In particular these methods of analysis are based on two approximations that are presented and discussed in the following chapters. 2.3.1 Pseudo-steady state approximation The rate of production or consumption of a specific chemical species can be expressed as the difference between the sum of the reactions that lead to its formation ( € R i prod ) and those that lead to its consumption ( € R i scomp ): € 1 V dn i dt = R i prod − R i scomp (2.21) When a species is extremely reactive, e.g. a radical or an ion, it usually happens that its formation and consumption rates are much greater than the accumulation terms. Thus two conditions are verified: € R i prod >> 1 V dn i dt ; € R i scomp >> 1 V dn i dt In such conditions, it is acceptable to simplify the term of accumulation from 2.21 and express the mass balance for the species i as: 0 = R i p r o d − R i c o n s u m p t i o n (2.22) The simplification of 2.21 in 2.22 is called Pseudo Steady State Approximation – PSSA, because it removes from the mass balance for the species i the term of accumulation. It is noted that the state of the species i is called ‘pseudo’ steady state, not just steady, as the application of 2.22 does not imply that the dependence of the concentration of the species i from time is null, i.e. it is independent from time, but only that this is smaller than the rates of production and consumption of the highly reactive compound. To realize this aspect, it is useful make an example. Consider the case of the production of compounds B and C from compound A through the series reactions seen in the previous paragraph: A ! B ! C. The mass balance of this reacting 14 system is given by equations 2.14-2.16. If it is assumed that the species B is highly reactive, it is possible to apply the PSSA to the mass balance of B. Thus it becomes: € 0 = k 1 C A − k 2 C B (2.23) The set of equations given by 2.14, 2.16 and 2.23 can be easily integrated by parts. We get: € C A = C A 0 exp − k 1 t ( ) (2.24) € C B = k 1 k 2 C A = k 1 k 2 C A 0 exp − k 1 t ( ) (2.25) € C C = C A 0 1 − exp − k 1 t ( ) ( ) (2.26) It is easy to verify that equations 2.25 and 2.26 can be derived directly from 2.19 and 2.20 by imposing that k2>>k1, which is equal to impose a high reactivity for compound B. The level of error associated to the application of the PSSA can be noted by comparing the graphs of the evolution in time of the concentrations obtained with the analytical solution and the approximated one, reported in Figure 2.4 for a constant k1 = 10 s-1 and a constant k2 = 1000 s-1. 15 a) b) Figure 2.4. Variation in time of concentrations of compounds A, B and C for a mechanism of series reaction obtained using the analytical solution (a) and that based on the PSSA. As it can be noted, the comparison between the results obtained applying the two methodologies of calculation shows how the PSSA gives excellent results when the assumption of high reactivity of the compound at which the PSSA is applied holds. It is noted also how the concentration of compound B, as anticipated before, is not 0.001 0.01 0.1 1 0 0.05 0.1 0.15 0.2 0.25 t (s) C (mol/m3) CA CB CC 0.001 0.01 0.1 1 0 0.05 0.1 0.15 0.2 0.25 t (s) C (mol/m3) CA CB CC 16 constant in time, and it has a dependence equal to that of compound A. Finally, it is noted that a typical error associated to the introduction of the PSSA is the lack of the capacity of estimate, also qualitatively, the behavior of the system in the first instants, i.e. before the pseudo-stationary state is reached. 2.3.2 Rate Determining step approximation An intrinsic property of a chemical reacting system is that it can be described as a succession of reactions, parallel or in series, characterized by a direct proportionality with regard to the concentration of the reactants. As seen in the previous chapter, kinetic constants are characterized by a strong non-linearity with regard to temperature, interpreted by means of the Arrhenius law as an exponential dependence from a parameter defined as activation energy. This is often equal to the energetic barrier that has to be overcome so that the reaction can proceed. Often the reactions in a reagent system have activation energies that are significantly different between them. This implies that the kinetic constants of the different reactions that form the kinetic scheme can be considerably different, even of many orders of magnitude. In these conditions, it can happen that a specific reaction among all is characterized by a reactive flux (i.e. number of molecules per unit time and unit volume that pass through the transition state to form the products, thus without considering the inverse processes) significantly lower than that of the other reactions, and considerably greater than the inverse flux. If this reaction represents a fundamental step of the reactive process (i.e. it is needed to convert the reactants to products), then it becomes a bottleneck for the entire reactive system and conditions its global rate. This reaction is called rate-determining step (RDS) of the set of reactions, and it coincides with the reaction whose reactive flux is minimum. To better understand this important concept and asses how it can be quantitatively used to determine kinetic laws, we will now study the properties of a mechanism of reactions in series that leads to transformation of a reactant A to a product B through a set of N reactions. Let Xi be the N-1 intermediates of reaction. Thus, the reacting system is representable through the following set of reactions: A X1 1) X1 X2 2) … 17 Xk-1 Xk K) … XN-1 B N) Consider now the rates of the different reactions ri as the sum of their direct and inverse fluxes of reaction. It follows that: € r 1 = ! r 1 − " r − 1 € r 2 = ! r 2 − " r − 2 … € r k = ! r k − " r − k … € r N = ! r N − " r − N Consider now the behavior of the system at steady state. In such conditions, the rates of production and consumption of the intermediates Xi will be the same. It follows the condition: € r 1 = r 2 = ... = r K = ... = r N = r (2.27) Relation 2.27 shows that for a system of reactions in series at the steady state, the net rates of every elementary reaction are equal between them. It is now interesting to determine what admitting a rate determining step means for this system. At a first look, in fact, it seems that every reaction has the same rate and it is not immediately intuitive to understand what the presence of a slow stage means, or even if it is possible. To examine in depth this important aspect we report in Fig. 2.5 the rate of every reactive stage using vectors with module equal to the intensity of the direct and inverse fluxes of reaction. Suppose that the system has a RDS and that this is reaction k. 18 r r1 r-1 r2 r-2 rN r-N rk r-k Fig. 2.5. Representation of the reaction fluxes for a set of N reactions in series that lead to the conversion of reactant A to product B. Stage K represents the RDS of this system The graphical representation reported in figure 2.5 explicitly shows what it means for the RDS to have a reactive flux € ! r k minor than those of the other reactions and significantly greater than the inverse flux € ! r − k . In particular it can be noted that the net flux of reaction r is approximately equal to the direct rate of the process k € ! r k , which is the reason why this reaction is called rate determining step, and that the direct and inverse rates of any other processes are approximately equal, i.e. € ! r i ≅ " r − i for i ≠ k. The condition of equality between direct and inverse rate of a specific chemical reaction implies that this reaction is at the equilibrium. In such conditions this reaction is not affected by kinetic effects, as an increase of the rate of the direct process would involve an increase by the same amount of the rate of the inverse process. This has the important consequence that any intervention (e.g. an increase of temperature or the use of a catalyst) on a reacting system characterized by the presence of a RDS will 19 increase its global rate only if it will increase the kinetic constant of the RDS, as all the other reactions are at equilibrium. This is a fundamental concept for the chemical kinetics, as it shows how it is often not necessary to understand in detail the kinetics of a reacting system, but it can be enough just to identify which stage is the determining step of the process. Once it has been identified it will be possible to optimize the operative conditions by simply finding the conditions (e.g. a specific catalyst) that maximize the rate of this process. From a quantitative point of view, the presence of a RDS can be used to determine the kinetic law of the process in a very simplified way with respect to the solution of the material balance for each chemical species. This is possible by matching the rate of the global process to that of the rate determining step, and by expressing the concentration of all the intermediates of reaction by applying the equilibrium conditions. For the case under exam, by defining the kinetic constant of reaction i as ki (with the minus sign if the inverse reaction is intended), and the concentration of the species i as Xi, we get: € R B = r = k K X K − 1 − k − K X K (2.28) € k eq 1 = X 1 A € k eq 2 = X 2 X 1 … € k eqK − 1 = X K − 1 X K − 2 … € k eqK + 1 = X K + 1 X K … € k eqN = B X N − 1 Thus it is easy to demonstrate that by substituting in succession in 2.28, the global rate of the reaction becomes: 20 € R B = r = k K k eqK − 1 k eqK − 2 ... k eq 1 A − k − K k eqK + 1 k eqK + 2 k eqN B (2.28) It becomes thus possible to express the global rate of the process as a function of the concentration of reactants and products of the reaction, and of the product of sequences of some equilibrium constants (it is left to the reader to demonstrate that the product of sequences of the equilibrium constants can be substituted with the single equilibrium constant between the first and the last term of the product). That is an important result that we will use in the study of heterogeneous reacting systems. 2.4 Reaction Mechanisms As the concepts introduced in the previous paragraphs introduce some instruments that can be used to analyze kinetic systems, however they do not provide any starting point to understand why a kinetic mechanism has a particular form. For example they do not help to understand the reason why it is necessary to form a specific intermediate of reaction, or why a gas phase reaction needs the presence of a catalytic solid phase to proceed in the reaction. A systematic analysis of all the typologies of the possible mechanisms is beyond the scope of this course, which discusses the topic in a simplified way. However it is possible to classify the mechanisms of reaction at a first level of complexity as direct mechanisms, that proceed through a concerted path, and mechanisms that require the formation of at least one reaction intermediate, which here we will call indirect mechanisms. Direct mechanisms include all those reactions where the breakage and the formation of bonds that involve the transition from reagent to product of a reaction occur at the same time in one reaction. For example, Diels Alder reactions and the three/four-center reactions (so called for the number of atoms involved in the reactive process, e.g.: H2 + Cl2 ! 2 HCl and C2H5Cl ! C2H4 + HCl). Reactions that proceed through an indirect mechanism require the formation of one or more highly reactive intermediates of reaction. Among the principal indirect mechanisms there are radical mechanisms, surface mechanisms, and ionic mechanisms. The reaction path followed by a specific reacting system, either concerted, radical, or heterogeneous, is the one that maximizes the reactive flux, i.e. the number of molecules converted to product per unit time and volume. Even if different reaction paths can coexist in parallel for the same reacting system, usually only one is dominant. Direct mechanisms of reactions are, in some ways, the simplest 21 to study and require only a good ability of estimation of the kinetic constant, topic that is discussed in the fifth chapter of these notes. Indirect mechanisms instead can be studied, analyzed, and understood using some special precautions, such as PSSA and RDS approximations. The methods to study radical and surface mechanisms are the subject of discussion of the next two sections, while for ionic mechanisms, important for organic chemistry and electrochemistry, please refer to other references. 2.4.1 Radical Chain reactions Radical mechanisms of reaction are characterized by the formation of at least two radical species among the reaction intermediates. Radicals are molecules (or atoms) characterized by having one of the valence orbitals occupied by only one electron. Molecules with this electronic structure are called ‘doublets’, because of the spin number and are usually indicated with a dot after the molecular formula (in general R·). Such characteristic makes these chemical species highly reactive, decreasing significantly the activation energy for the reactions in which they can be involved. The formation of a radical in a reactive environment can promote a determined chemical reaction if it takes part in it as a catalyst, i.e. contributing actively to the reactivity without being consumed. This is possible through a series of at least two reactions, named propagation reactions, whose net result is the transformation of the reactants into products, and the regeneration of the original radical, even its chemical composition may change several times. The radical mechanism requires, besides the propagation phase, a phase in which the radicals are generated, called initiation, and one in which they are consumed, called termination. A radical mechanism for a generic reaction A + B ! C + D can be thus schematized by the following set of reactions. Initiation: I2 ! 2 R· Propagation: R· + A ! C + R1· R1· + B ! D + R· Termination: 22 R· + R· ! T1 R· + R1· ! T2 R1· + R1· ! T3 The set of propagation reactions includes all those reactions that have at least one radical among their reactants, and whose number of radicals does not change during the reaction. A particular subset of the propagation reactions is the catalytic cycle, which is formed by those reactions through which the radical, generated from the initiation reaction, is initially transformed and then regenerated. The minimum number of reactions in a catalytic cycle is two. The number of times a radical takes part to a catalytic cycle before being consumed in a termination reaction is called chain length. A similar concept is that of quantum yield, used when the initiation reaction occur by irradiation with light. The quantum yield of a reaction is defined as the number of molecules reacted per absorbed photon. One of the first examples of radical mechanism is given by the mechanism of conversion of hydrogen and chlorine to hydrochloric acid. The path followed by this reaction, first proposed by Nernst in 1918, is the following: Initiation: : Cl2 ! 2 Cl· Propagation: Cl· + H2 ! HCl + H· H· + Cl2 ! HCl + Cl· Termination: Cl· + H· ! HCl The reaction is usually initiated photochemically. The quantum yield is of 106 molecules converted per absorbed photon. A fourth typology of reactions that can occur in a reacting system through a radical chain mechanism are the branching reactions. It is a chemical reaction whose number of radicals produced by the reaction is greater than the number of the reactant ones. Systems where branching reactions are present are characterized by an extremely high reactivity, usually associated to risks of explosion. An example of reacting system where branching reactions are active is given by the reaction between hydrogen and oxygen. This mechanism is characterized by two branching reactions that occur in series: 23 R1) H· + O2: ! OH· + O: R2) O: + H2 ! OH· + H· where O2: and O: are characterized by having two unpaired electrons, a molecular structure called triplet. The sum of the two reactions of propagation shows how the reaction of atomic hydrogen with O2 and H2 leads to formation of three radicals, increasing considerably the reactivity of the system. R3) H· + O2: + H2 ! 2OH· + H· If we add to reaction R3 the propagation reaction OH· + H2 ! H2O + H· multiplied by two, we get: H· + O2 + 3 H2 ! 2 H2O + 3H· which shows that in this reacting system the number of radicals increases by 2 at every step of the reaction. It is possible to have an approximate measure of the propensity of a system to explode, by doing a balance on the rates of generation and consumption of the radicals. If we define the rate of production of radicals in initiation reactions as ri, and the rate of multiplication and consumption of the same in propagation and termination reactions respectively as rb and rt, the global balance of formation and consumption of radicals takes the following form: € dR ⋅ dt = r i + r b − r t If the pseudo-steady state approximation is applied, and it is assumed that the rb and rt terms are proportional through the kinetic constants kb and kt to the radical concentration (i.e. rb = kb R· and rt = kt R·), we get: € R ⋅ = r i k t − k b 24 When the termination constant becomes equal to that of branching, the concentration of radicals tends to diverge. Such condition is defined explosive limit of a branching reaction. The application of the PSSA to the study of radical mechanisms, as just demonstrated, allows obtaining very important results for what concerns the interpretation of the kinetics of reacting systems. In particular the systematic application of the PSSA to the study of the kinetics of radical mechanisms allows estimating what is the dependence of the rates of the global reaction from the reactant concentration, i.e. the kinetic law of the system, regardless of the concentration of radicals formed during the reaction. An example of the application of this powerful method of study of radical systems to a specific problem is the subject of the next section. 2.4.1.1 Study of Radical chain Mechanisms through the PSSA One of the principals aims of the study of a reacting system is the determination of its kinetic law i.e. the dependence of the production rate of the desired compound, Ri, from operative conditions (i.e. temperature and pressure), and concentration of the compounds present in highest concentration and of which it is generally possible to measure the concentration in the reactor. In the case of a system reacting through a radical mechanism, the last of these requisites makes particularly difficult the deduction of the kinetic law, as the concentration of radicals is often difficult to measure, as these are present in very low concentrations because of their high reactivity. A solution to this problem is possible if, once known the kinetic scheme, the PSSA is applied to the mass balances of the radical species. Using this approach it is possible to obtain a system formed by a number of equations equal to that of the radical species which, if solved with respect to the concentration of the radicals, allows expressing their value as a function of the concentration of the non-radical species and of the kinetic constants of the reactions in which the radicals are involved. The substitution of the resulting expressions in the mass balances of the non-radical species allows obtaining the kinetic law in a form that is not dependent on the concentration of the radical species. In order to understand the power of this method of analysis of kinetic schemes, it is useful to consider an example. It can be the reaction of synthesis of hydrobromic acid, whose kinetic law has already been presented in the first chapter of these notes (1.9) and that we report here: 25 € R HBr = k 1 C H 2 C Br 2 ( ) 1 / 2 1 + k 2 C HBr C Br 2 (2.29) Though this kinetic law expresses a rather complex dependence from the concentration of reactants and products of the reaction, it is however interesting to observe that this has actually been derived from experimental data. The procedure followed for the deduction of the 2.29 i the application of the rules of asymptotic combination to the kinetic laws of production of HBr, deduced experimentally in form of power law for high and low concentrations of HBr. In particular we find that: if CHBr is low (i.e. when the reaction starts): € R HBr bassa = k b C H 2 C Br 2 ( ) 1 / 2 (2.30) if CHBr is high (i.e. at the end of the reaction): € R HBr alta = k a C H 2 C Br 2 ( ) 3 / 2 C HBr (2.31) Relations 2.30 and 2.31 can be combined by the asymptotic combination law as: € 1 R HBr = 1 R HBr alta + 1 R HBr bassa (2.32) By substituting 2.30 and 2.31 into 2.32, we get an expression equivalent to 2.29, that becomes equal with a convenient redefinition of the constants ka and kb. The kinetics of formation of hydrobromic acid has been among the first to be analyzed in depth by the first physical-chemists who studied the laws of kinetics. The first one who gave a satisfactory explanation of the reason because the kinetic law of formation of HBr has the form reported in 2.29 was Bodenstein in 1912. It is one of the first scientific studies in which it is assumed that the reaction mechanism requires 26 the formation of radicals as intermediate species. The following analysis of this mechanism by means of the PSSA lead to the formulation of a kinetic law with the same dependence from the concentrations of reactants and products of reaction of 2.29. It was a result of extreme importance, as it confirmed the assumption that a system can react through formation and consumption of radical species. To obtain such result the PSSA assumption has been applied to radical species, it is interesting to study here the details of the followed procedure. The kinetic scheme proposed by Bodenstein to explain the conversion of H2 and Br2 to HBr is composed by the following 5 reactions: 1) Br2 ! 2 Br· 2) Br· + H2 ! HBr + H· 3) H· + Br2 ! HBr + Br· 4) H· + HBr ! H2 + Br· 5) 2 Br· ! Br2 These 5 reactions can be interpreted as follows: the first one is an initiation reaction, the second, the third, and the fourth are propagation reactions, the last one is a termination reaction. The second and the third reactions form the catalytic cycle. The law of production of HBr defined by this mechanism is: € R HBr = ν HBr , i r i = k 2 C Br ⋅ C H 2 + k 3 C H ⋅ C Br 2 − k 4 C H ⋅ C HBr i = 1 Nreaz ∑ (2.33) To express 2.33 as a function of the concentration of the non-radical species it is possible to apply the PSSA to the mass balances of the radical species: € dC Br ⋅ dt = 2 k 1 C Br 2 − k 2 C Br ⋅ C H 2 + k 3 C H ⋅ C Br 2 + k 4 C H ⋅ C HBr − 2 k 5 C Br ⋅ ( ) 2 = 0 (2.34) 27 € dC H ⋅ dt = k 2 C Br ⋅ C H 2 − k 3 C H ⋅ C Br 2 − k 4 C H ⋅ C HBr = 0 (2.35) 2.34 and 2.35 can be easily processed to express the concentrations of H· and Br·. By summing the two equations, making explicit the resulting equation in Br·, the concentration of H· can be obtained from 2.35. We get: € C Br ⋅ = k 1 k 5 C Br 2 # $ % & ' ( 0.5 (2.36) € C H ⋅ = k 2 C H 2 k 3 C Br 2 + k 4 C HBr C Br ⋅ = k 2 C H 2 k 3 C Br 2 + k 4 C HBr k 1 k 5 C Br 2 # $ % & ' ( 0.5 (2.37) Also, by subtracting 2.35 from 2.33, a simple expression for the production rate of HBr is obtained: € R HBr = 2 k 3 C H ⋅ C Br 2 (2.38) The desired kinetic law can be now easily obtained by substituting equation 2.37 into 2.38: € R HBr = 2 k 3 C Br 2 k 2 C H 2 k 3 C Br 2 + k 4 C HBr k 1 k 5 C Br 2 " # $ % & ' 0.5 (2.39) Equqation 2.39 can be easily modified to a form more similar to 2.29 by dividing numerator and denominator by k3CBr2: € R HBr = 2 k 2 k 1 k 5 " # $ % & ' 0.5 C H 2 C Br 2 ( ) 0.5 1 + k 4 C HBr k 3 C Br 2 (2.40) It can be noted that 2.40 and 2.29 are formally identical. 28 The procedure followed here shows not only how the application of the PSSA allows analyzing a kinetic mechanism and to deducing the desired kinetic law, but also how this procedure allows attributing a physical meaning to the constants k1 and k2 from equation 2.29, which can be interpreted as a combination of kinetic constants of the elementary acts. As the kinetic constants of the elementary reactions 1-5 can be measured experimentally in an independent way (or estimated theoretically), relation 2.40 is susceptible of a direct verification through comparison with the experimental data from which eq. 2.29 has been derived. Such check is the proof, often sought by who is interested in kinetic mechanisms, that the proposed mechanism is able to describe the kinetics of the examined system. 2.4.2 Surface mechanisms: Langmuir method The study of the kinetics of chemical reactions that occur through surface mechanisms, i.e. through reactions that imply the interaction of the reactant molecules in the gas (or liquid) phase with a surface, coincides for many aspects with that of the kinetics of radical systems. The similarity is essentially due to the fact that surface reactions proceed fast thanks to the formation of chemical species adsorbed on highly reactive surfaces that assume the role of intermediates of reaction. The first person who proposed this interpretation of surface mechanisms and who formalized his intuition in a systematic methodology of study of surface processes was Langmuir. The model of Langmuir is based on the assumption that on every surface a definite and constant number of sites exists with which the chemical species present in the gas or liquid phase are able to form chemical or physical bonds, and thus to adsorb on them. Concepts as surface site, adsorbed species, and adsorption reaction are discussed in section 1.11, to which the reader is referred to deepen their meaning. On the basis of this interpretation of surface reactivity, the model of Langmuir introduces the following fundamental concepts: - the species adsorbed and the surface sites are involved in the chemical reactions as any other chemical species present in the homogeneous phase. Their concentration has the dimension of moles per unit of surface area (usually mol/m2). This implies that the law of conservation of mass holds (i.e. the conservation of the elements between reactants and products), according to which a null molecular weight can be assigned to any chemical site. In the 29 following discussion we will indicate a free site with the symbol σ and an adsorbed one as Ci*. - The sum of the concentration of free sites and adsorbed species is constant and equal to the total number of sites (Γ) present on the surface when no chemical species is adsorbed: € σ + C i ads = Γ i = 1 Nads ∑ - As the adsorbed chemical species are highly reactive, it is possible to apply the pseudo-steady state approximation to their variation in time: € dC i * dt = 0 The application of these three rules to the study of surface processes allows to considerably simplify the analysis and has helped, occasionally, to obtain kinetic laws that are still of great importance for the characterization of the reactive properties of heterogeneous processes. In the following two sections two examples are reported of the application of such method to the study of adsorption isotherms of gaseous chemical species on surfaces. More in general, it is important to note that the schematization of Langmuir is still at the foundation of the kinetic study of any surface reaction mechanism. 2.4.2.1 Langmuir isotherms The first application of the method of Langmuir to the study of the surface processes regarded the determination of the concentration of the relative quantity of chemical species adsorbed on a surface kept at a constant temperature as a function of the concentration of the gaseous species precursor to the adsorption. On the basis of the schematization of Langmuir, the reversible adsorption of a species A on a surface to give the adsorbed species A* can be expressed by the following reactions: adsorption: A + σ ! A* desorption: A* ! A + σ If it is assumed that the two reactions occur with kinetic constants kads and kdes at a rate proportional to the concentration of the reactants, the concentration of the reacted 30 species can be easily derived as a function of the concentration of the gaseous species by imposing the PSSA on the adsorbed species, and imposing the conservation of the surface sites: € dC A * dt = k ads C A C σ − k des C A * = 0 (2.41) € C σ + C A * = Γ (2.42) By combining 2.41 and 2.42 it is possible to express the concentration of the adsorbed species as a function of the concentration of A: € C A * = k ads C A Γ k ads C A + k des (2.43) Equation 2.43 can be expressed also as a function of the fraction of chemical species adsorbed on the surface by remembering that this is defined by the ratio between the concentration of the adsorbed species and the total concentration of sites: € ϑ A = C A * Γ = k ads C A k ads C A + k des (2.44) Equations 2.44 and 2.43 represent two analogous forms of the Langmuir adsorption isotherm, in this case valid for the adsorption of a single component through a single site mechanism. It is noted that to obtain the adsorption isotherm just one PSSA condition has been imposed, even if there are two surface species, i.e. the free sites σ and the adsorbed species A*. The reason is that as the total number of sites is conserved, the number of linearly independent PSSA conditions is equal to the total number of surface species diminished by 1. It is indeed easy to verify that by applying the PSSA on σ, a relation similar to equation 2.41 is obtained. The dependence of € ϑ A from the concentration of A calculated for a collisional adsorption constant (refer to the third chapter of these notes for its calculation) assuming that A has the molecular weight of nitrogen, and for a desorption constant with a pre-exponential factor of 1013 s-1, and an activation energy of 20 kcal/mol is 31 reported in figure 2.6. As it can be noted, the molar fraction of adsorbed species has a strong dependence on temperature, due to the activation energy of the desorption process. Figure 2.6. Variation of the molar fraction of adsorbed species (theta A) as a function of the concentration of precursors in the gas phase calculated for three different temperatures. It is reminded that the average concentration of gaseous species calculated for an ideal gas at 1 bar is of about 24 moles per m3. The procedure now adopted to determine the surface coverage fraction for the adsorption of a single species is immediately extensible to the competitive adsorption of N chemical species, provided that the reaction of adsorption is single site. Consider at this scope a gas phase where N chemical species are present with a concentration Ci, and that they can be adsorbed on a surface to give adsorbed species Ci*. The N PSSA conditions on the adsorbed species and the condition of conservation of the sites are: € dC i * dt = k i ads C i C σ − k i des C i * = 0 (2.45) 0.0001 0.001 0.01 0.1 1 0 10 20 30 40 50 CA (mol/m3) Theta A CA* (500 K) CA* (400 K) CA* (600 K) 32 € C σ + C i * i = 1 N ∑ = Γ (2.46) By making explicit Ci* from 2.45 and substituting into 2.45, an equation in σ is obtained, which allows making explicit the concentration of free sites as: € C σ = Γ 1 + k i ads k i des C i i = 1 N ∑ (2.47) By substituting 2.47 into 2.45 is thus immediate to derive the adsorption isotherm of Langmuir for every species Ci: € C i * = k i ads k i des C i 1 + k i ads k i des C i i = 1 N ∑ Γ (2.48) It can be noted that 2.48 is fundamentally similar to isotherm 2.43, derived for the system with one component, from which it differs only for the extension of the denominator to the sum on every adsorbed species. 2.4.2.2 BET isotherms An important application of the method of Langmuir to study the processes of physical adsorption has been performed in 1938 by Braunauer, Emmett, and Teller with the scope of explaining some experimental data that showed a dependence not easily interpretable between the volume of a gas adsorbed on a surface and the pressure of the gaseous precursor. It was indeed demonstrated that it was possible to measure the quantity of adsorbed gas on a solid surface (even if porous) as a function of the pressure of the gas phase, that had to be maintained in every case below the saturation vapor pressure of the gas P0 to avoid the formation of a liquid film on the surface of the solid. The adsorption curves, measured at a fixed temperature and thus isothermal, had the shape reported in Figure 2.7. By analyzing the physical adsorption 33 curve, Braunauer, Emmett, and Teller found that it was possible to extrapolate from this data a piece of information of fundamental importance: the surface area of the solid. It is an essential data for any study of reactivity of surfaces, as it is explicitly required in the formulation of any mass balance in heterogeneous reacting systems (refer for example to section 1.3 of these notes). Also, the BET procedure is applicable to porous solids too, for which it is virtually impossible to derive a measure of surface area starting from geometrical considerations. As the result of the BET treatment is of significant interest for chemical kinetics, and the procedure followed is instructive as an example of application of the method of Langmuir, even if it is a little more ‘elaborate’, we will see now how it was possible for Braunauer, Emmett, and Teller to derive the adsorption law that is now known with the initials of their names. P/P0 0 1 V (cm3/g) Figure 2.7. Example of correlation between the volume of an adsorbed gas on the surface of a solid and the pressure of the gas phase. P0 is the saturation vapor pressure. 34 The procedure of deduction of the BET isotherm is based on the concept of site introduced by Langmuir, according to which on the surface a determined number of sites is present, and the adsorption of gaseous species can occur only on one of the available sites. In addition to that, in the BET treatment it is assumed that it is possible also the adsorption of one or more molecules on others already adsorbed. The model of surface coverage can be thus schematized as reported in Figure 2.8. Site Number 1 2 3 N Adsorbed Molecules 2 0 1 4 Figure 2.8. BET schematization of the surface adsorption. On the basis of this schematization of the adsorption process, the following variables can be introduced: σm = surface density of molecules corresponding to a monolayer, and total number of sites when the surface is completely free (mol/m2). σ = surface density of all the adsorbed molecules (mol/m2). ϑi = fraction of sites on which the molecules are adsorbed (value between 0 and 1). On the basis of these definitions, the following relations hold: € σ = σ m i ϑ i i = 1 ∞ ∑ (2.49) € 1 = ϑ 0 + ϑ i i = 1 ∞ ∑ (2.50) Also, the ratio between the volume of the adsorbed molecules V and those adsorbed on a monolayer Vm must be equal to that between the correspondent surface densities: 35 € σ σ m = V V m (2.51) By defining the surface species on which are adsorbed 0, 1, 2, ... molecules and S0, S1, S2, ... , and as G the gaseous species, in steady conditions the following reactions must be at the equilibrium: G + S0 = S1 G + S1 = S2 … By assuming that the molecules are adsorbed with collisional efficiency (i.e. every collision with a surface species Si is effective and leads to the adsorption of the molecule) and that the reactions of desorption proceed with a kinetic constant kdes1 for the adsorbed molecules at direct contact with the surface, and with kinetic constant kdes2 for those adsorbed on another molecule, the following relations are derived (it is assumed that in the gas phase just the precursor to adsorption is present): € k ads P RT ϑ 0 = k des 1 ϑ 1 σ m € k ads P RT ϑ 1 = k des 2 ϑ 2 σ m € k ads P RT ϑ 2 = k des 2 ϑ 3 σ m … For an infinite number of layers, i.e. when the surface is covered by a uniform monolayer of liquid, it must hold: € k ads P 0 RT = k des 2 σ m (2.52) By combining between them the equilibrium relations and 2.52, and imposing x=P/P0 the following recursive relations are obtained: 36 € x = ϑ 2 ϑ 1 € x = ϑ 3 ϑ 2 That produce the law of dependence of € ϑ i from x for i > 1: € ϑ i = x i − 1 ϑ 1 (2.53) While € ϑ 1 can be expressed with respect to € ϑ 0 as a function of a constant C as: € ϑ 1 = 1 σ m k ads k des 1 P RT ϑ 0 = 1 σ m k ads k des 1 P 0 RT P P 0 ϑ 0 = Cx ϑ 0 (2.54) From which: € ϑ i = Cx i ϑ 0 (2.55) By substituting 2.55 in 2.49 and 2.50, we get: € σ = σ m iCx i ϑ 0 i = 1 ∞ ∑ (2.56) € 1 = ϑ 0 + Cx i ϑ 0 i = 1 ∞ ∑ (2.57) By making explicit € ϑ 0 from 2.57 and substituting it into 2.56, we get: € σ = σ m iCx i i = 1 ∞ ∑ 1 + Cx i i = 1 ∞ ∑ (2.58) By observing that: 37 € x i i = 1 ∞ ∑ = x i i = 0 ∞ ∑ − 1 = 1 1 − x − 1 = x 1 − x (2.59) and that (by deriving 2.59 and making some easy algebraic passages): € ix i i = 1 ∞ ∑ = x 1 − x ( ) 2 (2.60) 2.58 can be expressed as: € σ = σ m C x 1 − x ( ) 2 1 + C x 1 − x (2.61) So, it is simple to demonstrate that 2.61 can be rewritten as: € x 1 − x 1 σ = 1 C σ m + x C − 1 C σ m (2.62) Remembering the definition of x and by substituting 2.51, the desired relation is obtained: € P P 0 − P 1 V ads = 1 CV m + P P 0 C − 1 CV m (2.63) Expressing experimental data of adsorption as a function of P/P0 and (P/P0-P)/Vads (the line reported in Figure 2.7 should become at this point a straight line), it becomes possible to determine the parameters 1/CVm and (C-1) /CVm by means of a simple linear regression. The desired parameter, the surface area, is at this point easily estimable by dividing the volume of a monolayer Vm by the average molecular diameter of the adsorbed molecule. 38 2.4.3 Surface mechanisms: Langmuir-Hinshelwood kinetics The analysis of the surface reaction mechanisms has shown that most of these can be interpreted by a model known with the name of who proposed and used it first: Irving Langmuir and Cyril Hinshelwood (both Nobel prize winners for chemistry for their studies on chemical kinetics). Based on such model, a surface process is described by the succession of the following three sets of reactions: 1) adsorption (and desorption) of the reactants; 2) surface reaction between the adsorbed species; 3) desorption of the products of reaction; One of the elements that characterizes the Langmuir-Hinshelwood mechanisms is the assumption that the chemical reaction, which converts the reactants into products, occurs between adsorbed species, an aspect from which the Eley-Rideal mechanism, according to which the conversion from reactants to products occurs through a reaction between a gaseous species and an adsorbed one, differs substantially. Another element is to consider all the reactions as reversible, except one that takes the role of Rate Determining Step of the process. A Langmuir-Hinshelwood mechanism for a generic heterogeneous reaction A+B ! C+D is generally described by a kinetic surface scheme that has a form similar to the following (be σ the free sites and A*, B*, C*, and D* the adsorbed species): Adsorption of the reactants: R1) A + σ ↔ A* R2) B + σ ↔ B* Surface reaction: R3) A* + B* ↔ C* + D* Desorption of the reactants: R4) C* ↔ C + σ R5) D* ↔ D + σ 2.4.3.1 Study of surface kinetics with RDS approximation The study of surface kinetics that proceed through Langmuir-Hinshelwood mechanisms, allows obtaining simple