Chemical Engineering - Chemical Reaction Engineering and Applied Chemical Kinetics

Chapter 1

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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 3 Chapter 1. Introductory aspects and classification of chemical reactions 1.1 Introduction Chemical kinetics is the study of the chemical evolution of a reacting system in time. It assumes a complementary role with respect to thermodynamics, the scope of which is the study of equilibrium properties of a macroscopic system, independently from time and energy barriers that have to be overcome to reach the equilibrium. Notwithstanding this fundamental difference, chemical thermodynamics and kinetics are closely related, as will be shown during this course. For this reason, a good understanding of thermodynamics is needed to face the study of kinetics. From a historical point of view, chemical kinetics is a relatively young science, born in the second half of the XIX century with the works of Wilhelmy on the rate of inversion of sucrose, and with the study of Barthelot and Saint Gilles on the reaction between ethanol and acetic acid. It was in the beginning of the XX century that the studies on chemical kinetics acquired liveliness, benefiting from the developments of the kinetic theory of gases, and statistical thermodynamics, that provided the theoretical basis through which to interpret the experimental results. The field of study of kinetics pertains to all those systems in which a transformation occurs with a modification of the molecular composition of the chemical species involved, through the disruption or the formation of one or more chemical bonds. Such transformation is called chemical reaction. It is important here to note that by system we mean, in the thermodynamic sense, a portion of space defined by a control surface. The definition of chemical bond used here is relatively broad, meaning not just covalent bond responsible for the structure of molecules (intramolecular bonds), but also electrostatic and van der Waals interactions. These two types of interactions are energetically weaker and include hydrogen and ionic bonds, which cause the formation of complexes in solution and non-covalent adsorption of chemical species on reactive surfaces (intermolecular bonds). The focus of chemical kinetics is the estimation of the rate with which chemical reactions take place. 4 1.2 Reaction rates A reaction rate is defined as the rate at which a chemical species is produced inside the reacting system. Depending on whether the reaction occurs in a homogeneous phase or at the interface between two phases, we define: - homogeneous reaction rate: R i V = n i u d V ⋅ u d t = m o l m 3 s (1.1) where RiV defines the number of moles ni of species i produced per unit time and unit volume. RiV is measured, in SI units, in moles per cubic meter per second. - heterogeneous reaction rate: R i S = n i u d S ⋅ u d t = m o l m 2 s (1.2) where RiS defines the number of moles ni of species i produced per unit time and unit area. RiS is measured, in SI units, in moles per square meter per second. It can be noted that the reaction rate is defined as an intensive property, and thus it is to be intended in a local sense as a property of space and its intensive state variables. For this reason, the reaction rate is to be understood as a function of temperature, pressure and concentration of the compounds present in that portion of space in which the rate is being measured. Thus: RiS/V = Ri(T,P,Cj(j=1,Nc)) (1.3) Besides the definition of homogeneous and heterogeneous reaction rates, others are sometimes used for convenience, and derive from the method used for their measurement or from their application. In particular, here we report the definition of reaction rate sometimes used for mixed gas-solid or liquid-solid systems: 5 R i m i x v o l ( ) = n i u o V s ⋅ u o t = m o l m s 3 s R i m i x m a s ( ) = n i u o V m ⋅ u o t = m o l k g s s (1.4) In those systems, the rate of production of compound i is referred to the volume (or mass) of the solid reactant and are particularly convenient to use when it is not possible to evaluate the surface area of the solid reactant (e.g. a catalyst). 1.3 Reaction rate and mass balances The knowledge of the reaction rate expression is not sufficient to define the chemical evolution of a particular reagent system, defined as the knowledge of the change in space and time of its chemical composition. It requires the solution of the mass balance relative to the species of which you want to study the evolution. If the temperature of the system or the fluid dynamics field are unknown, it is needed to solve also the energy and momentum balances, besides the mass balance of the chemical species involved. How to write and solve these balances for reagent chemical systems is the core of chemical reaction engineering, and will not be examined in depth here. However, it is of interest to consider briefly the mathematical expression of the mass balances frequently used to study reagent systems, i.e. ideal reactors (CSTR, PFR and Batch reactor) and the general form of the material balance: CSTR € Q in C i in − Q out C i out + V ⋅ R i V T , P , C j = 1 , Nc ( ) + S ⋅ R i S T , P , C j = 1 , Nc ( ) = 0 (1.5) PFR € dQC i dV = R i V ( T , P , C j = 1 , Nc ) + S V R i S ( T , P , C j = 1 , Nc ) (1.6) BATCH € 1 V dn i dt = R i V ( T , P , C j = 1 , Nc ) + S V R i S ( T , P , C j = 1 , Nc ) (1.7) General form of the steady material balance € v • ∇ C i = D i ∇ 2 C i + R i V T , P , C j = 1 , Nc ( ) (1.8) 6 The solution of equations 1.5-1.8, under adequate boundary and validity conditions of the assumptions from which the equations have been derived, allows to calculate the concentration of species i as a function of the operative conditions (e.g. temperature and pressure) of the process. It is of interest now to make some considerations. First, the expression of the rate of reaction is the same in every one of the material balances. This is due to the intensive nature of this quantity and shows that its calculation is separated, from a mathematical and physical point of view, from the particular form of the considered material balance. Second, it is interesting to note that in the general form of the material balance there is not the term of the surface reaction rate. That is because, when the mass balance is made indefinitely, the production of chemical species happens only at the interface between two domains and appears in the differential equations as a boundary condition. Based on these considerations, the field of study of chemical kinetics may seem rather limited, being its scope the evaluation of the expression Ri reported in (1.3). Actually the problem is very complex, as we will see during this course. 1.4 Elementary reactions and complex reagent systems The estimation of Ri is made particularly complex by the implicit dependence of the chemical species involved in the reagent system. This dependence is usually not linear and can assume very complex forms. For example, in a relatively simple system such as the synthesis of hydrobromic acid from molecular hydrogen and bromine. The rate of formation of the hydrobromic acid has the following expression: R H B r = k 1 C H 2 C B r 2 ( ) 1 / 2 1 + k 2 C H B r C B r 2 (1.9) Such complexity can be explained, from a molecular point of view, by the observation that a generic chemical reaction, consisting in a rearrangement of chemical bonds of the molecules involved, does not happen in a single 7 physical event, such as the collision of two reagent molecules, but in an often very complex succession of elementary acts. In this context, we define as an elementary act a chemical reaction where its molecularity, i.e. number and nature of the molecules involved in the physical event that originates the reaction, coincides with the stoichiometry, which defines the proportion between the reacting molecules. To make this concept more concrete, consider the reaction of combustion between hydrogen and oxygen, the product of which is water. While the global stoichiometry of the process is given by the following global reaction: R1) H2 + ½ O2 ! H2O from a molecular point of view, the reacting process is originated by the reaction between hydrogen and oxygen to give two hydroxyl radicals: R2) H2 + O2 ! 2OH⋅ Followed by a cascade of reactions of which the most important are: R3) OH⋅ + H2 ! H2O + H⋅ R4) H⋅ + O2 ! O⋅ + OH⋅ R5) H⋅ + O2 ! HO2⋅ R6) H⋅ + O⋅ ! OH⋅ R7) 2H⋅ ! H2 R8) 2O⋅ ! O2 … Extrapolating this example to a more general context, the reaction R1 is defined global reaction, while the set of reactions R2-R8 is called kinetic scheme. The single reactions Ri (i=2,8) are called elementary acts and the matrix νij is called matrix of the stoichiometric coefficients. In the stoichiometric matrix, i varies between 1 and the number of components, and j varies between 1 and the number of chemical reactions. The introduction of a kinetic scheme complicates 8 the definition of reaction rate introduced in the previous paragraphs, as a generic species i can appear as reactant or as product in more than one chemical reaction. However, the reaction rate Ri can be linked to that of the single elementary act in which the generic species i appears, by defining: rij : number of moles of species i produces in reaction j. ri = rij/νij : rate of reaction j. The independence of the ratio between rij and νij from the index i, derives from the stoichiometric constraint, according to which the ratio between the rate of production of different chemical species involved in a single elementary act must be equivalent to the ratio between the respective stoichiometric coefficients. Thus Ri, rij and ri are linked from the following relation: R i = r i j j = 1 N r ∑ = ν i j r j j = 1 N r ∑ (1.10) The relation 1.10 holds both for reactions in homogeneous phase and for surface reactions. 1.5 Dependence of reaction rate from the concentration of the chemical species The definition of elementary act simplifies considerably the problem of the calculation of the functional dependency of Ri from Nc concentrations of reactant species. In fact, if the calculation of Ri is led to that of the rate of a single elementary act ri, it is seen experimentally (and it is confirmed theoretically) that the reaction rate rj is proportional, by a constant kj, to the concentration of the chemical species to their stoichiometric coefficients taken in absolute value: r j = k j ( T , P ) ⋅ C i ν i j i = 1 N r e a g e n t i ∏ (1.11) the proportionality constant k is called kinetic constant and maintains the functional dependency from temperature and pressure. 9 Because of its simplicity, the expression (1.11) has been generalized to describe the dependency of the reaction rate from the concentration of the reactants also for non-elementary reactions, taking the name of power law: r j = k j ( T , P ) ⋅ C i θ i j i = 1 N c ∏ (1.12) Unlike from expression 1.11, in the case of the power law, the product is extended to all the chemical species. These are generically elevated to coefficients, named orders of reaction, that are usually determined on the basis of interpolation of experimental data, or by means of regression from complex kinetic schemes. Because of this, the orders of reactions of a reaction rate expressed by power law can assume non-integer or, at times, negative values. For example, the kinetics of combustion of methane can be described effectively using a power law with the following form: CH4 + 2O2 ! CO2 + 2H2O k = k 0 C H 4 [ ] 0 . 2 O 2 [ ] 1 . 3 Though a reaction rate expressed in the form of power law can be of simpler and more direct application than implementing a full kinetic scheme, it is necessary to remember that any mathematical relation determined on the basis of regression of experimental data, is valid only within the interval of parameters for which the regression has been carried. Thus, it is advisable the cautious application and only after having evaluated carefully the way it has been obtained. 1.6 Dependence of reaction rate on the temperature For an elementary act, the functional dependency of its reaction rate from the temperature is accounted for primarily in its kinetic constant, and secondly in the dependence of the reactant concentration from the temperature (for a gas, at constant composition and pressure, the molar concentration is inversely proportional to the temperature). Since the early experimental studies on 10 chemical kinetics, this dependence has been found to be strongly non-linear, and it can be expressed for most of the chemical reactions in a form known as Arrhenius equation: € k = A ⋅ exp − Ea RT $ % & ' ( ) (1.13) In this expression, the term A is called pre-exponential factor, the quantity Ea is defined as activation energy, R is the ideal gas constant and T is the temperature expressed in Kelvin degrees. To obtain a better agreement with experimental data, the Arrhenius equation is often used in a modified form, where dependence from temperature is introduced in the pre-exponential factor too: € k = A ⋅ T α ⋅ exp − Ea RT % & ' ( ) * (1.14) A rational explanation of the reason because the dependence of the kinetic constant from the temperature is well represented both from the Arrhenius equation and, for some reactions, from its modified form will be given in chapters 3 and 5 of these notes. 1.7 Dependence of the reaction rate on the pressure From what said in sections 1.5 and 1.6, it seems that the dependence of the reaction rate on the pressure is negligible, or at most contained in an indirect way in the dependence of the concentration of the reactants on the pressure of the system. If this can be true for some reactant systems, it is not however in an absolute way. It is well known indeed that the kinetic constant can depend, even significantly, from the pressure at which the reaction is conducted. In particular, this dependence is present for reactions that need a third body to which transfer their energy in order to relax the energy of the system, in a process known as transfer of vibrational-translational energy. The functional form assumed by the kinetic constant when it is dependent from the pressure will be discussed in chapter 5. 11 1.8 Thermodynamic consistency and microscopic reversibility principle The formulation of the problem of the estimation of the rate of a specific chemical reaction in terms of development and solution of a kinetic scheme formed by elementary reactions, complicates heavily the kinetic study of a reactant chemical system. A partial simplification is offered by thermodynamics, and it allows linking the direct and the inverse rate of a specific chemical reaction, using a functional relation based on the equilibrium constant of the reaction. Consider a generic chemical reaction: A + B ⇔ C From a microscopic point of view, the transformation of the compounds A and B in the compound C follows a physical path that can be followed in reverse, thus giving a path to transform the compound C in the compounds A and B. That is independent from the fact that the reaction happens through a single elementary act or not, thanks to the symmetry of the physical laws that determine the development of the path of the transformation. This is the reason because a generic chemical reaction is usually written connecting reactants and products with a symbol (usually a double arrow, or an equal sign) that indicates, implicitly, that the reaction can be followed both ways: from the reactants to the products or from the products to the reactants. The two semi-reactions are usually denoted as direct reaction and inverse reaction: Direct: A + B ⇒ C Inverse: C ⇐ A + B Suppose that we leave the reagent system evolve until a time tending to infinite. A condition where the compounds A, B, and C are in equilibrium will be reached. In such conditions the rates of direct and reverse reactions will not be null, but they will have the same value. The net rate of transformation of the reactants in products, given by subtraction of direct and inverse rates, is zero. Thus, the following relation holds: 12 € r d eq = r i eq k d C A eq C B eq = k i C C eq (1.15) it follows that the kinetic constant of the reverse reaction is linked to that of the direct reaction from the relation: € k i = k d C A eq C B eq C C eq (1.16) The ratio between the equilibrium concentrations elevated to the stoichiometric coefficients can be correlated directly with the equilibrium constant of the chemical reaction, remembering that: € k eq T , P ( ) = a i ν i i = 1 Nc ∏ (1.17) where ai is the activity of compound i. For and ideal gaseous system, the relation 1.17 becomes: k e q = e x p − Δ G r 0 ( T , P r i f ) R T #$%% &'(( = C i C r i f #$%% &'(( ν i i = 1 N c ∏ (1.18) where Crif is the reference concentration, evaluated as Crif =Prif/RT for an ideal gas. The reference pressure is usually the atmospheric pressure. Substituting 1.18 into 1.16, it gives: k i = k d C A e q C B e q C C e q = k d C A e q / C r i f C B e q / C r i f C C e q / C r i f ⋅ C r i f = k d ⋅ C r i f ⋅ e x p Δ G 0 T , P r i f ( ) R T #$%% &'(( (1.19) Generalizing 1.19 to a generic gas phase reaction where there is a variation of number of moles ∆n, and where the direct and inverse reaction rates are 13 proportional to the concentration of the reactants, the following expression is obtained: € k i = k d ⋅ C rif ( ) − Δ n ⋅ exp Δ G 0 T , P rif ( ) RT % & ' ' ( ) * * (1.20) The relation 1.20 is called thermodynamic consistency of a chemical reaction. It allows computing the inverse constant of a chemical reaction starting from thermodynamical data and the direct constant. As said before, the dissertation developed is applicable not only to elementary reactions, but also to complex reactions. It is left as an exercise the estimation of the inverse reaction constant for the non-elementary reaction of combustion of methane described in section 1.5. A law strictly linked to thermodynamical consistency of a chemical reaction, but different for field of validity, is the microscopic reversibility principle. Such principle, valid for an elementary reaction, states that, for a direct reaction, the physical path followed by the atoms during the reaction in the transition from the conformation of the reactants to that of the products is the same followed during the reverse path. This concept is strictly correlated to that of potential energy surface, which will be introduced in the following paragraph. 1.9 Potential energy surface for an elementary reaction From a microscopic point of view, a chemical reaction is a rearrangement of the relative position of some atoms that form the molecules involved. In this context, a chemical reaction can be seen as a motion of atoms in a multidimensional space defined by the relative coordinates of every atom and from its vectorial velocity. Such space is known as phase space. The physical laws that can be used to describe the motion of the atoms are two: Newtonian mechanics and quantum mechanics. Although very different, both theoretical contexts allow calculating the dynamics of the reagent system deducing the laws of motion of the atoms starting from the interatomic interaction energy expressed as a function of the spatial coordinates of the atoms involved. The function that links the interaction energy of the atoms to their coordinates is 14 defined Potential Energy Surface (PES). Thus, a chemical reaction can be described as motion of the atoms on this surface. From a graphical point of view, a potential energy surface is representable only if the geometry of the reagent system is function of not more than two coordinates. That is the case of the reaction between molecular and atomic hydrogen: H2 + H ! H + H2. The potential energy surface for this reaction is reported in Figure 1.1. Fig. 1.1 Qualitative PES of the reaction between H2 and H. Because of the difficulty of drawing and reading a tridimensional graph, a PES like that in Fig. 1.1 is usually represented in two dimensions, reporting only the energy isolevel curves. An example is the one reported in Figure 1.2. Fig. 1.2 PES of the reaction between H2 and H obtained projecting the isolevel curves from Fig. 1.1 on the plane of the molecular coordinates, taken from the url http://www.dur.ac.uk/eckart.wrede/QCT. 15 A further simplification of the graphical representation of a PES is possible. In fact the experimental and theoretical study of chemical reactions has shown that, usually, a reagent system moves on its potential energy surface following the path with the minimum energy (Minimum Energy Path, MEP). If we define the physical coordinate that describes the motion along the MEP of the PES of the reagent system as the reaction coordinate λ, it’s possible to represent the energy variation as a function of λ on a simple bidimensional graph, such as the one reported in Figure 1.3. The shape of the PES projected along the MEP for a specific reaction will be one of the criteria that will be adopted to classify homogeneous and heterogeneous chemical reactions in the next two sections. Reactants Products Reaction Coordinate Energy Fig. 1.3. PES for the reaction between H2 and H reported as a function of the reaction coordinate λ. 1.10 Classification of homogeneous reactions Many methodologies have been proposed in literature to classify chemical reactions as a function of their typology. Among them, here we chose to differentiate chemical reactions at a first level as a function of their global order of reaction, defined as the sum of the reaction orders. At a second level, we differentiate chemical reactions as a function of the characteristics of their potential energy surface. Thus, homogeneous reactions can be classified as: 16 -Order 0 reactions. Reactions whose rate is independent from the concentrations of the reactants. They are rather infrequent reactions that usually imply a complex mechanism. A typical situation where a 0 order reaction can be found, is when the reaction requires the adsorption of the reactants on a catalyst, the sites of which are filled of adsorbed species. The reaction rate is expressed as: k r = the dimensions of k are those of a reaction rate (mol/m3/s). -First order reactions, also known as unimolecular reactions. They are largely present in reagent systems and their study, both experimental and theoretical, constitutes one of the most important topics in chemical kinetics. They can assume two forms: A ! B A ! B+C The reaction rate is expressed as: ] [ A k r = where [A] represents the concentration of the reactant. The dimensions of k are those of a frequency (1/s). Unimolecular reactions can be further differentiate as a function of their PES, as activated reactions when the reaction path passes through a maximum (Fig. 1.4a), or non-activated (barrierless) when the PES does not show a maximum (Fig. 1.4b). 17 Reactants Products Reaction Coordinate Energy Reactants Products Reaction Coordinate Energy A) B) Fig. 1.4 Potential energy surfaces for activated and non-activated (barrierless). Activated unimolecular reactions include the following reactions: - Isomerization (e.g.: toluene ! cycloheptatriene - Elimination (e.g.: SiHCl3 ! SiCl2 + HCl) - Beta-scission (e.g. R-C2H5 ! R-H + C2H4) Non-activated unimolecular reactions are typically reactions where a single covalent bond is broken, such as homolytic scissions (e.g. CH4 ! CH3 + H). Second order reactions, also known as bimolecular reactions. Similarly to unimolecular reactions, these reactions are also among the most prevalent reactions in reagent systems and among the most studied in literature. They can assume two forms: A +B ! C+D A +B ! C The reaction rate is expressed as: ] ][ [ B A k r = where A and B represent the concentration of the reactants. The dimensions of k, in SI, are (m3/s/mol), even if the bimolecular kinetic constants are often expressed in (cm3/s/mol) or (cm3/s/molecule). The potential energy surface of bimolecular reactions is similar to those reported in Figure 1.4 (in the case of non-activated reactions, usually the potential energy surface reported in Figure 1.4b is followed from right to left). 18 In particular it is noted that many bimolecular reactions can be seen as inverse processes of unimolecular reactions. Bimolecular reactions include the following reactions: - radical recombination (usually non-activated, e.g. CH3 + CH3 ! C2H6) - salt methatesis reaction (e.g. NaCl + AgNO3 ! NaNO3 + AgCl) - extraction (usually activated e.g. H + CH4 ! H2 + CH3) - substitution (e.g. H + Al(CH3)3 ! HAl(CH3)2 + CH3) - dissociative recombination (e.g. CH3 + CH3 ! C2H5 + H, an adduct is formed and then it disassociates before to be energetically stabilized) - unimolecular reactions in fall off regime (see section 5.4) Third order reactions, also known as termolecular reactions. They usually are bimolecular reactions for which a direct proportionality of the kinetic constant from the gas concentration is experimentally measured, and thus also from the pressure. They usually are reported in the form: A + B + M ! AB + M The reaction rate is expressed as: ] ][ ][ [ B A M k r = where A and B represent the reactants concentration and M, named third body, represents the total gas concentration. The kinetic constant k has the dimensions of m6/mol2/s. The physical reason that determines the third order dependence of the kinetic constant from the concentration is that the reaction proceeds in two elementary acts. The first one is the formation of a complex energetically excited. The second one is the stabilization of the complex by hitting a third molecule (from which the name of M, third body). Further details about this important class of reactions are given in chapter 5.4. Potential energy surfaces for termolecular reactions have the shape reported in Figure 1.5. 19 Reactants Products Reaction Coordinate Energy Fig. 1.5 Potential energy surfaces for termolecular reactions. Reactions that can have a termolecular nature are typically recombination of radicals, e.g. 2CH3 + M ! C2H6 +M. However it is important here to note that the same reaction, such as the one reported in the example, can act as a bimolecular or a termolecular reaction depending on the pressure of the system. See section 5.4 to further analyze this important aspect. - nth order reactions. Some reactions can have non-integer or even negative global orders of reaction. Such reactions are always non-elementary and are to be understood simply as stoichiometric relations that link between them reactants and products of a global process that can be very complex. Usually the reaction rate expression is derived by fitting of experimental data or by regression made from a complex kinetic scheme. 1.11 Classification of heterogeneous, or surface reactions The classification of heterogeneous reactions, or of reactions that happen at the interface between two phases, follows a criterion that is less systematic compared to the one seen for reactions in homogeneous phase. Such criterion differentiates the reactions on the basis of their nature. It is necessary to introduce some concepts and definitions, before we begin to list the surface reactions that are more often seen in reagent systems. The first concept is that of surface site. By surface site is intended a part of the solid surface (e.g. of a catalyst) that is capable of forming a chemical bond with a chemical species present in the fluid phase in contact. Surface sites are present 20 on the surface with a certain concentration usually indicated with Γ and with the dimensions of mol/m2. An example of surface and surface site for a surface of monocrystalline silicon (100) reconstructed 2x1 is reported in Fig. 1.6. Fig. 1.6. Atomic structure and surface site for a Silicon (100) surface reconstructed 2x1. The surface density of the sites is 6.8x1014 sites/cm2. Hereafter we will refer to a surface site that is not occupied by an adsorbed chemical species with the symbol σ. For adsorbed species, we intend a surface site that is connected with a chemical species present in the fluid phase through a chemical bond. We will after refer to adsorbed species denoting the name of the chemical species with an asterisk (e.g. the species A adsorbed on a surface will be defined as A*). Once the nature of the chemical species present on a generic chemical surface is defined, it is possible to classify the surface reactions on the basis of their nature as: -Inclusion or wall reactions. They are reactions where a molecule hits a wall and is included in the solid phase of which the wall is the boundary (inclusion reaction). They can also be reactions where, as a consequence of the collision, the molecule changes its molecular conformation (wall reaction). They are usually reported in the form: A + surf ! Abulk A + surf ! B + surf The reaction rate is expressed as: 21 r s = k [ [ A ] where A is the concentration of the reactant. The kinetic constant k has the dimensions of a velocity (m/s). Examples of this type of reactions are the inclusion of carbon atoms produced during the combustion that give rise to carbon deposits, and also the neutralization of the charge of an ion produced in a plasma when it collides with a wall: - inclusion: C + surf ! Cbulk - wall: CH4+ + surf ! CH4 + surf - Adsorption reactions. They are reactions where a gas molecule interacts with one or more surface sites and forms one or more stable chemical bonds adsorbing on the surface of the material. They can be of two types, depending on the number of active sites involved: single site or double site. They are usually reported as: A + σ ! A* AB + 2σ ! A* + B* The reaction rate is expressed as: € r s = k [ A ] σ [ ] € r s = k [ AB ] σ [ ] 2 where A and AB represent the concentration of the reactants, while σ represents the concentration of the active sites. The dimensions of the kinetic constant k depend on the way the concentration of the active sites is expressed. It can be expressed in the form of number (or moles) of sites per surface unit (i.e. mol/m2) or in the form of fraction of sites (i.e. adimensionally). When the concentration of the sites is expressed in adimensional units (as a fraction), the adsorption constant has the dimensions of a velocity, both for single and double site adsorptions. When this formalism is applied, the reaction rate is often expressed as the product of the molar flux of the molecules towards the surface (given by the rate of collisions times the concentration of gas reactants) and by 22 an adimensional coefficient named sticking coefficient, which is a probability of reaction. See section 3.7 to further analyze this important concept. Instead, in the case where the concentration of the sites is expressed in dimensional units, the constant k is expressed in such dimensions that its product with the concentration of the reactants gives a quantity with the dimensions of mol/m2/s. Adsorption reactions are differentiated depending on the type of interaction between the reactant and the surface. The interaction energy can be due to van der Waals or weak electrostatics interactions (physical adsorption), or can be due to the formation of one or more covalent bonds between the adsorbed molecule and the surface (chemical adsorption), forming a bond of high energy (exceeding 15 kcal/mol). The potential energy surface for a physical adsorption reaction, usually characterized by an adsorption energy of 3-5 kcal/mol, is reported in Fig. 1.7. Gas Phase Molecule Adsorbed State Reaction Coordinate Energy 3-5 kcal/mol Fig. 1.7. Potential energy surface for a physical adsorption reaction. For chemical adsorptions, two types of reactions are possible, depending if a barrier that has to be surpassed to make the adsorption happen is present (Fig. 1.8b) or not (Fig. 1.8a). If the barrier is smaller than the energy gained form the formation of the physisorbed state, chemical adsorption is likely to take place contextually to the collision between gas phase molecule and surface (Fig 1.8a). If this is not the case (Fig. 1.8b), then only the molecules colliding with a sufficient energy may reach the chemisorbed state. 23 a) Gas Phase Molecule Chemisorbed state Reaction Coordinate Energy 10-80 kcal/mol b) Gas Phase Molecule Chemisorbed State Reaction Coordinate Energy 10-80 kcal/mol Fig. 1.8. Potential energy surface for a not activated (a) and activated (b) chemical adsorption reaction. - Desorption reactions. They are processes opposite with respect to adsorption processes. They can involve two molecules adsorbed in near sites (recombinative desorption) or a single adsorbed species (unimolecular desorption). They are usually reported in the form: A* ! A + σ A* + B* ! AB + 2σ The reaction rate is expressed as: € r s = k [ A *] € r s = k [ A *] B * [ ] where A* and B* represent the concentration of the species adsorbed on the surface. Differently from what seen for adsorption reactions, the concentration of the adsorbed species are usually reported in dimensional units (mol/m2). The dimensions of the chemical constant k are those of a frequency in the case of unimolecular desorption, and of m2/mol/s in the case of recombinative desorption. 24 - Reactions of adsorbed species. They are bimolecular reactions whose reactants are adsorbed species or isomerizations that lead to a conformational modification of the adsorbed molecules. One of the two reactants can be a free site. They are usually reported in the form: A* ! B* A* + B* ! C* + D* The reaction rate is expressed as: € r s = k [ A *] € r s = k [ A *] B * [ ] where A* and B* represent the concentration of the species adsorbed on the surface. Similarly to what seen for desorption reactions, the concentration of the adsorbed species is usually reported in dimensional units (mol/m2). The dimensions of the kinetic constant k are those of a frequency in the case of isomerizations, and of m2/mol/s in the case of bimolecular reactions. - Eley-Rideal reactions. They are reactions where one reactant is a molecule in the gas (or liquid) phase while the other is adsorbed, and the product is an adsorbed molecule with a different conformation and composition, and, at times, also a gas molecule with a composition different than that of the reactant. They are usually expressed as: A + B* ! C* + D The reaction rate is expressed as: € r s = k [ A ][ B *] where A and B represent the concentration of the reactants in the fluid and adsorbed phase, respectively. Similarly to what seen for adsorption reactions, the dimensions of the kinetic constant k depend on how the concentration of the adsorbed species is expressed. It can be in the form of number (or moles) per 25 unit of surface area (i.e. mol/m2) or in the form of fraction (i.e. adimensional). The kinetic constant is often expressed as a probability of reaction times the flux of gas molecules to the surface times the fraction of adsorbed sites on the surface.