Chemical Engineering - Chemical Reaction Engineering and Applied Chemical Kinetics

Chapter 3

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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 Chapter 3. Kinetic theory of Gases 3.1 Introduction The kinetic theory of gases, anticipated by intuitions of Newton, Herepath and Bernoulli, was developed and has been widely accepted by the scientific community in the second half of the 19th century thanks to two fundamental works published by Clausius in 1857 and by Maxwell in 1860. Subject and motivation of these studies was the attempt to describe the properties of a gas on the basis of the structural hypothesis that it is composed by a set of particles (atoms or molecules) that have a determined velocity and interaction energy. The results they wanted to achieve with this study were conceptually far-reaching. The aims of the study were to obtain a fundamental comprehension of what heat and temperature are, to find an explanation of the inverse proportionality between pressure and volume that was predicted by the ideal gas law, and to obtain a way to calculate the fundamental properties of gases, such as heat capacity, heat conductivity, viscosity and diffusion coefficient. The proposed path, extremely innovative at the time, was based on the hypothesis that the macroscopic properties of a thermodynamic system (like temperature and pressure) can be described from the comprehension of the characteristics of the elements that constitute such set, and thus from the study of its macroscopic properties. At the time these studies were considered with great skepticism, both because of the high number of simplifying hypotheses adopted to formulate a problem that is mathematically simple, and because of the reluctance to describe thermodynamic systems in terms of sets of molecules (it has to be reminded that at those times the existence of molecules was not universally accepted). Among the many results of the kinetic theory of gases there is the first formulation reported in literature of an expression for the kinetic constant of an elementary bimolecular chemical reaction. Although the application of the kinetic theory of gases to reacting systems is in many ways obsolete and inappropriate to describe the complexity of a reacting system, it is however possible to apply it successfully to estimate, at a first level of approximation, the kinetic constant of some particular classes of reactions, such as bimolecular recombination reactions that do not present an energetic barrier, and gas-surface collision reactions. Also the concepts introduced into the kinetic theory of gases constitute an ideal introduction to more advanced theories, such as transition state theory, that for certain aspects can be considered an extension of the reactive kinetic theory of gases. For such reasons it will be here presented a formulation of the kinetic theory of gases, short but self-consistent, and the main results that can be obtained when it is applied to the study of reacting systems. 3.2 Fundamentals and cardinal equation of kinetics The kinetic theory of gases, in the formulation proposed by Clausius and Maxwell, is based on seven hypotheses: 1) High number of molecules per unit of volume; 2) Molecules separated by distances that are great with respect to their dimensions; 3) Molecules in continous motion with velocity vector v, its magnitude included between 0 and ∞, and velocity distribution function f(v) (i.e. how the velocity is distributed among the considered population of molecules) that is not known a priori; 4) No intermolecular interaction energy; 5) Homogeneous distribution; 6) Elastic collisions; 7) All the directions are equally probable (isotropy); The third hypothesis is of fundamental importance, because it introduces the concept that not all the molecules in a gas move with the same velocity, but that it is statistically distributed. The determination of a velocity distribution function is one of the principal results of the kinetic theory of gases. For this purpose we introduce the following definition: dNv = Nf(v)dv (3.1) which defines the number of molecules in a set of N molecules with velocity included between v and v + dv. Where v is the velocity vector, v is its magnitude, and f(v) is defined as the velocity distribution function. As the velocity distribution is isotropic, f(v) = f(v). It is reminded that dv and dv are linked by the relation dv = 4πv2dv. We define: dNvx = Nf(vx)dvx (3.2) as the number of molecules with velocity in x direction included between vx and vx + dvx. f(vx) is the distribution function of velocities in x direction, and dNvx/N defines the fraction of molecules with velocity included between vx and vx+dvx. Hypotheses 1-7 are reasonably satisfied for a rarified gas, except for hypothesis 4. More advanced formulations of the kinetic theory of gases allowed removing the assumption of absence of intermolecular interactions. For the level of discussion here addressed however such hypothesis can be accepted, as its removal does not imply significant modifications of the principal results we are interested in. The first result that can be obtained by applying the kinetic theory of gases is the estimation of the number of collisions between N molecules contained in a recipient of volume V and the walls that contain it. For this purpose, consider an arbitrary portion of dimension A of one of the walls containing the gas, and define as x the axis perpendicular to that wall. Such situation is represented in Figure 3.1. We will try now to determine the number of collisions that occur among all the molecules with a velocity included between vx and vx+dvx, and the surface A in a time interval ∆t. For this purpose, suppose to count from an instant t = 0 to an instant t = ∆t. In this time interval the molecules with velocity vx will cover a distance L = vx∆t. It follows that a generic molecule will be able to collide with the wall in the time interval ∆t only if it is at a distance from the wall lower than L at the instant t = 0. The total number of collisions between the molecules in the gas and the surface A will be then equal to the number of molecules with velocity vx that are in the volume of the parallelepiped with base A and side L at the instant 0. Defining as n the number of molecules per unit volume (n = N/V), the number of collisions dNcoll is given by: dNcoll = 1/2 A L n dNvx/N = 1/2 A vx∆t n dNvx/N = ½ A vx∆t n f(vx)dvx (3.3) The expression 3.3 is multiplied by a statistical coefficient ½ to take into account that the molecules have a probability of only 50% of having a velocity vx directed toward the surface A. The total number of collisions is obtained by integrating 3.3 for vx included between – ∞ and + ∞. Thus it requires the knowledge of the distribution function f(vx). A L vx x z y Fig. 3.1. Schematic representation of the volume that defines the set of molecules with velocity comprised between vx and vx+dvx that in the time interval ∆t collide with the surface A. The knowledge of the number of collisions of the molecules of a gas with a surface allows determining some quantities of particular interest. For this purpose it is convenient to think in terms of molecular flux, i.e. the number of collisions per unit time and unit surface area, Zsurf, defined as: dZsurf = dNcoll/A/∆t = ½ vx nf(vx)dvx (3.4) Once Zsurf is known, it is possible to calculate the amount of momentum ∆M transferred between gas and wall per unit surface area and unit time. Assuming that every collision is elastic, the momentum transfer ∆M is comparable, for the law of conservation of angular momentum in a collision, to the change of momentum of the impacting molecule. In an elastic collision with a surface, a molecule of mass m changes its velocity vector by keeping constant the components vy and vz, parallel to the plane of the collision, and by changing the verse of the perpendicular component vx. Such situation is represented in Figure 3.2. V1 V2 x vx -vx vy vy Fig. 3.2. Result of an elastic collision between a molecule and a surface. The component of the velocity perpendicular to the surface (vx) changes its verse, while the parallel component maintains the same direction and sense. The total momentum transferred for the collision of a single molecule with the surface is: ΔM = 2mvx (3.5) the total momentum dΔMtotA-1Δt-1 transferred by all the molecules with velocity included between vx and vx + dvx that collide with the surface A in the unit time ∆t can be then calculated by multiplying 3.4 and 3.5: dΔMtotA-1Δt-1 = mvxvx nf(vx)dvx (3.6) It can be noted that the momentum transferred per unit time is equals to the force exerted by the gas on the walls, and that the force per unit surface represents a pressure. It is obtained then that the pressure exerted on the walls by the molecules with velocity included between vx and vx+dvx equals to: dP= mvx2 nf(vx)dvx (3.7) thus the total pressure is: P = n m v x2 f v x ( ) d v x = n m v x2 - ¥ + ¥ ∫ (3.8) where the integral represents the mean square velocity in the x direction of all the gas molecules. Noting that the system is isotropic, i.e. = = , and that the square velocities are linked by the relation: € v 2 = v x 2 + v y 2 + v z 2 (3.9) it follows = 1/3, and thus: € P = nm v 2 3 (3.10) Reminding that n=N/V, and that, from the ideal gas law, PV=N/NavoRT, the following two relations are obtained: € PV = Nm v 2 3 (3.11) Reminding that the ratio between R and the Avogadro number is equal to the Boltzmann constant: € 1 2 m v 2 = 3 2 RT N avo = 3 2 k b T (3.12) Equations 3.11 and 3.12 are known as the cardinal equations of kinetics, and they link between them temperature, molecular motion, and pressure of a gas. They represent one of the principal results of the kinetic theory of gases and will be later used to solve the problem of the determination of the velocity distribution function. 3.3 Velocity distribution function Maxwell has solved the problem of the estimation of the velocity distribution function in 1860 on statistical bases. As the results he obtained are of fundamental importance for the application of the kinetic theory of gases to systems of reacting gases, we will report here the mathematical derivation. The aim of this discussion is the analytic calculation of the functions f(v) and f(vx), defined by equations 3.1 and 3.2. For this purpose we define the following quantities: dNvx/N = f(vx)dvx (3.13a) d2Nvxvy/N2 = dNvx/N dNvy/N = f(vx) f(vy)dvxdvy (3.13b) d3Nvxvyvz/N3 = dNvx/N dNvy/N dNvz/N = f(vx) f(vy)f(vz)dvxdvydvz (3.13c) the adimensional quantities dNvx/N, dNvxvy/N2, and dNvxvyvz/N3 represent the number of molecules with velocity comprised between vx and vx+∆vx, those with velocity comprised between vx and vx+∆vx, and between vy and vy+∆vy, and those with velocity comprised between vx and vx+∆vx, vy and vy+∆vy, and vz and vz+∆vz, respectively. It is now necessary to defined the space of velocities as the tridimensional space with variables vx, vy, and vz where it is reported the velocity vector of every considered molecule. Expression 3.13c, expressed as a function of volume unit of the velocity space, i.e. quantity dvxdvydvz, defines the density of phase space, and represents its population for every velocity v with components vx, vy, vz: € ρ = dN vxvyvz 3 N 3 dv x dv y dv z = f ( v x ) f ( v y ) f ( v z ) (3.14) To respect the condition of isotropy, the density of the velocity space must be constant for every assigned velocity v. Referring to Figure 3.3, this is equal to impose that ρ is constant on the spherical shell comprised between the surfaces 4πv2 and 4πv2 + dv. v dv vx vy vz Figure 3.3. Representation of a velocity vector v in the velocity space, and of the spherical shell that defines the set of molecules with velocity included between v and v + dv. The condition of isotropy for the velocity distribution among the molecules with velocity comprised between v and v+dv leads to the following two conditions: € ρ = f ( v x ) f ( v y ) f ( v z ) = constant (3.15a) € v 2 = v x 2 + v y 2 + v z 2 = constant (3.15b) Equations 3.15a and 3.15b express the condition that all the molecules with velocity components vx, vy, and vz that satisfy 3.15b, i.e. belong to the spherical shell reported in Figure 3.3, are distributed in the velocity space so that their density is independent from the direction of motion (isotropy condition). Passing to the derivative form and applying the Lagrange multipliers method, equations 3.15a and 3.15b can be solved. Remembering that the derivative of a constant is 0, we get: € df ( v x ) dv x dv x f ( v y ) f ( v z ) + f ( v x ) df ( v y ) dvy dv y f ( v z ) + f ( v x ) f ( v y ) df ( v z ) dv z dv z = 0 (3.16a) € 2 v x dv x + 2 v y dv y + 2 v z dv z = 0 (3.16b) that can be rewritten as: € 1 f ( v x ) df ( v x ) dv x dv x + 1 f ( v y ) df ( v y ) dvy dv y + 1 f ( v z ) df ( v z ) dv z dv z = 0 (3.17a) € v x dv x + v y dv y + v z dv z = 0 (3.17b) Using the Lagrange multipliers method with parameter λ, and gathering dvx, dvy, and dvz, 3.17a and 3.17b can be combined to give one equation: 1 f ( v x ) d f ( v x ) d v x + λ v x !"# $%& d v x + 1 f ( v y ) d f ( v y ) d v y + λ v y !"## $%&& d v y + 1 f ( v z ) d f ( v z ) d v z + λ v z !"# $%& d v z = 0 (3.18) The three terms that appear in equation 3.18 are derivatives, functions of independent variables. The equation is thus satisfied if and only if the three multipliers of the derivatives are identically zero, i.e. the following three equations hold: € 1 f ( v x ) df ( v x ) dv x + λ v x = 0 (3.19a) € 1 f ( v y ) df ( v y ) dv y + λ v y = 0 (3.19b) € 1 f ( v z ) df ( v z ) dv z + λ v z = 0 (3.19c) Equations 3.19a, 3.19b, and 3.19c are between them equivalent, and their solution will be thus identical. We will focus from now on only on 3.19a, the integration of which can be done indefinitely by parts. After few algebraic manipulations, the solution we get is: € f ( v x ) = Cexp(- β v x 2 ) (3.20) where C is a constant of integration and β is a constant obtained by redefinition of λ. Constants C and β can be calculated imposing appropriate boundary conditions. First, remembering that dNvx represents the number of molecules with velocity included between vx and vx+dvx, it is possible to impose the condition of normalization: € dN vx −∞ + ∞ ∫ = Nf ( v x ) dv x −∞ + ∞ ∫ = NC exp − β v x 2 ( ) dv x −∞ + ∞ ∫ = N (3.21) Condition 3.21 is equal to impose that the number of molecules with velocity comprised between -∞ and +∞ is equal to the total number of considered molecules. To integrate 3.21 it is useful to remind the integration rules of gaussian functions, as reported in table 3.1. As the number of molecules with velocity included between 0 and +∞ is half the number of those with velocity between -∞ and +∞, integration of 3.21 gives: € NC exp − β v x 2 ( ) dv x −∞ + ∞ ∫ = 2 NC exp − β v x 2 ( ) dv x = 2 NC 1 2 π β ' ( ) * + , 1 / 2 0 + ∞ ∫ (3.22) Table 3.1. Integrals for gaussian functions Integral Form Solution € exp − ax 2 ( ) dx 0 ∞ ∫ € 1 2 π a # $ % & ' ( 1 / 2 € x exp − ax 2 ( ) dx 0 ∞ ∫ € 1 2 a € x 2 exp − ax 2 ( ) dx 0 ∞ ∫ € 1 4 π a 3 # $ % & ' ( 1 / 2 € x 3 exp − ax 2 ( ) dx 0 ∞ ∫ € 1 2 a 2 Equating 3.22 to N (condition 3.21), we can express C as a function of β: € C = β π $ % & ' ( ) 1 / 2 (3.23) β can be finally determined using the cardinal equation of kinetics 3.12, remembering the definition of root mean square velocity reported in 3.8, and that = 1/3. Thus we get: € 3 2 k b T = 1 2 m 3 v x 2 β π $ % & ' ( ) 1 / 2 exp − β v x 2 ( ) dv x −∞ + ∞ ∫ (3.24) Using table 3.1 is then easy to demonstrate that the solution of 3.24 is the condition: € β = m 2 k b T (3.25) The velocity distribution function in the x direction assumes the following analytical expression: € f ( v x ) = m 2 π k b T # $ % & ' ( 1/2 exp - 1 2 m v x 2 k b T # $ % & ' ( (3.26) Finally, as f(v)dv, which has the physical meaning of number of molecules with velocity comprised between v and v+dv, must be equal to the product of density of space of velocities ρ times the volume unit 4πv2dv, and that f(vy) and f(vz) have the same functional form of f(vx), and that v2=vx2+vy2+vz2, it is possible to express the velocity distribution function f(v) as: € f ( v ) d v = m 2 π k b T # $ % & ' ( 3/2 exp - 1 2 mv 2 k b T # $ % & ' ( 4 π v 2 dv (3.27) Thus, the velocity distribution function in a gas is a Gaussian whose exponential is proportional to the square of the velocity or, considering also the term 1/2m, to the kinetic energy, and is divided by a term kbT with the dimensions of thermal energy of the system. The distribution function represented as a function of velocity is reported in Fig. 3.4. It is here important to note some of the properties of the velocity distribution function: 1) even if the distribution functions f(vx) and f(v) are both normalized to one when are integrated in dvx and dv (it is left to the student to verify numerically or analytically the accuracy of such statement), both have the dimensions of the inverse of a velocity (s/m). 2) the velocity distribution function has a maximum that with increasing temperature becomes less pronounced, and shifts to higher values. This means that, while at low temperatures most of the molecules of a gas have similar velocities, with the increase of temperature the distribution of velocities is more uniform and thus distributed between low and high values. Thus, in these conditions to think in terms of a single average velocity is not significant. Figure 3.4. Velocity distribution function for N2 calculated at 1000K and 2000K. 3) the velocity distribution function can be used to determine the average velocity of the molecules of a gas: € v = vf ( v ) d v 0 ∞ ∫ = v m 2 π k b T % & ' ( ) * 3/2 exp - 1 2 mv 2 k b T % & ' ( ) * 4 π v 2 dv = 8 k b T π m 0 ∞ ∫ (3.28) The average velocity of N2 and H2 is reported in figure 3.5 as a function of temperature. It is interesting to note that the decrease of mass has a significant effect on the average velocity of the two molecules. This increase of velocity is in part responsible for the high reactivity of hydrogen, both in its molecular form and, in a more accentuated way, in the atomic form. Finally it is important to note that also at room temperature, the molecules of a gas have a high average velocity. 4) Combining information of Fig. 3.4 and 3.5, it is apparent that with an increase of temperature there will be an increasing fraction of molecules with a velocity that is extremely high with respect both to the reference system and to the other molecules that on average are slower. The kinetic energy owned by these molecules is an important source of energy that, in an inelastic intermolecular collision, can act as principal promoter of a chemical reaction that requires to overcome an energetic barrier to proceed. The study of intermolecular collisions will be the topic of discussion of the next paragraph. Figure 3.5. Average velocity of nitrogen and hydrogen calculated as a function of temperature. 3.4. Intermolecular collisions Bimolecular reactive processes are determined by effective inelastic collisions between molecules that move freely in a gas contained in a recipient. With effective we mean that the result of the collision is a molecular reorganization that leads to the formation of one or more products of the reaction. To develop a kinetic theory of the reactive processes, it is necessary to determine the total number of intermolecular collisions that occur per unit time and unit volume in a gas. Such quantity, usually indicated with Z, has the dimensions of a reaction rate (number of collisions/m3/s), and represents the superior limit of the rate of a chemical reaction (as it is not possible that the reactive acts are more than the total number of collisions) and, in the situation in which all the collisions are reactive, it gives a good estimate, as first approximation, of the reaction rate. The estimation of the total number of collisions can be done at different levels of approximation, characterized by an increasing complexity. In this exposition we will consider two different approaches, both with important applicative implications. Approach A. The total number of collisions of a molecule can be calculated directly by assuming that it moves in an environment where all the other molecules are still. To simplify the calculation and without loss of generality, it is possible to consider the problem in two dimensions, x and z, assuming that the molecule, spherical and with diameter dA moves in direction x. The situation here described is represented schematically in Fig. 3.6. x z A B (dA+dB)/2 a ab c Fig. 3.6. Representation of the motion of A in direction x in a gas where molecules B are present. In a time interval ∆t A covers a distance c. In time interval ∆t A covers a distance c and collides with all the molecules B that are below the line a, defined as the tangent to the diameter of A parallel to the direction of motion. Molecules of B that cross a are those whose center of mass is at distance d of less than half the sum of the diameters of A and B, defined in figure 3.6 by line ab. In can be then concluded that, disregarding the scattering due to the collisions, in the time interval ∆t A collides with all the molecules B contained into the volume of the cylinder that has a base with radius (dA+dB)/2 and height c, that being equal to the distance covered in the time interval ∆t, which is equal to the product of the average velocity of A, , times the time interval ∆t. Thus the number of collisions of A is given by: € N collA = π d A + d B 2 # $ % & ' ( 2 v A Δ t N B V (3.29) where NB/V is the number of molecules of B present per unit volume. The total number of collisions Z can be then calculated by multiplying NcollA by all the molecules A in the unit of volume and dividing by ∆t: € Z AB = π d A + d B 2 # $ % & ' ( 2 v A N B V N A V (3.30) The quantity π(dA+dB)2/2 is usually defined as cross section, it has the dimensions of Å2, and it is indicated with the symbol σ. Expression 3.30 is strictly valid only when the velocity of B is so smaller than that of A that molecules B can be considered as still. This is for example the case of collisions between electrons and gas molecules in a plasma, as the electrons, that have a mass that is much lower than that of the molecules, move at velocities that can be even of orders of magnitude greater than those of the molecules. For this class of reactions, known as electronic impact processes, the cross section is usually a function of the kinetic energy of the incident electron. Indicating the number of electrons present per unit of volume with the symbol ne, the rate of an electronic impact process can be expressed as: € R el = Z el = v e σ el v e ( ) f ( v e ) d v e 0 ∞ ∫ N B V n el (3.31) or, thinking in terms of kinetic energy of the electrons Eel rather than electron velocity (Eel = 1/2meve2) and imposing f(ve)dve = f(Eel)dEel: € R el = Z el = 2 E el m e σ el v e ( ) f ( E el ) dE el 0 ∞ ∫ N B V n el (3.32) The solution of 3.32 requires that both the dependency of the cross section from the energy of the incident electron, and the energy distribution function of the electrons, which is usually a complex function of the electromagnetic field used to generate and sustain the plasma, be known. Approach B. The estimation of the total number of collisions between two molecules A and B can be done rigorously at a higher level of theory integrating the velocity distribution function for A and B, times the relative velocity of approach between the two molecules vr, and times a cross section σ. The number of collisions that occur in time ∆t between all the molecules A with velocity comprised between vA and vA + dvA, and all the molecules B with velocity comprised between vB and vB + dvB, can be calculated following a reasoning analogous to that exposed in approach A, that leads to the following expression, formally analogous to 3.30: € dN coll = v r σ Δ t N A V f v A ( ) d v A N B V f v B ( ) d v B (3.33) It is important to note that in 3.33 vr, vA, and vB appear in vector notation. To solve 3.33 is thus necessary to switch to the scalar form by referring the velocities of A and B with respect to those of the center of mass (vM), and remembering the following vector relations: € v r = v A − v B (3.34a) € m A + m B ( ) v M = m A v A + m B v B (3.34b) It follows (it is left to the reader the demonstration, reminding that vA=vM+mB/(mA+mB)vr and that vB=vM-mA/(mA+mB)vr): € 1 2 m A v A 2 + 1 2 m B v B 2 = 1 2 µ v r 2 + 1 2 m A + m B ( ) v r 2 (3.35a) € d v A d v B = d v r d v M (3.35b) where µ is the reduced mass and it is defined as 1/µ = 1/mA + 1/mB. Dividing 3.33 by ∆t, substituting 3.34 and 3.35 and solving between 0 and ∞ we get: € Z = dN coll dt = v r σ ∫ m A 2 π k b T % & ' ( ) * 3 / 2 exp − m B v B 2 2 k b T % & ' ( ) * m B 2 π k b T % & ' ( ) * 3 / 2 exp − m B v B 2 2 k b T % & ' ( ) * d v r d v M ∫ ∫ N A V N B V (3.36) Remembering that dv = 4πv2dv and solving in vM using the formulas in table 3.1, we get: € Z = v r σ ∫ µ 2 π k b T % & ' ( ) * 3 / 2 exp − µ v r 2 2 k b T % & ' ( ) * d v r N A V N B V (3.37) Equation 3.37 allows calculating Z once the functional dependency of the cross section from the relative velocity vr is assigned, and thus it represents its more general formulation. Often, equation 3.37 is reported as a function of the relative kinetic energy of the incident molecules. Being E = ½ µvr2 we get: € Z = 1 k b T 8 π k b T µ # $ % & ' ( 1 / 2 E σ ( E ) ∫ exp − E k b T # $ % & ' ( d E N A V N B V (3.38) Equation 3.37 and 3.38 can be solved once it is known the functional dependence of σ from vr or from E. Among the possible solutions, two deserve a particular attention for their specific applications: the case in which σ has constant values, and that in which σ is a function of an activation energy that has to be overcome to make the reaction happen. Situation 1: σ=constant If σ is independent from the relative velocity of A and B, i.e. when it is not necessary to overcome an activation energy for the reaction to occur, the solution of 3.37 is: € Z = σ 8 k b T π µ $ % & ' ( ) 1 / 2 N A V N B V = σ v N A V N B V = k coll N A V N B V (3.39) In 3.39 the term that is multiplied by the concentration of the reactants is called collisional constant, and it is equal to the product of the cross section times the average velocity calculated using the reduced mass. Situation 2: σ = σ(E) If the reaction is activated, it is necessary to determine a functional expression, that is coherent with Newtonian dynamics of collisions between particles with different masses, and whose interaction energy is function of both the reciprocal distance and the relative velocity of approach. To determine the functional law of interaction, the collisional process can be schematized as reported in Figure 3.7. r ϕ A B x vr b Figure 3.7. Schematization of the collision process between a molecule A and a molecule B. To study the reactive process, it can be used a reference system centered in B, and that is moving with a velocity vr parallel to the x axis reported in figure. We define as r the distance between the centers of mass of A and B at the instant t, and as ϕ the angle formed by the vector r with the x axis. In such conditions, the law of conservation of energy of the system A-B at the generic instant t is: € 1 2 m dr dt " # $ % & ' 2 + p ϕ 2 2 mr 2 + U r ( ) = 1 2 mv 0 2 (3.40) Equation 3.40 takes explicitly into account that, after the collision, the molecule A will change its own trajectory and thus it will gain a component of kinetic energy associated with the angular rotation. The collisional process between A and B must also satisfy the law of conservation of angular momentum, thus: € mv 0 b = p ϕ = cos t (3.41) Defining as distance of greatest approach rm the distance at which the velocity of variation of the distance r assumes the minimum value, the following condition holds for rm: € dr dt " # $ % & ' rm = 0 (3.42) thus in correspondence of rm we get: € p ϕ 2 2 mr m 2 + U r m ( ) = 1 2 mv 0 2 (3.43) Substituting 3.41 into 3.43 we get: € b 2 r m 2 = 1 − U r m ( ) 1 2 mv 0 2 (3.44) Defining rm as the minimum distance at which the reaction occurs, and U(rm) as the energy needed to make the reaction happen, we can determine the minimum distance b at which two molecules with relative energy Er must be so that they can bring their centers at the distance rm that is necessary to make the reaction happen. The section πb2 assumes then the meaning of reactive cross section for the system A-B, and is expressed by: € π b 2 = σ R = π r m 2 1 − E a E r % & ' ( ) * 1 / 2 (3.45) Where Ea represents the repulsive intermolecular energy reached at distance rm that must be overcome for the system to react. Substituting 3.45 into 3.38 it is possible to determine the total number of collisions per unit time and unit volume for an energetically activated system. The solution of the resulting equation is: € Z = π r m 2 8 k b T π µ # $ % & ' ( 1 / 2 exp − E a / k b T ( ) N A V N B V = σ v exp − E a / k b T ( ) N A V N B V (3.46) Expression 3.46 is formally identical to 3.39, from which it differs only for the exponential term, which gives a dependency of the kinetic constant from the temperature analogous to what predicted from the Arrhenius law. Equations 3.46 and 3.39 give expressions of the rate of an elementary bimolecular reaction that are consistent with the Arrhenius law, and predict a direct proportionality with respect to the concentration of the reagents that is in full agreement with experimental observations, of which they give a physical justification. However, the systematic application of these relations to estimate kinetic constants of bimolecular reactions has been marked by systematic failures, except for some particular cases. The reason of this failure is due in part to the difficulty of estimating accurately and systematically the reactive cross section for a generic reaction, but mainly to the fact that the kinetic theory of gases does not consider in any way the influence of the intramolecular motions on the kinetics of the reaction. The removal of such simplification implies a great complication of the problem, thus it will be necessary to use a level of theory that is significantly higher to adequately describe the dynamics of a reagent system. However, the kinetic theory of gases gives a simple and effective method to estimate in a first approximation the kinetic constant of non activated bimolecular reactions, and of their inverse processes too. 3.5 Non activated bimolecular and unimolecular reactions A non activated bimolecular reaction usually has a kinetic constant whose value is substantially independent, or weakly dependent, on temperature. For this class of reactions is usually a good approximation to use equations 3.39 to estimate the kinetic constant. In this case it is said that the kinetic constant is assumed equal to its collisional limit. Application of 3.39 requires performing an estimation of the reactive cross section for the collisional process. This can be performed in two ways: using directly as parameter σ the molecular diameter given by the Lennard-Jones potential, or starting from geometrical considerations on the structure of the molecule. We will use this second approach to estimate the kinetic constant of the recombination reaction between a methyl (CH3) and atomic hydrogen (H) to give methane (CH4). To estimate the molecular diameter of methyl and hydrogen, suppose to inscribe them inside a sphere. To take into account the influence of electrostatic interactions, we increase the interaction radius of hydrogen atoms by a quantity equal to the Bohr radius (0.52 Å). The procedure that has been followed is schematized in Figure 3.8. C H H H H 0.52 1.09 0.52 120 dCH3 = 3.32 Å dH = 1.04 Å Figure 3.8. Estimation of molecular diameters starting from structural data. Once known the molecular diameters of CH3 and H, it is possible to calculate the mean collisional diameter (which is in fact a radius) as: dm = (dA + dB)/2 = 2.19 Å the collisional cross section: σ = πdm2=15.06 10-20 m2 the reduced mass µ: µ = mAmB/(mA+mB)=1*15/(1+15)*10-3/6.023*1023 kg = 1.55 10-27 kg and finally the collisional constant (at 300K): kcoll = σ (8kbT/πµ)0.5 = 3.9 10-16 m3/s/molec = 2.4 108 m3/s/mol = 2.4 1014 cm3/s/mol It is now interesting to compare the resulting collisional constant with the experimental data, kexp = 1.2 1015 T-0.4 cm3/s/mol, that at 300K is equal to 1.2 1014 cm3/s/mol. The agreement between the calculation and the experimental data is very good, in particular if we consider the numerous arbitrary decisions that have been assumed to estimate the collisional diameter. The approach we adopted can be used in an effective way to calculate the kinetic constant of the inverse process, i.e. the dissociation reaction of methane to give methyl and H. To estimate this constant we can use the principle of thermodynamic consistency, according to which the direct and inverse constants of a chemical reaction are bound between them by the equilibrium constant, as seen in detail in chapter 1. For a generic process: A B + AB kd ki the direct constant kD and the inverse one kI are bound by the relation: € k I = k D P 0 RT exp Δ G 0 ( T , P 0 ) RT # $ % & ' ( = σ v P 0 RT exp − Δ S 0 ( T , P 0 ) R # $ % & ' ( exp Δ H 0 ( T , P 0 ) RT # $ % & ' ( (3.47) Where P0 is the reference pressure for the calculation of the ∆G0 of the reaction. The first 4 terms of expression 3.47 allow estimating the pre-exponential factor, while the last term represents the activation energy of the process, that for non activated unimolecular reactions is usually very similar to the variation of enthalpy of the reaction, and is pften left expressed as a function of temperature. In the case of methane, at 300 K the ∆H0 is -104.8 kcal/mol, and the ∆S0 is -29.3 cal/mol/K. The kinetic constant calculated for the decomposition reaction of CH4 is: kI = 2.4 1014 101325/8.314/300 10-6 exp(29.3/1.987)exp(-104.8/RT) = 2.51016exp(-104.8/RT) s-1. The resulting constant is very similar, both in value and activation energy, to the experimental data: kdecCH4= 2.41016 exp(104.8/RT) s-1 Notwithstanding the good agreement with the experimental data found in the estimation of the kinetic constant of barrierless reactions using the kinetic theory of gases, this theory should be used with caution, remembering that it allows estimating in a rigorous way only a superior limit of the kinetic constant, and that intramolecular phenomena can significantly affect the rate of the reactive process. 3.6 Collisional frequency and mean free path Besides function Z, that defines the total number of collisions that occur in a gas per unit time and unit volume, two additional quantities exist that strongly characterize the reactive properties of a gas. They are the collision frequency ν of a molecule (number of collisions per second, s-1) and its mean free path λ (distance covered between a collision and the next one, m). The frequency of collisions is obtained by dividing Z by the density of particles of which it is wanted to calculate the collisional frequency. For particles A dispersed in a gas B, we get: € ν = Z /( N A / V ) = σ v N A V N B V / N A V = σ v N B V = σ v P B RT (3.48) where with PB we mean the partial pressure of B. The mean free path can instead be estimated by dividing the average velocity of the gas particles by the frequency of collision, and is thus given by: € λ = v ν = 1 σ P B RT (3.49) It is left to the reader as exercise the estimation of Z, ν, and λ for a gas made by N2 at 300 K. 3.7 Collisional constant with a surface The kinetic theory of gases can also be applied to estimate the kinetic constant of the collisional process between a gas molecule and a surface. The flux of molecules that collide with a surface per unit time and unit surface area can be in fact be calculated integrating 3.4, using the distribution function 3.26 and the integrals of Table 3.1: € Z surf = 1 2 n v x f v x ( ) dv x −∞ + ∞ ∫ = n k b T 2 π m = 1 4 v n = k coll surf n (3.50) where n represents the density of the gas molecules. The surface collisional constant has the dimensions of a velocity (m/s) and is in fact equal to one fourth of the average velocity of the gas molecules. The surface collisional constant can be used to estimate the rate of adsorption reactions of chemical species on a surface. In such form it is usually multiplied by the fraction of free sites present on the surface, and by a multiplicative coefficient γ known as sticking coefficient, so that rads is expressed as: r a d s = g k c o l l s u r f n q s u r f = k a d s n q s u r f (3.51) Where the adsorption constant is expressed as the product of the collisional constant and of the sticking coefficient. Usually, the sticking coefficient, adimensional, is determined experimentally and is expressed in the Arrhenius form: € γ = A exp( − Ea / RT ) (3.52) The activation energies of adsorption processes are often very low (3-5 kcal/mol).