Principles of Chemical Kinetics

Transcription

1 Kinetic Theory of Gases Rates of Chemical Reactions Theories of Chemical Reactions Summary Principles of Chemical Kinetics Ramon Xulvi-Brunet Escuela Politécnica Nacional

2 Kinetic Theory of Gases Rates of Chemical Reactions Theories of Chemical Reactions Summary Outline 1 Kinetic Theory of Gases Statistical mechanics of gases The Maxwell-Boltzmann distribution of speeds Molecular collisions 2 Rates of Chemical Reactions Differential and integrated laws Reactions mechanisms Important types of chemical reactions 3 Theories of Chemical Reactions Collision theory Activated complex theory

3 Kinetic Theory of Gases Rates of Chemical Reactions Theories of Chemical Statistical Reactions mechanics Summary of gases Chemical reactions and statistical mechanics The Maxwell-Boltzmann distribution of In other to reactions to happen molecules need to come close to be able to interact. Per unit time, the probability of interaction depends on the probability of molecules to come close. That probability is computed by means of the statistical mechanics. The simplest statistical models that allow the calculation of that probability are based on the kinetic theory of gases.

4 Kinetic Theory of Gases Rates of Chemical Reactions Theories of Chemical Statistical Reactions mechanics Summary of gases Presure of an ideal gas The Maxwell-Boltzmann distribution of Let v x be the velocity component in the x direction of molecules having a mass m. All molecules located in the volume v x t A, and moving to the wall of area A, will strike the wall. If n is the number of molecules per unit volume, the number of molecules that will strike the wall in the time t will be 1/2n Av x t.

5 Kinetic Theory of Gases Rates of Chemical Reactions Theories of Chemical Statistical Reactions mechanics Summary of gases Presure of an ideal gas The Maxwell-Boltzmann distribution of The force on the wall due to the colision of a molecule is F = d(mv)/dt, which yields F t = (mv). If molecules rebound elastically, (mv) = 2mv x. Therefore: F t = 2mv x. Consequently, for the total number of molecular colisions, F t = (1/2n Av x t)(2mv x ) p = F/A = n mv 2 x. If the motion of the gas is isotropic, then v 2 x = v 2 y = v 2 z = v 2 /3. therefore p = 1 3 n m v 2 Note that presure is proportional to v 2.

6 Kinetic Theory of Gases Rates of Chemical Reactions Theories of Chemical Statistical Reactions mechanics Summary of gases Temperature and energy of an ideal gas The Maxwell-Boltzmann distribution of Using ɛ = m v 2 /2 and n = nn A /V, where N A is the Avogrado s number and n is the number of moles, the above equation for presure can be written as pv = 2nN A ɛ /3. Comparing this result with the empirical ideal gas law pv = nrt, and defining k as Boltzmann s constant k = R/N A, it is easy to conclude: T = pv nr = 2 N A 3 R ɛ ɛ = 3 2 kt v2 = 3 kt m

7 Kinetic Theory of Gases Rates of Chemical Reactions Theories of Chemical Statistical Reactions mechanics Summary of gases Fundamental hypotheses The Maxwell-Boltzmann distribution of The distribution of molecular speeds we want to compute is F(v x, v y, v z )dv x dv y dv z, which gives us the probability of finding the x, y, z components of speed between v x and v x + dv x, v y and v y + dv y, v z and v z + dv z, respectively. 1 In each direction, the velocity distribution is an even funcion of v. 2 In any particular direction, the velocity distribution is independent and uncorrelated with the distributions in orthogonal directions. 3 The v 2 obtained from the distribution function should agree with v 2 = 3kT/m. 4 The three dimensional velocity distribution depends only on the speed v and not on the direction.

8 Kinetic Theory of Gases Rates of Chemical Reactions Theories of Chemical Statistical Reactions mechanics Summary of gases Derivation of the distribution The Maxwell-Boltzmann distribution of hypothesis 1 The number of molecules moving in a given direction must be equal to the number of molecules moving in the oposite direction. (Otherwise, the presure on one side of the volume would be greater than on the other). Therefore, the distribution function should satisfy F(v x ) = F( v x ), F(v y ) = F( v y ), F(v z ) = F( v z ). one posibility is... F(v x ) = f (v 2 x) F(v y ) = f (v 2 y) F(v z ) = f (v 2 z )

9 Kinetic Theory of Gases Rates of Chemical Reactions Theories of Chemical Statistical Reactions mechanics Summary of gases Derivation of the distribution The Maxwell-Boltzmann distribution of hypothesis 2 There should be no relationship between x axis, y axis, and z axis velocities. The three components of velocity should be independent. F(v x, v y, v z ) = F(v x )F(v y )F(v z ). From hypothesis 1, F(v x, v y, v z ) = f (v 2 x)f (v 2 y)f (v 2 z ). The only function that has the property f (a + b + c) = f (a) f (b) f (c) is the exponential function. ( κ ) 1/2 F(v x ) = f (v 2 x) = exp (±κv 2 π x ) ( κ ) 1/2 F(v y ) = f (v 2 y) = exp (±κv 2 π y ) ( κ ) 1/2 F(v z ) = f (v 2 z ) = exp (±κv 2 π z )

10 Kinetic Theory of Gases Rates of Chemical Reactions Theories of Chemical Statistical Reactions mechanics Summary of gases Derivation of the distribution hypothesis 3 v 2 should agree with the ideal gas law: v 2 = 3kT/m. v 2 x = v 2 xf(v x )dv x = 1 2κ From v 2 = v 2 x + v 2 y + v 2 z = 3 v 2 x v 2 = 3/2κ. The Maxwell-Boltzmann distribution of 3 2κ = 3kT m κ = m 2kT F(v x, v y, v z ) dv x dv y dv z = F(v x )F(v y )F(v z ) dv x dv y dv z = ( m ) 3/2 mv 2 = exp ( 2πkT 2kT ) dv xdv y dv z

11 Kinetic Theory of Gases Rates of Chemical Reactions Theories of Chemical Statistical Reactions mechanics Summary of gases Derivation of the distribution The Maxwell-Boltzmann distribution of hypothesis 4 From hypothesis 3, F(v x, v y, v z ) depends on v 2, not on the directional property of v. That is, it depends on the speed, not on the velocity. The probability that the speed is between v and v + dv is equal to the probability that velocity vectors terminate within the volume of a spherical shell between the radius v and the radius v + dv. Thus, dv x dv y dv z = 4πv 2 dv. Maxwell-Boltzmann distribution: ( F(v)dv = 4πv 2 m 2πkT ) 3/2 exp ( mv 2 2kT )dv

12 Kinetic Theory of Gases Rates of Chemical Reactions Theories of Chemical Statistical Reactions mechanics Summary of gases Energy distribution The Maxwell-Boltzmann distribution of Note that our previous result is consistent with the Boltzmann distribution law, which states that the probability of finding a system with energy ɛ is proportional to exp ( ɛ/kt). This fact validates our choice F(v x ) = f (v 2 x). Taking into account that ɛ = mv 2 /2 and dɛ = mvdv = 2mɛdv, we can convert velocies to energies. Energy distribution: G(ɛ)dɛ = 4π ( ) 2ɛ ( m ) 3/2 ( exp ɛ ) dɛ = m 2πkT kt 2mɛ ( ) 1 3/2 ( = 2π ɛ exp ɛ ) dɛ πkt kt

13 Kinetic Theory of Gases Rates of Chemical Reactions Theories of Chemical Statistical Reactions mechanics Summary of gases Molecular collisions. The Maxwell-Boltzmann distribution of Since two molecules must have a close encounter to react, the molecular collision rate provides an upper limit to the reaction rate. The molecular collision rate between two species depends on: 1 The average relative speed of the molecules of the different species, v r. 2 The size of the molecules of each species. 3 The number of possible collision pairs.

14 Kinetic Theory of Gases Rates of Chemical Reactions Theories of Chemical Statistical Reactions mechanics Summary of gases Molecular collisions The Maxwell-Boltzmann distribution of The number of collisions one molecule of type 1 will experience with molecules of type 2 per unit time is Z 2 = πb 2 max v r n 2 Of course, for a molecule of type 1 moving through other molecules of the same type, Z 1 = πb 2 max v r n 1 = πd 2 v r n 1

15 Kinetic Theory of Gases Rates of Chemical Reactions Theories of Chemical Statistical Reactions mechanics Summary of gases Molecular collisions The Maxwell-Boltzmann distribution of The total number of collisions of molecules of type 1 with those of type 2 per unit time and per unit volume is simply: Z 12 = Z 2 n 1 = πb 2 max v r n 1n 2 By a similiar argument, if there were only one type of molecule, the total number of collisions per unit time and per unit volume is: Z 11 = 1 2 Z 1n 1 = 1 2 πb2 max v r (n 1) 2 The factor 1/2 appears because, if there are n molecules, then the number of pairs is n(n 1)/2, which tends to n 2 /2 as n 1.

16 Kinetic Theory of Gases Rates of Chemical Reactions Theories of Chemical Statistical Reactions mechanics Summary of gases Molecular collisions The Maxwell-Boltzmann distribution of How do we compute v r? Let us consider the average over all collision angles. The representative of the average angle of collision occurs when the molecules are traveling at right angles to one another. In this case (assuming equal masses for the particules), the relative velocity will have a magnitude of v r = 2 v. On the other hand v = v r = 2 v = 0 vf(v)dv = ( ) 8kT 1/2 = π(m/2) The last term is also the general solution. ( ) 8kT 1/2 πm ( ) 8kT 1/2 πµ

17 Kinetic Theory of Gases Rates of Chemical Reactions Theories of Chemical Differential Reactions andsummary integrated laws Reactions mechanisms Important typ Introduction Chemical reactions are described both at a macroscopic level and a microcospic level. At macroscopic level, the goal is to obtain an empirical description of the rates of the chemical reactions. Experiments tell us that these rates depend on: 1 the temperature, presure, and volume of the reaction vessel 2 the concentrations of the reactants and products 3 whether or not a catalyst is present At microcospic level, the goal is to understand the physics and dynamics of the reaction. Theories of chemical reactions attempt to obtain the macroscopic reaction rates and relate them with underlying mechanisms that govern the reaction at microscopic level.

18 Kinetic Theory of Gases Rates of Chemical Reactions Theories of Chemical Differential Reactions andsummary integrated laws Reactions mechanisms Important typ Differential laws aa + bb cc + dd Since 1 d[c] = 1 d[d] = 1 d[a] = 1 d[b], c dt d dt a dt b dt the rate of a reaction is defined as the rate of change of the concentration of a product or reactant over its stoichiometric number. The rate law for a reaction can therefore be written as, for instance, 1 d[c] = f ([A(t)], [B(t)], [C(t)], [D(t)]). c dt

19 Kinetic Theory of Gases Rates of Chemical Reactions Theories of Chemical Differential Reactions andsummary integrated laws Reactions mechanisms Important typ Differential laws In general, the function f ([A], [B], [C], [D]) might be a complicated function of the concentrations. However, it often occurs that f can be expressed as a simple product of a rate constant k, and the concentrations each raised to some power: 1 d[c] = k[a] m [B] n [C] o [D] p c dt When the rate law can be written in this simple way, the overall order of the reaction is defined as q = m + n + o + p the order respect to a particular species is defined as the power to which its concentration is raised.

20 Kinetic Theory of Gases Rates of Chemical Reactions Theories of Chemical Differential Reactions andsummary integrated laws Reactions mechanisms Important typ Differential laws important notes 1 d[c] = k[a] m [B] n [C] o [D] p c dt The left-hand side of the above equation has units of concentration per time, the rate constant k will then have units of time 1 concentration q. The overall order of a reaction cannot be obtained simply by looking at the overall stoichiometric numbers. It is determined experimentally. Example: however, its reaction rate is H 2 + Br 2 2HBr 1 d[hbr] = k[h 2 ][Br 2 ] 1/2 2 dt

21 Kinetic Theory of Gases Rates of Chemical Reactions Theories of Chemical Differential Reactions andsummary integrated laws Reactions mechanisms Important typ Integrated laws The rate law in its differential form describes how the rate of the reaction depends on the concentrations. integration It is often useful to determine how the concentrations themselves vary in time. This variation in time is computed by direct integration of the differential rate law. To do the integration, one needs to know the order of the reation. Only for simple reaction orders is the integration straitforward.

22 Kinetic Theory of Gases Rates of Chemical Reactions Theories of Chemical Differential Reactions andsummary integrated laws Reactions mechanisms Important typ Examples of integrated laws First-order reactions. A products d[a] dt = k[a] [A(t)] = [A(0)] exp ( kt) Second-order reaction. 2A products d[a] dt = k[a] 2 1 [A(t)] 1 [A(0)] = kt Second-order reaction. A + B products d[a] dt = k[a][b] ln [B(t)][A(0)] = ([B(0)] [A(0)])kt [B(0)][A(t)]

23 Kinetic Theory of Gases Rates of Chemical Reactions Theories of Chemical Differential Reactions andsummary integrated laws Reactions mechanisms Important typ Elementary steps postulate The macroscopic rate law is the consequence of a mechanism consisting of elementary steps, each one of which describes a process that takes place on the microscopic level. Three types of microscopic processes account for essentially all reaction mechanisms: unimolecular process: reaction of a single (energized) molecule. A products. Since the number of product molecules produced per unit time is proportional to the number of A molecules, then d[products] dt = d[a] dt = k[a]

24 Kinetic Theory of Gases Rates of Chemical Reactions Theories of Chemical Differential Reactions andsummary integrated laws Reactions mechanisms Important typ Elementary steps postulate Three types of microscopic processes account for essentially all reaction mechanisms: bimolecular process: collision between two molecular species. A + B products. Since the number of products per unit time is proportional to the number of collisions between A and B, then d[products] dt = k[a][b] termolecular process: collision between three molecular species. A + B + C products. Since the number of products per unit time is proportional to the number of collisions between A, B, and C, then d[products] = k[a][b][c] dt

25 Kinetic Theory of Gases Rates of Chemical Reactions Theories of Chemical Differential Reactions andsummary integrated laws Reactions mechanisms Important typ Elementary steps postulate Therefore, unimolecular processes follow first-order kinetics, biomolecular processes follow second-order kinetics, and termolecular processes follow third-order kinetics. The order of an elementary process is thus equal to its molecularity. This statement is only true for elementary processes, not necessarily for an overall process consisting of several elementary steps.

26 Kinetic Theory of Gases Rates of Chemical Reactions Theories of Chemical Differential Reactions andsummary integrated laws Reactions mechanisms Important typ Example of chemical reaction as combination of elementary reactions H 2 + Br 2 2HBr It has been shown that this reaction is what it is usually called a chain mechanism, which in this case consists of collisional production of Br atoms (fast): Br 2 + M 2Br + M Br + H 2 HBr + H H + HBr Br + H 2 H + Br 2 HBr + Br (slow reaction) (slow reaction) (fast reaction)

27 Kinetic Theory of Gases Rates of Chemical Reactions Theories of Chemical Differential Reactions andsummary integrated laws Reactions mechanisms Important typ Unimolecular decomposition (A products): The Lindemann mechanism How does a A molecule obtain enough energy to decompose? Lindemann suggested A + M A + M A products where M represents any molecule that can energize A by collision. (M might be A itself, or it might be a nonreactive molecule in which the reactant is mixed). The overall rate of the reaction is d[p]/dt = k 2 [A ]. The existence of the intermediate A complicates the quantitative solution substancially.

28 Kinetic Theory of Gases Rates of Chemical Reactions Theories of Chemical Differential Reactions andsummary integrated laws Reactions mechanisms Important typ Enzyme catalysis: The Michaelis-Menten mechanism Enzymes tend to speed up chemical reactions E + S X X products + E where X represents an intermediate and S is called the substrate.

29 Kinetic Theory of Gases Rates of Chemical Reactions Theories of Chemical Collision Reactions theorysummary Activated complex theory Collision theory remember The rate of molecular collisions provides an upper limit to the reaction rate, since two molecules have to come close to react. Z 12 = Z 2 n 1 = πb 2 max v r n 1n 2 however Not all collisions will lead to reaction. It is supposed that reactions take place if the relative energy of the collision (corresponding to a relative velocity v r ) is sufficiently large, larger than a particular value ɛ. πb 2 max is the cross section for hard-sphere collisions. From the above discussion, however, the cross section should depend on the relative energy. Let us denote it as σ(ɛ r ).

30 Kinetic Theory of Gases Rates of Chemical Reactions Theories of Chemical Collision Reactions theorysummary Activated complex theory Collision theory therefore The number of collisions between molecules of type 1 and type 2 can be written as Z 12 = k(ɛ r ) n 1n 2 where k(ɛ r ) = σ(ɛ r )v r is the rate constant for the reaction at ɛ r = µv 2 r /2. From v r = 2ɛ r /µ and ( ) 1 3/2 ( G(ɛ r )dɛ r = 2π ɛr exp ɛ ) r dɛ r, πkt kt we can compute k(t) = k(ɛ r ) = σ(ɛ r )v r G(ɛ r )dɛ r =

31 Kinetic Theory of Gases Rates of Chemical Reactions Theories of Chemical Collision Reactions theorysummary Activated complex theory Collision theory result k(t) = A ( ) 8kT 1/2 exp ( ɛ πµ kt ) remarks Although this expression accounts only for the relative translational energy of the reactants, it is possible to arrive to the same expression when the internal vibrational energy is also considered. This expression is very similar to the Arrhenius form k = A exp ( E a kt ) Reactants have favorable orientations for reaction. ( ) 8kT 1/2 k(t) = pa exp ( ɛ πµ kt )

32 Kinetic Theory of Gases Rates of Chemical Reactions Theories of Chemical Collision Reactions theorysummary Activated complex theory Activated complex theory result A theory that improves on the simple collision model by incorporating into the theory the partitions functions that describe the number of states accessible at the temperature of interest. It starts by modeling a chemical reaction by A + B AB products k(t) = kt h q q a q b exp ( ɛ kt ) where q is the partition function per unit volume for the activated complex, and q a and q b are the partition functions per unit volume for the reactants.

33 Kinetic Theory of Gases Rates of Chemical Reactions Theories of Chemical Reactions Summary Summary Statistical mechanics is the basic toolbox that is used to model chemical kinetics. Depending on the hypothesis considered, different models might be constructed to model the same chemical system. Also, for chemical systems with different structural features, different statistical hypotheses will need to be considered. Consequently, different models might be constructed, depending on the systems. The main function of these models is to related the empirical, macroscopic description of the chemical reactions with the underlying mechanisms for the reactions on the microscopic level.

34 Kinetic Theory of Gases Rates of Chemical Reactions Theories of Chemical Reactions Summary Acknowledgment PROYECTO PROMETEO Secretaría de Educación Superior Ciencia, Tecnología, e Innovación de la República del Ecuador Escuela Politécnica Nacional Proyecto Prometeo