SF Chemical Kinetics. Lecture 5. Microscopic theory of chemical reaction kinetics. Microscopic theories of chemical reaction kinetics. • A basic aim is to calculate the rate constant for a chemical reaction from first principles using fundamental physics. • Any microscopic level theory of chemical reaction kinetics must result in the derivation of an expression for the rate constant that is consistent with the empirical Arrhenius equation. • A microscopic model should furthermore provide a reasonable interpretation of the pre-exponential factor A and the activation energy EA in the Arrhenius equation. • We will examine two microscopic models for chemical reactions : – The collision theory. – The activated complex theory. • The main emphasis will be on gas phase bimolecular reactions since reactions in the gas phase are the most simple reaction types. 1 References for Microscopic Theory of Reaction Rates. • Effect of temperature on reaction rate. – Burrows et al Chemistry3, Section 8. 7, pp. 383-389. • Collision Theory/ Activated Complex Theory. – Burrows et al Chemistry3, Section 8.8, pp.390-395. – Atkins, de Paula, Physical Chemistry 9th edition, Chapter 22, Reaction Dynamics. Section. 22.1, pp.832-838. – Atkins, de Paula, Physical Chemistry 9th edition, Chapter 22, Section.22.4-22.5, pp. 843-850. Collision theory of bimolecular gas phase reactions. • We focus attention on gas phase reactions and assume that chemical reactivity is due to collisions between molecules. • The theoretical approach is based on the kinetic theory of gases. • Molecules are assumed to be hard structureless spheres. Hence the model neglects the discrete chemical structure of an individual molecule. This assumption is unrealistic. • We also assume that no interaction between molecules until contact. • Molecular spheres maintain size and shape on collision. Hence the centres cannot come closer than a distance δ given by the sum of the molecular radii. • The reaction rate will depend on two factors : • the number of collisions per unit time (the collision frequency) • the fraction of collisions having an energy greater than a certain threshold energy E*. 2 Simple collision theory : quantitative aspects. A(g) + B(g) ⎯⎯→k Products Molecular structure and Hard sphere reactants details of internal motion such as vibrations and rotations ignored. Two basic requirements dictate a collision event. • One must have an A,B encounter over a sufficiently short distance to allow reaction to occur. • Colliding molecules must have sufficient energy of the correct type to overcome the energy barrier for reaction. A threshold energy E* is required. Two basic quantities are evaluated using the Kinetic Theory of gases : the collision frequency and the fraction of collisions that activate molecules for reaction. To evaluate the collision frequency we need a mathematical way to define whether or not a collision occurs. A The collision cross section σ defines when a collision occurs. r δ A σ = πδ 2 = π r + r 2 ()A B rB Effective collision B diameter δ = rA + rB Area = σ The collision cross section for two molecules can be regarded to be the area within which the center of the projectile molecule A must enter around the target molecule B in order for a collision to occur. 3 Maxwell-Boltzmann velocity distribution function J.C. Maxwell • The velocity distribution curve 1831-1879 has a very characteristic shape. • A small fraction of molecules 3/ 2 2 move with very low speeds, a 2 ⎧ m ⎫ ⎡ mv ⎤ F(v) = 4π v ⎨ ⎬ exp⎢− ⎥ small fraction move with very high speeds, and the vast majority ⎩2π kBT ⎭ ⎣ 2kBT ⎦ of molecules move at intermediate speeds. F(v) • The bell shaped curve is called a Gaussian curve and the molecular speeds in an ideal gas sample are Gaussian distributed. • The shape of the Gaussian distribution curve changes as the temperature is raised. • The maximum of the curve shifts to higher speeds with increasing temperature, and the curve becomes broader as the temperature increases. • A greater proportion of the gas molecules have high speeds at high temperature than at low temperature. Gas molecules exhibit a spread or distribution v of speeds. The collision frequency is computed via the kinetic theory of gases. -3 -1 We define a collision number (units: m s ) ZAB. Z = π δ 2n n v Mean relative velocity AB A B r Units: m2s-1 n = number density of molecule j (units : m-3) j δ = rA + rB Mean relative velocity evaluated via kinetic theory. Average velocity of a gas molecule MB distribution of velocities enables us to statistically ∞ v = v F(v)dv estimate the spread of ∫0 molllecular velitilocities in a gas 3/ 2 2 2 ⎧ m ⎫ ⎡ mv ⎤ Some maths ! F(v) = 4π v ⎨ ⎬ exp⎢− ⎥ ⎩2π kBT ⎭ ⎣ 2kBT ⎦ 8k T v = B Mass of Maxwell-Boltzmann velocity π m molecule Distribution function 4 We now relate the average velocity to the mean relative velocity. If A and B are different molecules then 2 2 The average relative velocity is given by vr = vA + vB The expression across. 8k T v = B Hence the collision number 8kBT j v = π m j between unlike molecules r π μ can be evaluated. Reduced mass mAmB Z = π δ 2n n v μ = AB A B r mA + mB 1/ 2 For collisions between like molecules v = 2 v ⎧8k T ⎫ r Z = n n σ B The number of collisions per unit AB A B ⎨ ⎬ time between a single A ⎩ π μ ⎭ 1/ 2 molecule and other A ⎧8kBT ⎫ = ZnAnB Molecules. Z A = nAσ 2⎨ ⎬ ⎩π mA ⎭ Total number of collisions 1/ 2 between like molecules. Z n 2 ⎧8k T ⎫ Collision frequency Z = A A = n2 σ B We divide by 2 to ensure AA A ⎨ ⎬ 2 2 π mA factor That each A,A encounter ⎩ ⎭ Is not counted twice. E * Molecular collision is effective only if translational energy of reactants is greater than some threshold value. Fraction of molecules with kinetic energy ⎡ ε * ⎤ greater Than some F()ε > ε * = exp⎢− ⎥ minimum Threshold ⎣ kBT ⎦ value ε* 5 The simple collision theory expression for the reaction rate R between unlike molecules. dnA ⎡ ε * ⎤ R = − = ZnAnB exp⎢− ⎥ dt ⎣ kBT ⎦ 1/ 2 ⎧8k T ⎫ Z = σ ⎨ B ⎬ ⎩ π μ ⎭ The rate expression for a dc bimolecular reaction between A and B. R =−A =kc c dt AB We introduce molar variables EN**= Aε Hence the SCT rate expression. nnAB ccAB== NNAA dc⎡ E *⎤ RZNcc=−A =exp − Avogadro dn dc AAB ⎢ ⎥ AA= N dt⎣ RT ⎦ constant dtA dt The bimolecular rate constant for The rate constant for bimolecular collisions collisions between unlike molecules. between like molecules. 1/ 2 1/ 2 ⎧k T ⎫ ⎡ E *⎤ ⎧8k T ⎫ ⎡ E *⎤ B Collision k = N σ B exp − k = 2N Aσ ⎨ ⎬ exp⎢− ⎥ A ⎨ ⎬ ⎢ ⎥ Both of these ⎩π m ⎭ ⎣ RT ⎦ Frequency ⎩ π μ ⎭ ⎣ RT ⎦ expressions are factor ⎡ E *⎤ similar to the ⎡ E *⎤ = z exp − = z AA exp⎢− ⎥ AB ⎢ ⎥ Arrhenius equation. ⎣ RT ⎦ ⎣ RT ⎦ 6 We compare the results of SCT with the empirical Arrhenius eqn. in order to obtain an interpretation of the activation energy and pre-exponential factor. ⎡ E ⎤ k = A exp − A A,B encounters obs ⎢ RT ⎥ ⎣ ⎦ 1/ 2 1/ 2 ⎧8k T ⎫ ⎡ E *⎤ ⎧kBT ⎫ ⎡ E *⎤ B k = N Aσ ⎨ ⎬ exp − k = 2N Aσ ⎨ ⎬ exp⎢− ⎥ ⎢ ⎥ ⎩π m ⎭ ⎣ RT ⎦ ⎩ π μ ⎭ ⎣ RT ⎦ E * ⎡ E *⎤ ⎡ ⎤ = z exp − = z AA exp − AB ⎢ ⎥ ⎣⎢ RT ⎦⎥ ⎣ RT ⎦ A,A encounters Aobs = zAA = A'' T A Aobs = zAB = A' T A 8k 8k Pre-exponential A''= 2N σ B A'= N σ B •SCT predicts A π m A π μ factor that the pre-exponential factor should depend on temperature. SCT • The threshold energy and the d ln k E d ln k E *+ RT 2 activation energy can also be A = = 2 2 compared. dT RT dT RT • Activation energy exhibits Arrhenius RT EA = E *+ EA ≅ E * a weak T dependence. 2 SCT : a summary. • The major problem with SCT is that the threshold energy E* is very difficult to evaluate from first principles. • The predictions of the collision theory can be critically evaluated by comparing the experimental pre-exponential factor with that computed using SCT. • We define the steric factor P as the ratio between the experimental and calculated A factors. P = Aexp Acalc • We can incorporate P into the SCT expression for the rate constant. ⎡ E *⎤ • For many gas phase reactions k = PzAB exp⎢− ⎥ ⎣ RT ⎦ P is considerably less than unity. ⎡ E *⎤ • Typically SCT will predict that Acalc will be in k = PzAA exp⎢− ⎥ the region 1010-1011 Lmol-1s-1 regardless of ⎣ RT ⎦ the chemical nature of the reactants and products. • What has gone wrong? The SCT assumption of hard sphere collision neglects the important fact that molecules possess an internal structure. It also neglects the fact that the relative orientation of the colliding molecules will be important in determining whether a collision will lead to reaction. • We need a better theory that takes molecular structure into account. The activated complex theory does just that . 7 Transport Energy criterion Summary of SCT. property A,B encounters ⎡ E *⎤ k = Pz AB exp⎢− ⎥ 1/ 2 ⎣ RT ⎦ ⎧8kBT ⎫ ⎡ E *⎤ k = N Aσ ⎨ ⎬ exp⎢− ⎥ ⎩ π μ ⎭ ⎣ RT ⎦ ⎡ E *⎤ k = Pz AA exp⎢− ⎥ ⎡ E *⎤ ⎣ RT ⎦ = z AB exp − ⎣⎢ RT ⎦⎥ Steric factor (Orientation requirement) A,A encounters Weaknesses: • No way to compute P from molecular parameters 1/ 2 • No way to compute E* from first principles. ⎧k T ⎫ ⎡ E *⎤ B Theory not quantitat ive or predict ive. k = 2N Aσ ⎨ ⎬ exp⎢− ⎥ ⎩π m ⎭ ⎣ RT ⎦ Strengths: E * •Qualitatively consistent with observation ⎡ ⎤ (Arrhenius equation). = zAA exp⎢− ⎥ ⎣ RT ⎦ • Provides plausible connection between microscopic molecular properties and macroscopic reaction rates. • Provides useful guide to upper limits for rate constant k. Henry Eyring 1901-1981 Developed (in 1935) the Transition State Theory (TST) or Activated Complex Theory (ACT) of Chemical Kinetics.