Reaction Dynamics (2) Can we predict the rate of reactions?

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1 Reaction Dynamics (2) Can we predict the rate of reactions?

2 Reactions in Liquid Solutions Solvent is NOT a reactant

3 Reactive encounters in solution A reaction occurs if 1. The reactant molecules (A, B) meet 2. The molecules (A, B) have a certain minimum energy But 3. Reactants are surrounded by solvent molecules 4. Reactants migrate very slowly compared to gas phase 5. Solvent environment has to be included in the analysis: It exchanges energy (provides/removes) with the reactants It determines the migration speed of the reactants

4 Cage effect The solvent tends to slow the rate of approach of the reactants; they must diffuse toward one another through the solution But, it also keeps them together for many collisions once they come in contact. This phenomenon is known as the cage effect. Ex: Geminate recombination

5 Reactive encounters in solution (1) Formation of an encounter pair AB A + B AB υ = k d [A][B] where k d is the rate constant of diffusion The encounter pair AB can break up without reaction AB A+B υ = k d [AB] or it can go on to from products P AB P υ = k a [AB]

6 Reactive encounters in solution (2) d[ab] dt Assuming that [AB] is constant = k d [A][B] k d [AB] k a [AB] 0 therefore, [AB] = k d[a][b] k d + k a The rate of formation of products is d[p] dt k a [AB] = k 2 [A][B] and k 2 = k a[ab] [A][B] = k ak d k a + k d

7 Activation-controlled reaction If k a k d then k 2 k ak d k d = k a K where K is the equilibrium constant for A + B AB In this limit, the reaction proceeds at the rate at which energy accumulates in the encounter pair from the surrounding solvent. The activation energy is high.

8 Diffusion-controlled reaction (1) If k d k a then k 2 k ak d k a = k d The rate of reaction is governed by the rate at which the reactant molecules diffuse through the solvent. The activation energy is low. An indication that a reaction is diffusion-controlled is that its rate constant is of the order of 10 9 L mol -1 s -1 or greater.

9 Diffusion-controlled reaction (2) The rate constant of diffusion, k d, can be estimated by relating the diffusion constant with the temperature (T) and the viscosity (η) of the solvent k d = 8RT 3η Can we predict the rate of reaction, k 2, in this case? Yes, if we could know a priori which reactions are going to be diffusion-controlled!

10 Activated Complex Theory a.k.a. Transition State Theory Henry Eyring

11 Some important definitions (1) Elementary reaction. One step reaction. Reaction coordinate. Collection of motions, such as changes in interatomic distances and bond angles, that are directly involved in the formation of products from reactants. The reaction coordinate is typically chosen to follow the path of shallowest ascent/deepest descent of potential energy from reactants to products. Potential-energy profile. A curve describing the variation of the potential energy of the system of atoms that make up the reactants and products of a reaction as a function of one geometric coordinate, and corresponding to the energetically easiest passage from reactants to products ( ). For an elementary reaction the relevant geometric coordinate is the reaction coordinate ( ) (The reaction coordinate is sometimes approximated by a quasi-chemical index of reaction progress, such as degree of atom transfer or bond order of some specified bond.) Taken from IUPAC Compendium of Chemical Terminology

12 Some important definitions (2) Transition state ( ) In the formalism of transition state theory the transition state of an elementary reaction is that set of states (each characterized by its own geometry and energy) in which an assembly of atoms, when randomly placed there, would have an equal probability of forming the reactants or of forming the products of that elementary reaction. ( ) The assembly of atoms at the transition state has been called an activated complex. (It is not a complex according to the definition in this Compendium.) ( ) An activated complex, often characterized by the superscript, is defined as that assembly of atoms which corresponds to an arbitrary infinitesimally small (δ) region at or near the col (saddle point) of a potential energy surface. Taken from IUPAC Compendium of Chemical Terminology

13 Potential energy profile A + B AB P δ AB Transition state Activated complex E a, activation energy of the reaction, accounts for the change in potential energy associated with formation of the activated complex

14 Derivation of the Eyring Equation (1) Let s consider the gas-phase, elementary bimolecular reaction A + B P Reaction rate d[p] dt A + B AB P = k 2 [A][B] Assumptions: 1. The reaction between A and B proceeds through the formation of an activated complex, AB. 2. The activated complex and the reactants are in equilibrium with each other and we can model the reaction as a two-step process K (1) where AB falls apart by unimolecular decay into products.

15 Derivation of the Eyring Equation (2) The equilibrium constant expression (in terms of equilibrium concentrations) between the reactants and the activated complex is K c = [AB] /c [A]/c [B]/c = [AB] c [A][B] (3) c K c where is the standard-state concentration (often taken to be 1.00 mol L -1 )., as all equilibrium constants, is unitless.

Derivation of the Eyring Equation (3) The activated complex is assumed to be stable throughout a small region of width δ centered at the barrier top. The two step process given by eq.

16 Derivation of the Eyring Equation (3) The activated complex is assumed to be stable throughout a small region of width δ centered at the barrier top. The two step process given by eq. (2) predicts that the rate of the reaction will be the product of the concentration [AB ] of activated complex and ν c, the frequency with which these complexes cross over the barrier top, that is d[p] dt = ν c [AB] (3) combining equations (1), (2) and (3) we obtain d[p] dt AB = k 2 [A][B] = ν c [AB] = ν c [A][B]K c c or k 2 = ν c K c c (4)

17 Boltzmann Factor & Partition functions If a system has states with energies E 1, E 2, E 3,, the probability p j that the system will be in the state with energy E j depends exponentially on the energy of the state, that is p j (N, V, T ) e E j/k B T Boltzmann Factor where N is the number of particles, V is the volume and T is the temperature. The sum of the probabilities must equal 1, so the normalization constant for the above probability is 1/Q where Q(N, V, T )= j e E j/k B T Partition function Once it is known, a partition function can be used to calculate the macroscopic properties of the system: thermodynamic functions, heat capacities, entropies and equilibrium constants.

18 Derivation of the Eyring Equation (4) It can be shown that functions K c K c = q A q B q can be written in terms of partition (q /V )c (q A /V )(q B /V ) where the,, and are the partition functions of A, B, and, respectively. AB It is possible to write the partition function of the activated complex as q = q trans q int q trans (5) where is the partition function of the motion of the reacting system over the barrier top and q int accounts for all the other modes of the complex.

19 Derivation of the Eyring Equation (5) We can then rewrite K c as K c = q trans (q int /V )c (q A /V )(q B /V ) (6) The translational partition function corresponding to onedimensional translational motion is where m ( 2πm k B T ) 1/2 q trans = δ h is the mass of the activated complex. (7)

20 Derivation of the Eyring Equation (6) Combining equations (4), (6) and (7) we have k 2 = ν c c ( 2πm k B T ) 1/2 h δ (q int /V )c (q A /V )(q B /V ) (8) ν c δ and are not well defined and difficult to determine. Their product however can be equated to the average speed with which the activated complex crosses the barrier <u ac >= ν c δ

21 Derivation of the Eyring Equation (7) Using a (one-dimensional) Maxwell-Boltzmann distribution to calculate the average speed and replacing the result in (8), we obtain the transition-state theory expression for the rate constant k 2 = k BT (q int /V )c hc (q A /V )(q B /V ) = k BT hc K (9) K q Where is the equilibrium constant for the formation of the transition state from the reactants, but the motion along the reaction coordinate is excluded in int

22 Can we predict k 2 now? In very few cases! K To calculate we need to calculate the partition functions for the reactants and the activation complex. It is possible to calculate the partition functions for the reactants (eg: by using spectroscopic information about the energy levels), but it is very difficult to calculate the partition function of the activated complex (size, shape and structure have to be assumed). However, if it is possible to measure k 2 experimentally, we can obtain a lot of thermodynamic information about the reaction.

23 Thermodynamic interpretation (1) We define the standard Gibbs energy of activation,, to be the change in Gibbs energy in going from the reactants at a concentration c to the transition state at a concentration c. The relation between and is G K G therefore G RT G = RT ln K = ln K and e G /RT = K Combining equations (9) and (10) we have k 2 = k BT hc e G /RT (11) (10)

24 Thermodynamic interpretation (2) We can express the standard Gibbs energy of activation in terms of the the standard enthalpy of activation, and the standard entropy of activation G = H T S (12) Which if we combine with equation (11) gives k 2 = k BT hc e S /R e H /RT The formal definition of the activation energy is (13) E a = H +2RT (14)

25 Thermodynamic interpretation (3) Therefore if we combine equations (13) and (14) k 2 = e2 k B T hc e S /R e E a/rt A, pre-exponential factor Arrehnius equation! We have derived k 2 using an equilibrium constant expressed in concentration terms. If we had chosen instead to express the equilibrium constant in partial pressure terms, we would have obtained k 2 = e2 k B T h RT p e S /R e E a/rt where p is the standard state of one bar

26 Steric factor The entropy of activation is negative because two reactant species come together to form one species ( more order ). If the reduction in entropy is larger that expected for the simple encounter of A and B, this is an indication that the relative orientation of the reactants is important when forming the complex. It is possible to identify the additional reduction in entropy with the steric factor P in collision theory P = e S steric /R

Express the transition state equilibrium constant in terms of the partition functions of the transition state and the

Express the transition state equilibrium constant in terms of the partition functions of the transition state and the Module 7 : Theories of Reaction Rates Lecture 33 : Transition State Theory Objectives After studying this Lecture you will be able to do the following. Distinguish between collision theory and transition