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## Kinetics of Elementary Reactions

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Kinetics of Elementary Reactions

- A reaction is elementary if it takes place in a single irreducible act at the molecular level, just the way it is written in the stoichiometric equation.
- No intermediate between reactants and products can be detected (or visualized).
- The act of reaction is most often simple, where one bond is broken while another is formed.
- Although catalytic reactions are not elementary,
- they generally take place through a sequence of elementary steps
- their rate can, in principle, be predicted from a knowledge of the rates of the constituent elementary reactions.
- Therefore, before considering the overall kinetics of catalytic reactions, we must understand the dependence of elementary reactions on composition, temperature and pressure (volume).

Elementary Reactions

- Since an elementary reaction represents a molecular event, its equation may not be written arbitrarily, but the way it takes place.
- With this restriction, the molecularity of the reaction is identical to its stoichiometry.

Forward and Reverse Reactions

- The principle of microscopic reversibility suggests that a reaction and its reverse proceed by the same mechanism.
- Forward and reverse reactions must have the same intermediates and rate-determining transition states.
- Example: Alcohol Dehydration

Theories of Elementary Reaction Kinetics

- The rates of even the simplest reactions are very difficult to calculate from first principles. In engineering practice, you will rely on experimental data.
- While the basic science of reaction kinetics is not sufficiently developed for design purposes, existing models of reaction dynamics provide a means of understanding reaction phenomena, analyzing experimental data, and extrapolating knowledge to other systems.
- Atkins details three approaches to the calculation of rate constants:
- Collision Theory
- Transition State Theory
- Molecular Reaction Dynamics
- We will examine transition state theory.

Transition State Theory - Elementary Reactions

- Transition state theory is founded on the expectation that during the transition from initial reagents to final products, an activated complex of higher “energy” is formed.
- This transition state is not an intermediate, but a unique configuration of the system in transit from one state to another.
- Although this activated complex is inherently unstable, we often assume that it possesses thermodynamic properties (albeit ill-defined), and propose molecular structures.

Transition State of an SN2 Reaction

- You have seen the concept of a transition state in CHEM 245, where nucleophilic substitution reactions were introduced.
- In the example below, the alkoxide ion is the nucleophile (Lewis base) displaces iodide, the weaker base.
- The reaction is believed to be bimolecular, passing through a transition state as drawn below:
- Clearly this transition state is not a stable compound, and therefore is not a reaction intermediate, but an activated complex.

Potential Energy Surface for Hydrogen Exchange

- Owing to the complexity of potential energy calculations, one of the only systems to be analyzed is that of collinear hydrogen exchange.

Transition State Theory - Thermodynamic Formulation

- The Rate of an Elementary Step
- The number of elementary acts per unit time is determined the number of systems passing through the activated complex configuration.
- We express the elementary reaction as:
- At equilibrium, the activated complex Xy will be in equilibrium with the reactants and products, and the concentration can be calculated from thermodynamic principles.
- Where q is the reference concentration, usually 1 mole/litre.
- Transition state theory assumes that even when the system is not at equilibrium, activated complexes are at equilibrium with the reactants.

Transition State Theory - Thermodynamic Formulation

- Based on this assumption, the concentration of the activated complex is derived from a thermodynamic treatment:
- q = unit conc’n
- which, can be expressed in terms of the relative Gibbs energy of the activated complex,
- DGy represents the free energy of activation.
- The difference between the Gibbs energy of the activated complex, and the Gibbs energies of the reactants at the reference state
- This represents the free energy barrier to reaction that includes both potential energy (DH) and conformational restrictions (DS).

Transition State Theory - Thermodynamic Formulation

- The rate of the forward elementary reaction
- is expressed as:
- q = unit conc’n
- where n is the frequency of vibration of the activated complex in the mode that corresponds to decomposition into products.
- This is the frequency of the molecular vibration which leads the complex to dissociate into products C and D.
- For this diatomic reaction, statistical mechanics assigns
- :sec-1
- where kb = Boltzmann’s constant = 1.38066*10-23 J/K
- T = reaction temperature, K
- h = Planck’s constant = 6.6262*10-34 J s

Transition State Theory - Thermodynamic Formulation

- With a measure of the decomposition frequency, the rate of our elementary reaction takes the form:
- Given our elementary rate expression for the reaction,
- The rate constant, k, for the reaction is identifiable as:
- q = unit conc’n
- which ends our development of transition state theory. It correctly predicts the orders of the reaction, provides a means of interpreting the observed rate in terms of enthalpic and entropic contributions, and provides guidelines into the temperature dependence of k.

Temperature Dependence of Elementary Reactions

- The variation of elementary reaction rate constants with temperature is almost always expressed as:
- The term Ea is usually called the activation energy, although interpretations of this quantity differ between specific theories of reaction rate. The temperature exponent, m, does likewise.
- m = 0 corresponds to classical Arrhenius theory
- m = 1/2 is predicted by collision theory
- m = 1 is generated by transition state theory
- In practice, the dependence of the pre-exponential factor on temperature is usually much weaker than that of the activation energy.
- If gathered under kinetic control, reaction rate data plotted as ln(k) versus 1/T or ln(k/T) versus 1/T is usually linear.

Product

Reactant

‡

Reactant

Product

Early and Late Transition States- Endergonic reactions have transition states resembling the product in terms of energy and structure.
- This is called a “late” transition state or product-like t.s.

DG‡

DGo

- Exergonic reactions have a transition state more closely resembling the reactants in terms of both energy and structure.
- This is called an “early” transition state or reactant-like t.s.

DGo

DG‡

Hammond Postulate

- The use of transition state theory to describe chemical kinetics requires us to consider the structure of the transition state.
- By definition, the transition state cannot be isolated. How can we make meaningful inferences regarding its structure?
- While there is no universal relationship between the stability of a reaction product and its rate of formation, many reactions can be characterized by the Hammond Postulate.
- The position of the transition state along the reaction coordinate, its energy, and its geometry are related, and depend on the relative stabilities of the reactant and the product.
- A simple statement:
- The structure of a transition state resembles the structure of the nearest stable species.

Hammond Postulate: Examples

- Classify these reaction
- profiles in terms of:
- A. Product stability
- B. Transition State energy
- C. Position of the transition
- state (early/late)
- What generalizations can be made regarding the position of the transition state and the rate of reaction?

Food for thought…

- Consider the polymerization of methylmethacrylate to produce a transparent, glassy polymer (tradename plexiglass)
- the reaction proceeds with “head-to-tail” regioselectivity to give linear polymer chains
- as is the case for most polymerizations, it is strongly exothermic.
- What would the reaction profile look like for these reactions?
- Can product stability arguments (Hammond Postulate) be used to explain the head-to-tail preference?

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