2012 Reaction Dynamics

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Reaction Dynamics Proposal

Introduction

Reaction dynamics studied the physical aspects of chemical reactions. This includes finding out why some reactions occur, as well as predicting and controlling them. Reaction dynamics are similar to chemical kinetics, but it deals more with single reaction events over a shorter time period. In essence, reaction dynamic research seeks to understand chemical reactions at the simplest level.

Arrhenius Equation

Background

Svante Arrhenius (pronounced Svahn-tei Ahr-re-neus), a Swedish scientist, is considered to be one of the founders of physical chemistry. The Nobel Prize was given to him in 1903 for his work in the field. A little before this significant accomplishment, Arrhenius, in 1884, proposed an equation demonstrating the temperature dependence of the reaction rate constant (k); this resulted in remarkably accurate prediction of the rate of a chemical reaction. Later, the equation was justified by Van‘t Hoff, whose equation also explored the relationship between the change in temperature and the equilibrium constant.

Variables

The Arrhenius equation gives k, the reaction rate constant, and quantifies the speed of a chemical reaction and is a function of temperature and activation energy:

A is the pre-exponential factor (no units). R is the Universal gas constant, usually assigned the value of 8.314 J K-1 mol-1. Temperature is in absolute temperature scale of Kelvin. E_a is the activation energy and is usually in kJ/mol or J/mol. For a first order reaction, k is in s-1. For a second order reaction, k is in L/(mol*seconds) or M-1s-1. For more information on orders, please see the page for Rate Laws: https://260h.pbworks.com/w/page/61242630/Rate%20Laws

The Arrhenius equation can also be written in this format:

The only difference in using the Boltzmann constant (k_B) instead of the Universal gas constant is that this form of the Arrhenius equation relates the energy to the molecule directly, rather than the standard, energy (in kJ or J) per mole.

Recall that the rate constant, k, is measured in s-1 for a first order reaction; for a second order reaction, k is measured in L mol-1 s-1, or M-1 s-1. The reaction rate constant can be seen in the following equation, where A and B are products, and m and n are the moles of each, respectively:

For a single step reaction, we can insert the Arrhenius equation:

The Arrhenius pre-exponential factor or frequency factor, A, is an empirical constant describing the temperature dependence of the rate constant, k. The pre-exponential factor is interchangeable with the collision frequency, Z, for two molecules. Z is a function of time, the volume on the container, and the density of the particles. N is the number of atoms per unit volume, d is distance between molecules, and m_a is the mass of the atom. Consequently, for two of the same molecule, Z is defined as:

For two different molecules, Z is defined as :

Where μ represents the reduced mass. When the pre-exponential factor is determined experimentally, A is preferred. When it is determined mathematically, Z is preferred. Z is not ideal because it does not account for the steric effect of the molecules involved in the collision. The Z factor assumes that the molecules collide with a particular orientation, but this is not always the case. Consequently, a new variable, P, the probability of two molecules colliding in the proper orientation, may be introduced in direct proportion, but is very difficult to ascertain. More information on molecule orientation in collisions can be found in the Applied Reaction Dynamics section.

Activation energy, E_a, a term coined by Arrhenius in 1889, is defined as the energy required for a chemical reaction to occur, or the minimum energy needed to begin the reaction and is measured in kJ mol-1. Below is a reaction profile, which is a graph of free energy versus the reaction coordinate, or progress, of the reaction:

More information regarding activation energy can be found in the Transition State Theory section below.

Temperature is an independent variable in this reaction and is a measure of the average kinetic energy of the molecules and is measured in Kelvin (K).
The universal gas constant, R, is a physical constant and can be represented in many different units, but is generally given as 8.314 J K-1 mol-1 for the Arrhenius equation.
The Boltzmann Constant, kB, is a physical constant and is measured in J K-1.
Together, activation energy, temperature, and one of the physical constants, R or kB, form the term exp(-Ea/RT)or exp(-Ea/kBT), which is the probability than any given collision will result in a reaction.

Arrhenius Plot

To study how temperature affects reaction rate, an Arrhenius plot is made and analyzed. The plot of the natural log K (x-axis) to the inverse of temperature (y-axis). The negative slope, through some extrapolation, can give the activation energy.

Applied Reaction Dynamics

Molecular Orientation
There are three different kinds of collisions: Elastic collisions, inelastic collisions, and reactive collisions. The first two involve transfers of energy, but it is the third kind of collision that plays a major role in reaction dynamics.

Simply put, if the colliding molecules do not have a reactive collision, not reaction will occur. For example, when CH2=CH2 (ethene) and HCl collide, they form CH3CH2Cl (chloroethane). The two reactants form this product when the hydrogen from HCl bonds on to one of the carbons in ethene, causing the C=C bond to become a single bond, and the Cl to attach on to the other carbon. However, this reaction will only occur if the HCl collides with ethene hydrogen-first with a carbon; otherwise, the repulsion will cause the HCl molecule to simply “bounce off” the ethene molecule and continue on its way. This is most important in collisions/reactions involving unsymmetrical molecules.

Transition State Theory

The Transition State Theory (TST) describes the aptly named “transition state” in between the state where molecules are reactants and the state where molecules are products. In this state, the molecules form an “activated complex”. The activated complex is distinct from the reactant and product species, and is often drawn with dashed or dotted lines representing the transitory bonds. According to the TST, whether or not a reaction will occur is dependent upon three major factors:

The concentration of the activated complex
The rate at which the activated complex breaks apart upon exiting the transition state
The way in which the activated complex breaks apart: whether it breaks apart to reform the reactants or whether it breaks apart to form a new complex, the products.

In order to visually represent the reaction process, a reaction profile is used to plot the levels of energy required at various stages of the reaction. Apart from showing the initial (reactant) and final (product) energy levels, the reaction profile also shows the activation energy (E_a), which is the amount of energy required to reach the transition state for that reaction. If the transition state is not reached, the reaction will not occur. Some reactions have multiple transition states, with intermediate species forming in between the transition states.

The TST allows one figure out the amount of energy required for various steps of the raction. This in turn allows the calculation of reaction rates and rate constants. However, the TST is somewhat limited in that it does not work as well with reactions with low activation energy or reactions at high temperatures.

Marcus Theory of Electron Transfer

The Marcus theory states that the probability that an electron will transfer from an electron donor to an electron acceptor during a transition state will decrease with increasing distance. An electron donor is a compound that donates an electron to another compound. It is a reducing agents that is oxidized in the process. An electron acceptor is a compound that accepts electrons from another compound. It is an oxidizing agent that is reduced in the process. The components that control the rate constant of the electron transfer (k_et) include the distance between the donor-acceptor compounds, Gibbs free energy, and the reorganization of the energy. This is a monomolecular electron transfer and the probability of the transfer from a donor to an acceptor is given by kT/h. These components are described more in depth below:

The distance between the donor-acceptor complex (r) determines the probability
The Gibbs free energy of activation (ΔG) is determined by the standard reaction of Gibbs free energy (ΔG°) and the reorganization energy (λ).
The reorganization energy, denoted as λ, corresponds to the change in energy resulting from the molecular rearrangement. This rearrangement occurs as the charge is distributed throughout the donor-acceptor complexation in the medium.

The following shows the equation of the rate constant for an electron transfer where A is the pre-exponential factor, which contains the probability of crossing transition states (unitless), (J/mol) is the Gibbs free energy of the formation of the transition state, and the exponential term corresponds to the probability of the formation of the transition state.

The following equation shows the expression for the Gibbs free energy of the transition state where G⁰ is the standard Gibbs free energy (J/mol), λ is the reorganization energy, and λ_o is the initial reorganization energy:

Gas Phase vs. Solution Phase

Reactions carried out in solution are much more difficult to describe than in the gas phase. Molecules in the gas phase travel in straight lines after collisions whereas molecules in solution constantly collide with solvent molecules. Additionally, this collision leads to transport of the solutes by diffusion, meaning that reactions in solution occur much less frequently than in the gas phase.

In order for a reaction to take place in solution, the reactants must be temporarily “caged” by the solvent molecules in the hope that the reactants overcome the activation energy through collisions with the solvent. Reactions in solution are described using a steady-state approximation to help understand the phenomenon.

There are two distinctions given to reactions that occur in solution: diffusion-controlled and activation-energy-controlled. If the rate is limited by how often the reactants encounter each due to diffusion, the reaction is considered diffusion-controlled and the rate constant is conventionally denoted k_d. If the second step is the rate-limiting step (has a lower rate constant) and the first step in the reaction has reached equilibrium, the reaction is said to be activation-energy-controlled. The rate is now described as:

Where k₂ is the reaction constant for the second step and K₁is the equilibrium constant for the first reaction in the series.

Questions

1. What does the Arrhenius equation give? What does it do?

2. Which of the following statements is false?

a. Molecular orientation plays an important role in reactive collisions

b. Once an activated complex forms, the reaction will proceed to the reactants

c. A reaction will not occur if activation energy is not reached

d. There are three different kinds of collisions

3. All of the following are components that control the rate constant of the electron transfer except:

a. Distance between the donor-acceptor compounds

b. The area that the electron encompasses

c. Gibbs Free Energy

d. The reorganization of the energy

4. Which pre-exponential factor is most accurate? Is this factor determined experimentally or can it be mathematically determined?

Answers: 1. Arrhenius equation gives k, the reaction rate constant. It quantifies the speed of a reaction.

2. B

3. B

4. A is the most accurate constant, and it is determined experimentally.

Works Cited

Atkins, Peter and de Paula, Julio. New York (NY): W.H. Freeman and Company. p.323-325.

Barker, Brett. Peterson's Master AP Chemistry. Lawrenceville, NJ: Peterson's, a Nelnet company, 200.

Gross, Dixie J., and Ralph H. Petrucci. General Chemistry Principles and Modern Applications, Ninth Editon : Study Guide. Upper Saddle River: Prentice Hall PTR, 2006.

Pauling, Linus. General Chemistry. Minneapolis: Dover Publications, Incorporated, 1989.

Petrucci, Ralph H., William S. Harwood, and Geoff E. Herring. General Chemistry : Principles and Modern Applications. Upper Saddle River: Prentice Hall PTR, 2006.

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