Reaction Coordinate Diagrams

We can follow the progress of a reaction on its way from reactants to products by graphing the energy of the species versus the reaction coordinate. We will be vague in describing the reaction coordinate because its definition is a mess of other variables composed to best make sense of the progress of the reaction. The value of the reaction coordinate is between zero and one. Understanding the meaning of the reaction coordinate is not important, just know that small values of reaction coordinate (0-0.2) mean little reaction has taken place and large values (0.8-1.0) mean that the reaction is almost over. It is a kind of scale of the progress of a reaction. A typical reaction coordinate diagram for a mechanism with a single step is shown below:

Figure %: A reaction coordinate diagram for a single-step reaction

Note that the reactants are placed on the left and the products on the right. The choice of the energy levels of the reactants and products is dictated by their energies, those with higher energies are higher on the diagram and those with lower energies are lower on the diagram. The difference is energy between the reactants and the transition state is called the activation energy. The activation energy is the height of the energy barrier of the reaction. The transition state is the point of maximum energy on the diagram which represents a species possessing both reactant-like and product-like properties. Because it is so high in energy, the transition state is very reactive and can never be isolated due to its extremely short lifetime. The relative energy of the reactants and products, the ΔE on the diagram, determines whether the reaction is exothermic or endothermic. A reaction will be exothermic if the energy of the products is less than the energy of the reactants. A reaction is endothermic when the energy of the products is greater than the energy of the reactants. The is for an exothermic reaction. Below is a reaction coordinate diagram for an endothermic reaction.

Figure %: Reaction coordinate diagram for an endothermic reaction

If a reaction has n elementary steps in its mechanism, there will be n–1 minima between the products and reactants representing intermediates. There will also be n maxima representing the n transition states. For example, a reaction with three elementary steps could have the following reaction coordinate diagram.

Figure %: Reaction coordinate diagram for a three-step reaction

One confusing point about reaction coordinate diagrams is how to determine what the rate determining step is. Even experienced chemists consistently get this type of problem wrong. The rate determining step is not the one with the highest activation energy for the step. The rate determining step is the step whose transition state has the highest energy.

Activation Energy and the Arrhenius Equation

Intuitively, it makes sense that a reaction with a higher activation barrier will be slower. Think of how much harder you must roll a ball up a large hill than a smaller one. Let's consider chemical reactions more deeply to derive an equation which describes the relationship between the rate constant of a reaction and its activation barrier. To simplify our derivation, we will assume that the reaction has a one-step mechanism. This elementary step represents a collision as shown in . Therefore, the frequency of the collisions, f, will be important in our equation. Notice that only a certain orientation of the molecules will lead to a reaction. For example, the following collision will not lead to a reaction. The reagent molecules simply bounce off of one another:

Figure %: Only specific orientations during a collision will lead to a reaction.

Therefore, we will need to include an orientation factor (or steric factor), p, that takes into account the fact that only a certain fraction of collisions will lead to reaction due to the orientation of the molecules. Another factor we must consider is that only a certain fraction of the molecules colliding will have enough energy to overcome the activation barrier. The Boltzmann distribution is a thermodynamic equation that tells us what fraction of the molecules have a certain amount of energy. As you know, at higher temperatures the average kinetic energy of the molecules increases. Therefore, at higher temperatures more molecules have an energy greater than the activation energy--as shown in .