Kinetics, free energy of activation, first order reaction, second-order reaction, Arrhenius equation | Organic Chemistry
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Kinetics

Knowing whether a given reaction is exergonic or endergonic will not tell you how fast the reaction occurs, because the ∆G° of a reaction tells you only the difference between the stability of the reactants and the stability of the products; it does not tell you anything about the energy barrier of the reaction, which is the energy “hill” that must be climbed for the reactants to be converted into products. The higher the energy barrier, the slower is the reaction. Kinetics is the field of chemistry that studies the rates of chemical reactions and the factors that affect those rates.

The energy barrier of a reaction, indicated in Figure 3.4 by ΔG is called the free energy of activation. It is the difference between the free energy of the transition state and the free energy of the reactants:
free energy of activation
smaller free energy of activation faster reaction

Some exergonic reactions have small free energies of activation and therefore can take place at room temperature (Figure 3.4a). In contrast, some exergonic reactions have free energies of activation that are so large that the reaction cannot take place without adding energy above that provided by the existing thermal conditions (Figure 3.4b). Endergonic reactions can also have either small free energies of activation, as in Figure 3.4c, or large free energies of activation, as in Figure 3.4d.
Reaction coordinate diagrams

Figure 3.4 Reaction coordinate diagrams for (a) a fast exergonic reaction, (b) a slow exergonic reaction, (c) a fast endergonic reaction, and (d) a slow endergonic reaction. (The four reaction coordinates are drawn on the same scale.)

Kinetic stability

The rate of a chemical reaction is the speed at which the reacting substances are used up or the speed at which the products are formed. The rate of a reaction depends on the following factors:

  • The number of collisions that take place between the reacting molecules in a given period of time. The greater the number of collisions, the faster is the reaction.
  • The fraction of the collisions that occur with sufficient energy to get the reacting molecules over the energy barrier. If the free energy of activation is small, more collisions will lead to reaction than if the free energy of activation is large.
  • The fraction of the collisions that occur with the proper orientation. For example, 2-butene and HBr will react only if the molecules collide with the hydrogen of HBr approaching the π bond of 2-butene. If collision occurs with the hydrogen approaching a methyl group of 2-butene, no reaction will take place, regardless of the energy of the collision.

rate of a reaction

Increasing the concentration of the reactants increases the rate of a reaction because it increases the number of collisions that occur in a given period of time. Increasing the temperature at which the reaction is carried out also increases the rate of a reaction because it increases both the frequency of collisions (molecules that are moving faster collide more frequently) and the number of collisions that have sufficient energy to get the reacting molecules over the energy barrier.
For a reaction in which a single reactant molecule A is converted into a product molecule B, the rate of the reaction is proportional to the concentration of A. If the concentration of A is doubled, the rate of the reaction will double; if the concentration of A is tripled, the rate of the reaction will triple; and so on. Because the rate of this reaction is proportional to the concentration of only one reactant, it is called a first order reaction.
rate constantfirst-order rate constant
A reaction whose rate depends on the concentrations of two reactants is called a second-order reaction. If the concentration of either A or B is doubled, the rate of the reaction will double; if the concentrations of both A and B are doubled, the rate of the reaction will quadruple; and so on. In this case, the rate constant k is a second-order rate constant.
second-order rate constant
A reaction in which two molecules of A combine to form a molecule of B is also a second-order reaction: If the concentration of A is doubled, the rate of the reaction will quadruple.

Do not confuse the rate constant of a reaction (k) with the rate of a reaction. The rate constant tells us how easy it is to reach the transition state (how easy it is to get over the energy barrier). Low energy barriers are associated with large rate constants (Figures 3.4a and 3.4c), whereas high energy barriers have small rate constants (Figures 3.4b and 3.4d). The reaction rate is a measure of the amount of product that is formed per unit of time. The preceding equations show that the rate is the product of the rate constant and the concentration(s) of the reactants. Thus, reaction rates depend on concentration, whereas rate constants are independent of concentration. Therefore, when we compare two reactions to see which one occurs more readily, we must compare their rate constants and not their concentration-dependent rates of reaction. (Appendix III explains how rate constants are determined.)
Although rate constants are independent of concentration, they depend on temperature. The Arrhenius equation relates the rate constant of a reaction to the experimental energy of activation and to the temperature at which the reaction is carried out. A good rule of thumb is that an increase of 10 °C in temperature will double the rate constant for a reaction and, therefore, double the rate of the reaction.
Arrhenius equation
Where k is the rate constant, Ea is the experimental energy of activation, R is the gas constant (1.986 x 10–3 cal mol–1K–1, 8.314 x 10–3 kJ mol–1K–1), T is the absolute temperature (K), and A is the frequency factor. The frequency factor accounts for the fraction of collisions that occur with the proper orientation for reaction. The term e– Ea/RT corresponds to the fraction of the collisions that have the minimum energy (Ea) needed to react. Taking the logarithm of both sides of the Arrhenius equation, we obtain



How are the rate constants for a reaction related to the equilibrium constant? At equilibrium, the rate of the forward reaction must be equal to the rate of the reverse reaction because the amounts of reactants and products are not changing:

From this equation, we can see that the equilibrium constant for a reaction can be determined from the relative concentrations of the products and reactants at equilibrium or from the relative rate constants for the forward and reverse reactions. The reaction shown in Figure 3.3a has a large equilibrium constant because the products are much more stable than the reactants. We could also say that it has a large equilibrium constant because the rate constant of the forward reaction is much greater than the rate constant of the reverse reaction.

Reaction Coordinate Diagram for the Addition of HBr to 2-Butene

We have seen that the addition of HBr to 2-butene is a two-step reaction (Section 3.6). The structure of the transition state for each of the steps is shown below in brackets. Notice that the bonds that break and the bonds that form during the course of the reaction are partially broken and partially formed in the transition state—indicated by dashed lines. Similarly, atoms that either become charged or lose their charge during the course of the reaction are partially charged in the transition state. Transition states are shown in brackets with a double-dagger superscript.

A reaction coordinate diagram can be drawn for each of the steps in the reaction (Figure 3.5). In the first step of the reaction, the alkene is converted into a carbocation that is less stable than the reactants. The first step, therefore, is endergonic (∆G° is positive). In the second step of the reaction, the carbocation reacts with a nucleophile to form a product that is more stable than the carbocation reactant. This step, therefore, is exergonic (∆G° is negative).
Reaction coordinate diagrams for the two steps in the addition of HBr to 2-butene

Figure 3.5 Reaction coordinate diagrams for the two steps in the addition of HBr to 2-butene: (a) the first step; (b) the second step.

Because the product of the first step is the reactant in the second step, we can hook the two reaction coordinate diagrams together to obtain the reaction coordinate Because the product of the first step is the reactant in the second step, we can hook the two reaction coordinate diagrams together to obtain the reaction coordinate ∆G° for the overall reaction is the difference between the free energy of the final products and the free energy of the initial reactants. The figure shows that ∆G° for the overall reaction is negative. Therefore, the overall reaction is exergonic.
Reaction coordinate diagram for the addition of HBr to 2-butene

Figure 3.6 Reaction coordinate diagram for the addition of HBr to 2-butene.

A chemical species that is the product of one step of a reaction and is the reactant for the next step is called an intermediate. The carbocation intermediate in this reaction is too unstable to be isolated, but some reactions have more stable intermediates that can be isolated. Transition states, in contrast, represent the highest-energy structures that are involved in the reaction. They exist only fleetingly and can never be isolated. Do not confuse transition states with intermediates: Transition states have partially formed bonds, whereas intermediates have fully formed bonds.
We can see from the reaction coordinate diagram that the free energy of activation for the first step of the reaction is greater than the free energy of activation for the second step. In other words, the rate constant for the first step is smaller than the rate constant for the second step. This is what you would expect because the molecules in the first step of this reaction must collide with sufficient energy to break covalent bonds, whereas no bonds are broken in the second step.
The reaction step that has its transition state at the highest point on the reaction coordinate is called the rate-determining step or rate-limiting step. The rate determining step controls the overall rate of the reaction because the overall rate cannot exceed the rate of the rate-determining step. In Figure 3.6, the rate-determining step is the first step—the addition of the electrophile (the proton) to the alkene.
Reaction coordinate diagrams also can be used to explain why a given reaction forms a particular product, but not others. We will see the first example of this in Section 4.3.


Next Chapter : Reactions of Alkenes
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