We will assume an understanding of the postulates of the Kinetic Molecular Theory and of the energetics of chemical reactions. We will also assume an understanding of phase equilibrium and reaction equilibrium, including the temperature dependence of equilibrium constants.

We have carefully examined the observation that chemical reactions come to equilibrium. Depending on the reaction, the equilibrium conditions can be such that there is a mixture of reactants and products, or virtually all products, or virtually all reactants. We have not considered the time scale for the reaction to achieve these conditions, however. In many cases, the speed of the reaction might be of more interest than the final equilibrium conditions of the reaction. Some reactions proceed so slowly towards equilibrium as to appear not to occur at all. For example, metallic iron will eventually oxidize in the presence of aqueous salt solutions, but the time is sufficiently long for this process that we can reasonably expect to build a boat out of iron. On the other hand, some reactions may be so rapid as to pose a hazard. For example, hydrogen gas will react with oxygen gas so rapidly as to cause an explosion. In addition, the time scale for a reaction can depend very strongly on the amounts of reactants and their temperature.

In this concept development study, we seek an understanding of the rates of chemical reactions. We will define and measure reaction rates and develop a quantitative analysis of the dependence of the reaction rates on the conditions of the reaction, including concentration of reactants and temperature. This quantitative analysis will provide us insight into the process of a chemical reaction and thus lead us to develop a model to provide an understanding of the significance of reactant concentration and temperature.

We will find that many reactions proceed quite simply, with reactant molecules colliding and exchanging atoms. In other cases, we will find that the process of reaction can be quite complicated, involving many molecular collisions and rearrangements leading from reactant molecules to product molecules. The rate of the chemical reaction is determined by these steps.

We begin by considering a fairly simple reaction on a rather elegant molecule. One oxidized form of buckminsterfullerene C60C60 is C60O3C60O3, with a three oxygen bridge as shown in Figure 1.

Figure 1

Oxidized Buckminsterfullerene

C60O3C60O3 is prepared from C60C60 dissolved in toluene solution at temperatures of 0⁢°C0°C or below. When the solution is warmed, C60O3C60O3 decomposes, releasing O2O2 and creating C60OC60O in a reaction which goes essentially to completion. We can actually watch this process happen in time by measuring the amount of light of a specific frequency absorbed by the C60O3C60O3 molecules, called the absorbance. The absorbance is proportional to the concentration of the C60O3C60O3 in the toluene solution, so observing the absorbance as a function of time is essentially the same as observing the concentration as a function of time. One such set of data is given in Table 1, and is shown in the graph in Figure 2.

Figure 2

Oxidized Buckminsterfullerene Absorbance

The rate at which the decomposition reaction is occurring is clearly related to the rate of change of the concentration [C60O3][C60O3], which is proportional to the slope of the graph in Figure 2. Therefore, we define the rate of this reaction as

Figure 3

Rate of Decomposition

It is clear that the slope of the graph in Figure 2 changes over the course of time. Correspondingly, Figure 3 shows that the rate of the reaction decreases as the reaction proceeds. The reaction is at first very fast but then slows considerably as the reactant C60O3C60O3 is depleted.

The shape of the graph for rate versus time (Figure 3) is very similar to the shape of the graph for concentration versus time (Figure 2). This suggests that the rate of the reaction is related to the concentration of C60O3C60O3 at each time. Therefore, in Figure 4, we plot the rate of the reaction, defined in Equation 1 and shown in Figure 3, versus the absorbance of the C60O3C60O3.

Figure 4

Rate versus Concentration

We find that there is a very simple proportional relationship between the rate of the reaction and the concentration of the reactant. Therefore, we can write

As a second example of a reaction rate, we consider the dimerization reaction of butadiene gas, CH2CH2=CHCH-CHCH=CH2CH2. Two butadiene molecules can combine to form vinylcyclohexene, shown in Figure 5.

Figure 5

Dimerization of Butadiene to Vinylcyclohexene

Table 2 provides experimental data on the gas phase concentration of butadiene [C4H6][C4H6] as a function of time at T=250⁢°CT250°C.

We can estimate the rate of reaction at each time step as in Equation 1, and these data are presented in Table 2 as well. Again we see that the rate of reaction decreases as the concentration of butadiene decreases. This suggests that the rate is given by an expression like Equation 2. To test this, we calculate Rate[C4H6]Rate[C4H6] in Table 2 for each time step. We note that this is not a constant, so Equation 2 does not describe the relationship between the rate of reaction and the concentration of butadiene. Instead we calculate Rate[C4H6]2Rate[C4H6]2 in Table 2. We discover that this ratio is a constant throughout the reaction. Therefore, the relationship between the rate of the reaction and the concentration of the reactant in this case is given by

We would like to understand what determines the specific dependence of the reaction rate on concentration in each reaction. In the first case considered above, the rate depends on the concentration of the reactant to the first power. We refer to this as a first order reaction. In the second case above, the rate depends on the concentration of the reactant to the second power, so this is called a second order reaction. There are also third order reactions, and even zeroth order reactions whose rates do not depend on the amount of the reactant. We need more observations of rate laws for different reactions.

The approach used in the previous section to determine a reaction's rate law is fairly clumsy and at this point difficult to apply. We consider here a more systematic approach. First, consider the decomposition of N2O5(g)N2O5(g). 2N2O5(g)→4 NO2(g)+O2(g)2N2O5(g)→4 NO2(g)+O2(g) We can create an initial concentration of N2O5N2O5 in a flask and measure the rate at which the N2O5N2O5 first decomposes. We can then create a different initial concentration of N2O5N2O5 and measure the new rate at which the N2O5N2O5 decomposes. By comparing these rates, we can find the order of the decomposition reaction. The rate law for decomposition of N2O5(g)N2O5(g) is of the general form:

This approach to finding reaction order is called the method of initial rates, since it relies on fixing the concentration at specific initial values and measuring the initial rate associated with each concentration.

So far we have considered only reactions which have a single reactant. Consider a second example of the method of initial rates involving the reaction of hydrogen gas and iodine gas:

Following the same process we used in the N2O5N2O5 example, we write the general rate law for the reaction as

This procedure can be applied to any number of reactions. The challenge is preparing the initial conditions and measuring the initial change in concentration precisely versus time. Table 4 provides an overview of the rate laws for several reactions. A variety of reaction orders are observed, and they cannot be easily correlated with the stoichiometry of the reaction.

Once we know the rate law for a reaction, we should be able to predict how fast a reaction will proceed. From this, we should also be able to predict how much reactant remains or how much product has been produced at any given time in the reaction. We will focus on the reactions with a single reactant to illustrate these ideas.

Consider a first order reaction like A→productsA→products, for which the rate law must be

An interesting point in the reaction is the time at which exactly half of the original concentration of AA has been consumed. We call this time the half life of the reaction and denote it as t12t12. At that time, [A]=12⁢[A]0[A]12[A]0. From Equation 12 and using the properties of logarithms, we find that, for a first order reaction

These conclusions are only valid for first order reactions. Consider then a second order reaction, such as the butadiene dimerization discussed above. The general second order reaction A→productsA→products has the rate law

The half-life of a second order reaction differs from the half-life of a first order reaction. From Equation 15, if we take [A]=12⁢[A]0[A]12[A]0, we get

It is a common observation that reactions tend to proceed more rapidly with increasing temperature. Similarly, cooling reactants can have the effect of slowing a reaction to a near halt. How is this change in rate reflected in the rate law equation, e.g. Equation 9? One possibility is that there is a slight dependence on temperature of the concentrations, since volumes do vary with temperature. However, this is insufficient to account for the dramatic changes in rate typically observed. Therefore, the temperature dependence of reaction rate is primarily found in the rate constant, kk.

Consider for example the reaction of hydrogen gas with iodine gas at high temperatures, as given in Equation 6. The rate constant of this reaction at each temperature can be found using the method of initial rates, as discussed above, and we find in Table 5 that the rate constant increases dramatically as the temperature increases.

As shown in Figure 6, the rate constant appears to increase exponentially with temperature. After a little experimentation with the data, we find in Figure 7 that there is a simple linear relationship between lnkk and 1T1T.

Figure 6

Rate Constant

Figure 7

Rate Constant

From Figure 7, we can see that the data in Table 4 fit the equation

It is very important to note that the form of Equation 17 and the appearance of Figure 7 are both the same as the equations and graphs for the temperature dependence of the equilibrium constant for an endothermic reaction. This suggests a model to account for the temperature dependence of the rate constant, based on the energetics of the reaction. In particular, it appears that the reaction rate is related to the amount of energy required for the reaction to occur. We will develop this further in the next section.

At this point, we have only observed the dependence of reaction rates on concentration of reactants and on temperature, and we have fit these data to equations called rate laws. Although this is very convenient, it does not provide us insight into why a particular reaction has a specific rate law or why the temperature dependence should obey Equation 17. Nor does it provide any physical insights into the order of the reaction or the meaning of the constants aa and bb in Equation 17.

We begin by asking why the reaction rate should depend on the concentration of the reactants. To answer this, we consider a simple reaction between two molecules in which atoms are transferred between the molecules during the reaction. For example, a reaction important in the decomposition of ozone O3O3 by aerosols is O3 (g)+ Cl(g) →O2 (g)+ ClO(g ) O3 (g)+ Cl(g) →O2 (g)+ ClO(g ) What must happen for a reaction to occur between an O3O3 molecule and a ClCl atom? Obviously, for these two particles to react, they must come into close proximity to one another so that an OO atom can be transferred from one to the other. In general, two molecules cannot trade atoms to produce new product molecules unless they are close enough together for the atoms of the two molecules to interact. This requires a collision between molecules.

The rate of collisions depends on the concentrations of the reactants, since the more molecules there are in a confined space, the more likely they are to run into each other. To write this relationship in an equation, we can think in terms of probability, and we consider the reaction above. The probability for an O3O3 molecule to be near a specific point increases with the number of O3O3 molecules, and therefore increases with the concentration of O3O3 molecules. The probability for a ClCl atom to be near that specific point is also proportional to the concentration of ClCl atoms. Therefore, the probability for an O3O3 molecule and a ClCl atom to be in close proximity to the same specific point at the same time is proportional to the [O3][O3] times [Cl][Cl].

It is important to remember that not all collisions between O3O3 molecules and ClCl atoms will result in a reaction. There are other factors to consider including how the molecules approach one another. The atoms may not be positioned properly to exchange between molecules, in which case the molecules will simply bounce off of one another without reacting. For example, if the ClCl atom approaches the center OO atom of the O3O3 molecule, that OO atom will not transfer. Another factor is energy associated with the reaction. Clearly, though, a collision must occur for the reaction to occur, and therefore there rate of the reaction can be no faster than the rate of collisions between the reactant molecules.

Therefore, we can say that, in a bimolecular reaction, where two molecules collide and react, the rate of the reaction will be proportional to the product of the concentrations of the reactants. For the reaction of O3O3 with ClCl, the rate must therefore be proportional to [O3]⁢[Cl][O3][Cl], and we observe this in the experimental rate law in Table 4. Thus, it appears that we can understand the rate law by understanding the collisions which must occur for the reaction to take place.

We also need our model to account for the temperature dependence of the rate constant. As noted at the end of the last section, the temperature dependence of the rate constant in Equation 17 is the same as the temperature dependence of the equilibrium constant for an endothermic reaction. This suggests that the temperature dependence is due to an energetic factor required for the reaction to occur. However, we find experimentally that Equation 17 describes the rate constant temperature dependence regardless of whether the reaction is endothermic or exothermic. Therefore, whatever the energetic factor is that is required for the reaction to occur, it is not just the endothermicity of the reaction. It must be that all reactions, regardless of the overall change in energy, require energy to occur.

A model to account for this is the concept of activation energy. For a reaction to occur, at least some bonds in the reactant molecule must be broken, so that atoms can rearrange and new bonds can be created. At the time of collision, bonds are stretched and broken as new bonds are made. Breaking these bonds and rearranging the atoms during the collision requires the input of energy. The minimum amount of energy required for the reaction to occur is called the activation energy, EaEa. This is illustrated in Figure 8, showing conceptually how the energy of the reactants varies as the reaction proceeds. In Figure 8(a), the energy is low early in the reaction, when the molecules are still arranged as reactants. As the molecules approach and begin to rearrange, the energy rises sharply, rising to a maximum in the middle of the reaction. This sharp rise in energy is the activation energy, as illustrated. After the middle of the reaction has passed and the molecules are arranged more as products than reactants, the energy begins to fall again. However, the energy does not fall to its original value, so this is an endothermic reaction.

Figure 8(b) shows the analogous situation for an exothermic reaction. Again, as the reactants approach one another, the energy rises as the atoms begin to rearrange. At the middle of the collision, the energy maximizes and then falls as the product molecules form. In an exothermic reaction, the product energy is lower than the reactant energy.

Figure 8 thus shows that an energy barrier must be surmounted for the reaction to occur, regardless of whether the energy of the products is greater than (Figure 8(a)) or less than (Figure 8(b)) the energy of the reactants. This barrier accounts for the temperature dependence of the reaction rate. We know from the kinetic molecular theory that as temperature increases the average energy of the molecules in a sample increases. Therefore, as temperature increases, the fraction of molecules with sufficient energy to surmount the reaction activation barrier increases.

Figure 8

Reaction Energy
Endothermic Reaction (a) Exothermic Reaction (b)

Although we will not show it here, kinetic molecular theory shows that the fraction of molecules with energy greater than EaEa at temperature TT is proportional to e−EaR⁢TEaRT. This means that the reaction rate and therefore also the rate constant must be proportional to e−EaR⁢TEaRT. Therefore we can write

As a final note on Equation 19, the constant AA must have some physical significant. We have accounted for the probability of collision between two molecules and we have accounted for the energetic requirement for a successful reactive collision. We have not accounted for the probability that a collision will have the appropriate orientation of reactant molecules during the collision. Moreover, not every collision which occurs with proper orientation and sufficient energy will actually result in a reaction. There are other random factors relating to the internal structure of each molecule at the instant of collision. The factor AA takes account for all of these factors, and is essentially the probability that a collision with sufficient energy for reaction will indeed lead to reaction. AA is commonly called the frequency factor.

Our collision model in the previous section accounts for the concentration and temperature dependence of the reaction rate, as expressed by the rate law. The concentration dependence arises from calculating the probability of the reactant molecules being in the same vicinity at the same instant. Therefore, we should be able to predict the rate law for any reaction by simply multiplying together the concentrations of all reactant molecules in the balanced stoichiometric equation. The order of the reaction should therefore be simply related to the stoichiometric coefficients in the reaction. However, Table 4 shows that this is incorrect for many reactions.

Consider for example the apparently simple reaction

The key assumption of the collision model is that the reaction occurs by a single collision. Since this assumption leads to incorrect predictions of rate laws in some cases, the assumption must be invalid in at least those cases. It may well be that reactions require more than a single collision to occur, even in reactions involving just two types of molecules as in Equation 22. Moreover, if more than two molecules are involved as in Equation 20, the chance of a single collision involving all of the reactive molecules becomes very small. We conclude that many reactions, including those in Equation 20 and Equation 22, must occur as a result of several collisions occurring in sequence, rather than a single collision. The rate of the chemical reaction must be determined by the rates of the individual steps in the reaction.

Each step in a complex reaction is a single collision, often referred to as an elementary process. In single collision process step, our collision model should correctly predict the rate of that step. The sequence of such elementary processes leading to the overall reaction is referred to as the reaction mechanism. Determining the mechanism for a reaction can require gaining substantially more information than simply the rate data we have considered here. However, we can gain some progress just from the rate law.

Consider for example the reaction in Equation 22 described by the rate law in Equation 23. Since the rate law involved [NO2]2[NO2]2, one step in the reaction mechanism must involve the collision of two NO2NO2 molecules. Furthermore, this step must determine the rate of the overall reaction. Why would that be? In any multi-step process, if one step is considerably slower than all of the other steps, the rate of the multi-step process is determined entirely by that slowest step, because the overall process cannot go any faster than the slowest step. It does not matter how rapidly the rapid steps occur. Therefore, the slowest step in a multi-step process is thus called the rate determining or rate limiting step.

This argument suggests that the reaction in Equation 22 proceeds via a slow step in which two NO2NO2 molecules collide, followed by at least one other rapid step leading to the products. A possible mechanism is therefore

There are a few important notes about the mechanism. First, one product of the reaction is produced in the first step, and the other is produced in the second step. Therefore, the mechanism does lead to the overall reaction, consuming the correct amount of reactant and producing the correct amount of reactant. Second, the first reaction produces a new molecule, NO3NO3, which is neither a reactant nor a product. The second step then consumes that molecule, and NO3NO3 therefore does not appear in the overall reaction, Equation 22. As such, NO3NO3 is called a reaction intermediate. Intermediates play important roles in the rates of many reactions.

If the first step in a mechanism is rate determining as in this case, it is easy to find the rate law for the overall expression from the mechanism. If the second step or later steps are rate determining, determining the rate law is slightly more involved. The process for finding the rate law in such a case is illustrated in Exercise 11.