Elementary kinetics

We have seen how to estimate the energetics and feasibility of chemical reactions proceeding one way or the other using the methods of equilibrium thermodynamics. But equilibrium thermodynamics cannot tell us how rapidly a reaction will proceed.

This is the business of chemical kinetics which considers the details of the molecular collision and the intermediate complexes that can form during the event. For example, kinetics can provide a means of computing the characteristic time of the decay of the reactants in the atmosphere. One should keep in mind that negative Gibbs energy change for a reaction (thermodynamically favorable conditions) does not always mean that the reaction will proceed fast enough to be observed.

8.4.1 Reaction rate

A reaction rate can be defined intuitively as the rate at which the products of the reaction are formed, which is the same as the rate at which the reactants are consumed. As an example, consider a bimolecular reaction with molecules C and D as products and A and B as reactants:

The rate of this reaction (the rate of loss of A or B and the rate of increase of C and D) is d[A] d[B] d[C] d[D]

dt dt dt dt where [X] denotes the concentration of species X expressed in molecules cm-3 and k is the reaction rate coefficient. The units of k depend on the order of the reaction: for the bimolecular reaction (8.30) k is in cm3 s-1. The reaction rate coefficient k is unique for a given reaction at each given temperature. The temperature dependence k(T) is described by the Arrhenius equation:

[rate coefficient with activation energy]. (8.31)

Eact is called the activation energy. A large value of Eact usually implies a strong temperature dependence of the reaction rate coefficient. The constant A (not to be confused with the identity of the species A in (8.29)) before the exponential function is related to the frequency of molecular collisions and the probability for molecules to have an orientation in space favorable for a reaction. The dependence of A on temperature is usually weak compared to that of the exponential factor.

The idea of activation energy is shown schematically in Figure 8.5. The horizontal axis represents the reaction coordinate for the reactants. The reaction coordinate can be thought of as the distance between the molecules A and B in the reaction (8.29). The vertical axis is the potential energy of the reaction. AH is the standard enthalpy of formation for this reaction. Note that AH is negative, so the reaction is exothermic.

Figure 8.5 Schematic graph of energy change for an exothermic reaction. Reactants have to be energetic enough to overcome the barrier Eact.
Figure 8.6 Velocity distribution of molecules at two different temperatures. The velocity v is expressed in units of ^kB T/m0. The velocity v* corresponds to the kinetic energy equal to Eact.

For the products C and D to be formed by this reaction, the reactants A and B must have enough kinetic energy to overcome the energy barrier Eact. It should be noted that many reactions in the real atmosphere do not proceed because the activation barrier is too large. For example, C + O2 ^ CO2 does not take place in the atmosphere because of the large barrier.

Equation (8.59) implies that reactions proceed faster at higher temperatures.4 This can be explained with the help of kinetic theory. It follows from (2.26) that the higher the temperature of the gas, the greater the fraction of molecules that have kinetic energies that exceed a certain given energy. Figure 8.6 shows the velocity distribution for two temperatures. At higher temperatures more molecules have velocities higher than the threshold velocity v* corresponding to the kinetic energy 1 m0v*2 which is equal to Eact. This means that increasing the temperature of the gas increases the probability that molecules will overcome the barrier Eact and that the products will be formed at a higher rate.

4 For some reactions the activation energy is actually negative (no barrier). The rate of these reactions decreases with increasing temperature.

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