and the collision theory bimolecular rate coefficient is
As indicated, the terms multiplying the exponential are customarily denoted by A, the collision frequency factor, or simply the preexponential factor. Thus, the reaction rate coefficient consists of two components, the frequency with which the reactants collide and the fraction of collisions that have enough energy to overcome the barrier to reaction.
The quantity (8/cB7'/7iji)' /2 is a molecular speed; at ordinary temperatures its value is about 5 x 104cm s_1. The value of d for small molecules typical of atmospheric chemistry is around 3 x 10~~8cm. Thus an order of magnitude estimate for A is
The collision rate of one molecule with other molecules in a gas at standard temperature and pressure is ~ 5 x 109 collisions s-1. A typical molecular vibrational frequency is 3 x 1013s~'. The mean time between collisions is ~ 2 x 10~10s. Thus, a typical molecule undergoes on the order of 103-104 vibrational cycles between collisions.
Actual measured A factors are usually substantially smaller than those based on collision theory. Recall that the molecules have been assumed to be hard spheres, and molecular structure has been assumed not to play a role. In real molecules certain parts of the molecule are more "reactive" than others. As a result, some of the collisions will be ineffective if the colliding molecules are not pointing their reactive "ends" at each other. For example, in the reaction of hydroxyl (OH) radicals with CHBr3
OH + CHBr3 H20 + CBr3 ¿(298 K) = 1.8 x 10"13 cm3 molecule"1 s"1
the small hydrogen atom in CHBr3 is shielded by the three bulky bromine atoms, so only if the OH approaches in just the right direction so as to encounter the H atom will reaction occur. On the other hand, there are a number of reactions whose rate coefficients approach the collision limit. One example is
O + CIO —> CI + 02 ¿(298 K) = 3.8 x 1CT11 cm3 molecule^1 s"1
In summary, collision theory provides a good physical picture of bimolecular reactions, even though the structure of the molecules is not taken into account. Also, it is assumed that reaction takes place instantaneously; in practice, the reaction itself requires a certain amount of time. The structure of the reaction complex must evolve, and this must be accounted for in a reaction rate theory. For some reactions, the rate coefficient actually decreases with increasing temperature, a phenomenon that collision theory does not describe. Finally, real molecules interact with each other over distances greater than the sum of their hard-sphere radii, and in many cases these interactions can be very important. For example, ions can react via long-range Coulomb forces at a rate that exceeds the collision limit. The next level of complexity is transition state theory.
cm3 molecule 1 s
Evaluation of Collision Theory The reaction
has a measured rate coefficient
£oh+ho2 =4.8 x 10"11 exp(250/7") cm3 molecule"1 s~'
Calculate the collision theory rate coefficient A at T = 300 K and compare it to the measured rate coefficient:
fcoH+HOi (300 K) = 1.1 x 1(H° cm3 molecule"1 s"1
For the purpose of the collision theory estimate, let us assume that the radii of OH and H02 are each 2 x 10"s cm. Also calculate the fraction of collisions that lead to reaction.
The collision theory rate coefficient A requires the following quantities:
Kd2 = 7t(2x 10"10m + 2 x 10 10mf = 5.03 x I0",9m2 /woh/»hq; _ (2.82 x 10 -6)(5.4ij x 10 26) ^ ~ '"oh + mh0, ~ 2.82 x I0"26 + 5.48 x 10"26 = 1.86 x 10 26 kg molecule-'
A = iar- = (5.03 x 10 1 m- -----— ■■——---- -
Thus, the fraction of collisions that are reactive at 300 K is
3.2.2 Transition State Theory
Consider the hi molecular reaction
in which the preexisting chemical bond, B—C, is broken and a new bond, A—B, is formed. Reactions in which both A and BC are molecules are not important at atmospheric temperatures; the amount of electron rearrangement needed produces a large barrier to reaction. On the other hand, reactions in which A is a free radical and BC is a molecule are very important in the atmosphere. In this case, formation of the A—B bond and cleavage of the B—C bond occur virtually simultaneously. Essentially, the electronic energy contained in the first bond is transferred to the second.
The first step in the above reaction is the formation of a transient complex, called the activated complex or transition state:
The transition state can dissociate back to the reactants
or to the new products:
Transition state theory (also known as activated-complex theory) assumes that the transition state is much more likely to decay back to the original reactants than proceed to the stable products; if this is the case, then first two reactions can be assumed to be in equilibrium. The reactive process can then be represented as
The rate of reaction is that at which ABC* passes to products (as a result of translational or vibrational motions along the reaction coordinate).
We will not go through the full derivation of the transition state theory, for which there are many excellent references [e.g., Laidler (1987), Pilling and Seakins (1995)]. The result is that the rate coefficient is expressed as where A' is the collision theory preexponential factor and AS is the change in entropy, ¿■(ABC*) - 5(A)
— S(BC), which is a measure of the molecular rearrangement involved in forming the transition state and can itself be a function of temperature. Complex transition states in which energy can be distributed over many states have large AS. In this case the excess energy is less likely to be funneled into the channel that causes the transition state to decompose back to the original reactants.
In many cases the preexponential factor can be considered to be independent of temperature, and the rate coefficient is written as k=Acx p(-JP) (3.10)
where A is determined experimentally rather than from theory. Equation (3.10) is called the Arrhenius form for k.
3.2.3 Potential Energy Surface for a Bimolecular Reaction
Consider the potential energy surface for the biomolecular reaction (most elementary reactions can be considered as reversible)
b as shown in Figure 3.1. As the two reactant molecules approach each other, the energy of the reaction system rises. A point is reached, denoted by ABC*, beyond which the energy starts to decrease again. ABC*, the activated complex, is a short-lived intermediate through which the reactants must pass if the encounter is to lead to reaction. By estimating the structure of this transition state the activation energy E may be estimated (Benson, 1976). This point is a saddle point in the potential energy surface of the system. Figure 3.1 shows the relationship between the energies of the process. The activation energy for the forward reaction is Ef\ that for the reverse reaction is Er. The enthalpy of reaction is AHr. Note that
The forward reaction (left to right) sketched in Figure 3.1 is exothermic, and AHr = //products — //reactants ¡s negative. The reverse reaction (right to left) is endothermic and must have an activation energy, Er, at least as large as AHr. It is customary to identify the activation energy E in (3.10) with Ef (or Er.)
The height of the energetic barrier E depends on the amount of electronic rearrangement going from reactants to transition state. Molecule-molecule reactions involve a high degree of electronic rearrangement and have large activation energy barriers. Such reactions are usually unimportant at atmospheric temperatures. In radical-molecule reactions the amount of electronic rearrangement is small, often just the transfer of an atom. Activation energy barriers are much lower, and these reactions occur readily at ambient temperatures. In radical-radical reactions typically a single bond is formed from the addition of the two radicals, electronic rearrangement is minimal, and there is
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