## Kinetics Of Thermal Processes

Most chemical reactions are kjnetjeally complex. That is, they take place by means of a series of two or more consecutive steps rather than by means of a single encounter of the reacting species. Nevertheless, the overall reaction between, say, two reactants A and B can be represented by the generalized equation aA + bB ^ cC + dD + ••• (4-15)

where the stoichiometry of the reaction is given by the coefficients a, b, c, d, ... The rate, v, is then given by the differential equation

a dt b dt c dt d dt where nA is the number of moles of A, nB is the number of moles of B, and so on, and dnA/dt is the change in the number of moles of A per unit time. For reactions in which the volume of the system (V) is independent of time, the molar concentration of A is [A] = nA/V and d[A] = VdnA. Similarly, d[B] = VdnB, d[C] = VdnC, and d[D] = VdnD. It follows that the rate of the reaction per unit volume, r, usually referred to simply as the rate of the reaction, is

In certain cases it is convenient to express the rate with respect to the rate of change in concentration of a single reactant or product:

rA = rate of change of A = -*!£, rc = rate of change of C = df] ,etc.

dt dt

However, it is important to remember that in general rA = rc, although the two rates are related by the proportionality factor c/a.

The expression for the experimentally determined rate of reaction in terms of the composition of the system can be quite involved. However, in many cases at constant temperature it is of the form r = k[Af[Bf [C]7 [Df • • • [Xf • • • (4-19)

where k is the temperature-dependent rate constant and X is neither a reactant nor a product—that is, it is a catalyst. The overall order of the reaction is the sum of the exponents a + p + 7 + 8 + ••• + X + •••, and the order with respect to A is a, etc. For kinetically complex reactions there is in general no relationship between the exponents a, p, 7, and8 in equation (4-19) and the coefficients a, b, c, and d in the stoichiometric equation (4-15).

A special situation for equation (4-19) is the kinetically simple reaction that involves only the single step of reactants coming together. In this case the rate of the reaction at a given temperature is proportional only to the concentrations of the reacting species—that is, it does not depend on the nature or concentrations of the products of the reaction. The overall order of the reaction is then the number of reactant molecules (called the molecularity of the reaction) that must come together in a single encounter for the reaction to occur. Such reactions are called unimolecular, bimolecular, or trimolecular, depending on the number of reactant molecules; this number can be one, two, or three, respectively.12 Table 4-4 gives the rate expressions for the various possibilities of kinetically simple reactions.

The integrated solutions of the corresponding differential equations [defined by equation (4-17)] for the rates of these kinetically simple reactions, leading to reactant concentrations as a function of reaction time t, can be obtained by standard integration techniques.13 These integrated forms are used to obtain reaction rate constants from experimental time-dependent concentration data.

A chemically complex reaction is the combination of two or more kinetic-ally simple steps, giving the mechanism of the reaction. We will encounter complex reaction mechanisms in Chapter 5 in connection with many physical and chemical interactions occurring in the troposphere and stratosphere. Simple differential equations can be written for each kinetically simple step in a mechanism; however, integration of the new combined differential equation becomes difficult and in most cases impossible by the usual analytical techniques, since products of one kinetically simple step generally become reactants in one or more other steps in the mechanism.14 Numerical methods

12The probability that more than three molecules may come together in a single collision is so small that such events are not considered in kinetic treatments.

13See C. Capellos and B. H. J. Bielski, Kinetic Systems. Wiley-Interscience, New York, 1972. Two examples: For the unimolecular (first-order) reaction A ^ product, we have

and for the bimolecular (second-order) reaction A + B -i product, we write k =_1_ln IAI0IBI (4-21)

14Examples of some solutions of complex reactions are given by Capellos and Bielski (see note 13).

TABLE 4-4

Kinetically Simple Rates of Reaction

I. Unimolecular (first-order) A-> products r = k[A]

II. Bimolecular (second-order)

III. Trimolecular (third-order)

2A + B —^ products r = ¿[A]2 [B] A + B + C -¿^ products r = k [A][B][C]

of integration are now possible for very complex mechanisms utilizing the tremendous computing power of present-day digital computers. Of course, justification of a mechanism (or model) by computer simulation of this type requires good rate constants and spatial and temporal (space and time) behavior of all pertinent species.

Two time quantities are sometimes used to describe the extent of a chemical reaction. One is the half-life, t1/2, which is the time required for the reaction to be half-way completed; the other is the mean-life, t, which is the average of the lifetimes of all the reacting molecules. In general t1/2 and t depend on the concentrations of the species involved in the reaction. A unique case (and one for which t1/2 and t are frequently used) is that of a first-order reaction A ^ products (such as radioactive decay; see Chapter 13), where it can be shown that t1/2 and t are independent of concentration:

The mean-life t for a first-order reaction is the time required for the concentration of the reactant to decrease to 1/e (1/2.718) of its initial value, in contrast to t1/2, which is the time required for the concentration of the reactant to decrease to half of its initial value.

The dependence of the rate on temperature is embodied in the rate constant k. The most useful relationship expressing this dependence is the empirical Arrhenius equation k = Ae_E*/RT (4-24)

In this treatment the preexponential factor A and the activation energy Ea are assumed to be temperature independent, so that from equation (4-25) a plot of ln k vs 1/T should be linear, with a slope equal to —Ea/R. This behavior is found to be the case for many reactions, and this method involving the plot of ln k vs 1/T is the most common one for experimentally determining Ea. However, Arrhenius behavior will not necessarily be followed for kinetically complex reactions, and indeed reactions with very marked deviations from that predicted by equation (4-25) are encountered.

The simple collision theory of chemical kinetics, applicable to kinetically simple bimolecular gas-phase reactions between two unlike hard-sphere species 1 and 2, considers the rate of reaction r ( = kc1c2) to be proportional to the frequency of bimolecular collisions Z12. The proportionality constant equating r and Z12 includes the Boltzmann factor [equation (4-2)], which in this case is the fraction of molecules in two dimensions defined by the trajectories of motion of two colliding particles with energies equal to or greater than a potential energy barrier Ec. Note that this is the same form of exponential term found in the Arrhenius equation (4-24); the difference between the two is that Ea is the experimentally determined empirical activation energy, whereas Ec is associated with a specific potential energy barrier that must be overcome for the kinetically simple reaction to occur. The rate is also proportional to a temperature-independent steric factor p, which allows for the possibility that not all colliding particles with sufficient energy to overcome the potential barrier do actually react. (For example, the molecules might not react unless they are in a particular orientation.) Thus, r = pZi2e-Ec/RT = kcic2 (4-26)

c1 c2

where Z'12 is the specific collision frequency and is proportional to T1/2. The preexponential term in the Arrhenius equation (4-24) is therefore slightly temperature dependent, and this is in fact observed for reactions with low or zero activation energies, although any small temperature effect in this term is completely masked by the exponential term for reactions with large activation energies. For small species such as atoms and diatomic molecules, Z12 « 2 x 10-10 cm3 molecule-1s-1. If the reaction goes at every collision (i.e., Ec = 0, p = 1), then k = Z'12 and is called the collisional rate constant.

A kinetically simple trjmolecular gas-phase reaction requires the simultaneous encounter of three molecules. Collisions of this type (triple collisions) are very rare in comparison to binary collisions and are usually encountered only in two-body gas-phase combination reactions where simultaneous conservation of energy and momentum requires the presence of a third body. Trimole-cular atom or small-radical combination processes in the presence of inert third-body molecules typically have triple-collision rate constants of the order of 10~32 cm6 molecule-2 s-1. We will see several reactions of this type in Chapter 5.

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