## From Kinetics

The expressions for half-lives and lifetimes in Table 5.2 can be readily derived from the rate laws. For a first-order reaction of a pollutant species A, the rate law for the reaction

A -> Products is given by

Rearranging, this becomes -d[ A]

Integrating from time t = 0 when the initial concentration of A is [A]() to time t when the concentration is [A], one obtains

After one half-life (i.e., at / = t[/2) by definition

[A] = 0.5[A]0. Substituting into the integrated rate expression, one obtains

For second- and third-order reactions, if one assumes the concentrations of the reactants other than A are constant with time, the derivation is the same except that k is replaced by /c[B] (second order) or Jt[B][C] (third order).

In most practical situations, however, the concentration of at least one of the other reactants is not constant but changes with time due to reactions, fresh injections of pollutants, and so on. As a result, using half-lives (or lifetimes) of a pollutant with respect to second- or third-order reactions is an approximation that involves assumed constant concentrations of the other reactants. These half-lives for bimolecular and termolecular reactions are thus directly affected by the concentrations of the other reactant.

Derivation of the relationship between the rate constant k and the lifetime r follows that for tl/2, except that, from the definition of r, at t = t, [A] = [A]0/e.

and the oxidations of S02 and N02 via gas-phase OH reactions:

The reason for the pressure dependence of termolecular reactions can be seen by taking reaction (3) as an example. The exothermic bond formation between 0(3P) and 02 releases energy that must be removed to form a stable 03 molecule; if the energy remains as internal energy, the 03 will quickly fly apart to re-form O + 02. The third molecule, M, is any molecule that stabilizes the excited (03)* intermediate by colliding with it and removing some of its excess internal energy. Treating reaction (3) as an elementary reaction d[ 03]

Rate = MO][02][M] = + ——, dt one might expect the rate to increase with the concentration or pressure of the third body M. However, there clearly must be some limit since the rate cannot increase to infinity but only to some upper limit determined by how fast the two reactive species can combine chemically. As a result, one might intuitively expect the rates of reactions such as (3), (10), and (f f) to increase initially as the pressure of M is increased from zero and then to plateau at some limiting value at high pressures.

Let us take the reaction (10) of OH with S02 as an example of a termolecular reaction of atmospheric interest and examine how its pressure dependence is established. It is common in kinetic studies to follow the decay of one reactant in an excess of the second reactant. In the case of reaction (fO), the decay of OH is followed in the presence of excess S02 and the third body M, where M is an inert "bath" gas such as He,

Ar, or N2. Since it is assumed to be an elementary reaction, the rate law for reaction (10) can be written for low pressures:

If [M] is constant, and [M] can be combined to form an effective bimolecular rate constant, =

Since S02 is in great excess, its concentration does not change significantly even when all the OH has reacted and hence it remains approximately constant throughout the reaction at its initial value, [SO2]0. Rearranging the rate law and integrating from time t = 0 when the initial concentration of OH is [OH]0 to time t when the OH concentration is [OH], one obtains

Since the initial concentration of OH, [OH]0, is a constant, a plot of ln[OH] against reaction time t should be a straight line with slope or decay rate given by

A plot of these decay rates against [SO2]0 should thus be linear, with the slopes increasing with pressure since k ft depends on [M],

Figure 5.1 shows such a plot of the absolute values of the observed OH decay rates against [SO2]0 at total pressures of Ar from 50 to 402 Torr (Atkinson et al., 1976). As expected, the decay rates are linear with [SO2]0 and increase with the pressure of M.

To obtain the termolecular rate constant the effective bimolecular rate constant /eft = /c|"[M] is plotted in Fig. 5.2 as a function of total pressure (i.e., of [M]). As expected from the earlier discussion, /eft increases with [M] at low pressures but approaches a plateau at higher pressures.

Termolecular reactions can be treated, as a first approximation, as if they consist of several elementary steps, for example, for reaction (f0),

OH + S02 ^ HOSO*, (12, -12) HOSOf + M ^ H0S02 + M. (13)

HOSOf is the exited OH-SOz adduct that contains the excess internal energy from bond formation in (12), and H0S02 is the stabilized adduct resulting when some of this internal energy is removed by a collision with M.

If the system is treated as if the concentration of the energized adduct (HOSO*) remains constant with time, then its rates of formation and loss are equal. These rates can be written from Eqs. (12), (—12), and (13) since these are assumed to be elementary reactions. Thus

This is an example of the steady-state approximation, widely employed in gas-phase kinetics and mechanistic studies.

402 Torr 202 Torr

100 Torr

50 Torr

402 Torr 202 Torr

100 Torr

50 Torr

10"15[S02]0 (molecules cm"3)

FIGURE 5.1 Plots of the OH decay rates against the initial S02 concentration at total pressures of Ar from 50 to 402 Torr (adapted from Atkinson et al., 1976).

10"15[S02]0 (molecules cm"3)

FIGURE 5.1 Plots of the OH decay rates against the initial S02 concentration at total pressures of Ar from 50 to 402 Torr (adapted from Atkinson et al., 1976).

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