We will consider three types of chemical reaction:
First-order (unimolecular) A —> B + C Second-order (bimolecular) A + B —> C + D Third-order (termolecular) A + B+ M—>AB+M
The rate (in molecules cm 3 s ') of a first-order reaction is expressed as
where the first-order rate coefficient k\ has units of s_1 (reciprocal seconds).
Few reactions are truly first-order, in that they involve decomposition of a molecule without intervention of a second molecule. The classic example of a true first-order reaction is radioactive decay, such as 222Rn —> 218Po + a-particles. In the atmosphere, by far the most important class of first-order reactions is photodissociation reactions in which absorption of a photon of light (hv) by the molecule induces chemical change. Photodissociation, or photolysis, reactions are written as in which hv represents a photon of light of frequency v. In the photolysis of species A, the rate coefficient is customarily denoted by the symbol _/A.
Thermal decomposition of a molecule is often represented as first-order, but the energy required for decomposition is usually supplied through collision with another molecule. If the other molecule is an air molecule, it is denoted as M, and the actual reaction is A + M^B + C + M. Since M is at great excess relative to A, its concentration is constant, and the concentration of M can be implicitly included in the reaction rate coefficient; then, the reaction is written simply as A —> B + C.
The rate equation (3.1) can be integrated to give
Atmospheric Chemistry and Physics: From Air Pollution to Climate Change, Second Edition, by John H. Seinfeld and Spyros N. Pandis. Copyright (C) 2006 John Wiley & Sons, Inc.
Thus, specics A decays to Me of its initial concentration in time x = 1 ¡k\. This time is referred to as the e-folding time of the reaction, or the mean lifetime of A against this reaction.
The rate of a second-order, or bimolecular, reaction is
where the second-order rate coefficient k2 has units of cm3 molecule"1 s 1.
The termolecular reaction, which is written as A + B + M —♦ AB + M, actually does not take place as the result of the simultaneous collision of all three molecules A, B, and M. The probability of such an event happening is practically zero. Rather, what actually occurs is that molecules A and B collide to produce an energetic intermediate ABf (the dagger representing vibrational excitation):
In order for AB1 to proceed to the product AB, its excess energy must be removed through collision with another molecule denoted by M, to which the excess energy is transferred:
In the atmosphere, M is the background mixture of N2 and 02. Termolecular reactions are usually expressed as
We return to a derivation of the rate of termolecular reactions in Section 3,5.
Lifetime of a Species as a Result of a Chemical Reaction Let us determine the lifetime of a species undergoing a first-order reaction. When a species undergoes a first-order decay, its concentration as a function of time is given by (3.2)
A measure of the relative speed of a chemical reaction is the time required for A to decay to a certain fraction of its initial concentration. For example, the halflife, /1/2, of a reaction is the time needed for [A] — LA]o/2. A more commonly used measure of the speed of a reaction is derived from (3.2). The Lifetime, x, is the time at which [A]/[A]0 = e-1 = 0.368. Thus, T = \/k. Halflife and lifetime are related by tm=M (2)/k = 0.69/k
We will use lifetime i exclusively as a measure of the speed of a chemical reaction. When a substance participates in several chemical reactions, and we are interested in its lifetime as a result of reaction j, that lifetime is referred to as its lifetime against reaction i.
The concept of lifetime can be applied to reactions of any order. ¡Determine, for example, the lifetime of each of the reactants in the second-order reaction of nitric oxide and ozone:
NO + O3 NO2 + O2 ¿(298 K) = J.9 x 10"14cm3 molecule"1 s"1
The rate of the reaction is k [NO] ¡03 j and depends on the concentrations of both reactants. The lifetime of NO against this reaction is given by tno =
To calculate tno, it is necessary to specify the concentration of 03. This makes sense because the larger the 03 concentration, the shorter the lifetime of NO. At the Earth's surface at 298 K, if the Oi mixing ratio is 50ppb, then [O3] = (50 x 10"g)(2.5 x I019) = 1.25 x 1012 molecules cm"3. Then tNO =
(1.9 x (0~l4)(l .25 x I012) Conversely, the lifetime of 03 against this reaction is
and to calculate to3, a value of [NO] needs to be specified. For example, if the NO mixing ratio is lOppb, then the lifetime of 03 against this reaction is 3.5 mm.
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