D[ A

c dt

d dt

For example, in the thermal oxidation of NO by oxygen,

two molecules of NO disappear for each molecule of

02 reacted, and the rate of loss of NO is twice that of 02:

1 d[NO] -d[02] f d[N02] Rate = - - —;— = -:- = + -

2 dt dt

2 dt

While this convention is now widely used, it was not in some early kinetic studies. Thus one must be careful to note exactly how the rate is defined so that the reported rate constants are interpreted and applied correctly. In systems of atmospheric interest, the rate law or rate expression for a reaction, either elementary or overall, is the equation expressing the dependence of the rate on the concentrations of reactants. in a few reactions (mainly those in solution), products may also appear in the rate law.

For the general overall reaction (5), the rate law has the form

Rate = jk[A]",[B]"[C]"[D]?, where, depending on the mechanism of the reaction, m, n, p, and q may be zero, integers, or fractions. As noted earlier, in most gas-phase atmospheric reactions, the exponents of the product concentration (i.e., p and q) are zero and the rate laws involve only the reactant species, ft is important to stress here that in contrast to elementary reactions, in overall reactions the exponents in the rate laws (e.g., m, n, p, and q) do not necessarily bear a relationship to the stoichiometric coefficients of the reaction (e.g., a, b, c, and d).

The importance of distinguishing between elementary and overall reactions comes in formulating rate laws. For elementary reactions only, the rate law may be written directly from the stoichiometric equation. Thus for the general elementary gas-phase reaction aA + bB -> cC + dD, Rate = Jt[A]fl[B]\

where (a + b) < 3 by definition of an elementary reaction. For example, the rate expression for the elementary reaction (4) is given by

The rate constant, k, is simply the constant of proportionality in the expression relating the rate of a reaction to the concentrations of reactants and/or products, each expressed with the appropriate exponent. The order of a reaction is defined as the sum of the exponents in the rate law. Thus reaction (4) is (1 + 1) = second order. The order with respect to each species appearing in the rate law is the exponent of the concentration of that species; thus reaction (4) is first order in both 03 and NO.

The basis of predicting rate laws for elementary reactions from the stoichiometric equation lies in the fact that they must occur during a single collision (although the probability of reaction during any one collision is equal to or less than unity). Thus doubling the concentration of 03 in reaction (4) will double the number of collisions per second of 03 with NO. Assuming the probability of reaction per collision remains constant, then the number of 03 and NO molecules reacting, and 02 and N02 formed per unit time (i.e., the rate), must double.

The thermal oxidation of NO by molecular oxygen, reaction (6), is another example where the stoichiome-try and the molecularity of the reaction are directly related, and the rate law is

Thus the rate is proportional to the first power of the oxygen concentration and the square of the nitric oxide concentration and the reaction order is f + 2 = 3. However, in the troposphere, the 02 concentration is always so large relative to NO that it is effectively constant and thus can be incorporated into the rate constant k™. The rate law is now written

Rate = ¿¡^[NO]2, and the reaction is referred to as pseudo-second-order. We adopt the convention of writing a third-order rate constant as km, and a pseudo-second-order rate constant as khl, as illustrated in the preceding equations.

The rate law and the reaction order can often be used to show that a reaction cannot be an elementary reaction since, in the latter case, the exponents must be integers and the overall reaction order must be < 3. However, it should be noted that these kinetic parameters cannot be used to confirm that a particular reaction is elementary; they can only indicate that the kinetic data do not rule out the possibility that the reaction is elementary.

In gas-phase reactions, concentrations are usually expressed in molecules cm-3 and time in seconds, the convention we employ in this book. Thus the units of k are as follows: first order, s-1; second order, cm3 molecule-1 s-1; third order, cm6 molecule-2 s-1.

Concentrations of gaseous pollutants are often expressed in terms of parts per million (ppm) by volume, and time is expressed in minutes. Use of these concentration units must be reflected in the units used for the rate constants as well; for example, second-order rate constants are in units of ppm-1 min-1. Occasionally, gas concentrations are given in units of mol L-1 or in units of pressure such as Torr, atmospheres, or Pascals; these can be converted to the more conventional units

TABLE 5.1 Some Common Conversion Factors for Gas-Phase Reactions

Concentrations" 1 mol L-1 = 6.02 X 1020 molecules cm-' 1 ppm = 2.46 X 1013 molecules cm-3 1 ppb = 2.46 x 1010 molecules cm-3 1 ppt = 2.46 X 107 molecules cm- 3 1 atm = 760 Torr = 4.09 X 10-2 mol L- 1

= 2.46 X 1019 molecules cm-3

Second-order rate constants cm' molecule-'s- ' x 6.02 x 1020 = L mol-'s- ' ppm-1 min- 1 X 4.08 X 105 = L mol- 1 s- 1 ppm-1 min-1 X 6.77 X 10-16 = cm' molecule-1 s-1 atm-1 s-1 X 4.06 X 10-20 = cm3 molecule-1 s-1

Third-order rate constants cm6 molecule-2 s-1 x 3.63 x 1041 = L2 mol-2 s-1 ppm-2 min-1 X 9.97 X 1012 = L2 mol-2 s-1 ppm-2 min-1 X 2.75 X 10 - 29 = cm6 molecule-2 s-1

" The concentrations ppm, ppb, and ppt are relative to air at 1 atm and 25°C, where 1 atm = 760 Torr total pressure.

in tropospheric chemistry using the ideal gas law. Table 5.1 gives some common conversion factors for gas-phase concentrations and rate constants at f atm pressure (760 Torr total pressure) and 25°C.

For solution-phase reactions, we use concentration units of mol L 1, with units for the corresponding rate constants of L mol-1 s- 1 (second order) and L2 mol-2 s-1 (third order).

c. Half-Lives and Lifetimes

A rate constant is a quantitative measure of how fast reactions proceed and therefore is an indicator of how long a given set of reactants will survive in the atmosphere under a particular set of reactant concentrations. However, the rate constant per se is not a parameter that by itself is readily related to the average length of time a species will survive in the atmosphere before reacting. More intuitively meaningful parameters are the half-life (il/2) or the natural lifetime (r), the latter usually referred to simply as "lifetime," of a pollutant with respect to reaction with a labile species such as OH or N03 radicals.

The half-life (il/2) is defined as the time required for the concentration of a reactant to fall to one-half of its initial value, whereas the lifetime is defined as the time it takes for the reactant concentration to fall to \/e of its initial value (e is the base of natural logarithms, 2.718). Both t,/2 and r are directly related to the rate constant and to the concentrations of any other reactants involved in the reactions. These relationships are given in general form in Table 5.2 for first-, second-, and third-order reactions and are derived in Box 5.1.

TABLE 5.2 Relationships between the Rate Constant, Half-Lives, and Lifetimes for First-, Second-, and Third-Order Reactions

Reaction order

Reaction

Half-life of A

Lifetime of A

First

Second

Third

Products tA '1/2

A relevant example is the use of lifetimes to characterize the reactivity of organics. Compressed natural gas (CNG), for example, is a widely used fuel whose major component is methane, CH4. The only known significant chemical loss process for CH4 is reaction with OH:

Taking a typical average, daytime OH concentration of 1 X 106 radicals cm"3, the lifetime of CH4 with respect to this removal process is

0 0

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