Chemical kinetics is concerned with the speed or velocity of reactions. Many reactions have rates that at a given temperature are proportional to the concentration of one, two, or more of the reactants raised to a small integral power. For example, if a reaction is considered in which A, B, and C are possible reactants, then the rate equations that express the concentration dependence of the reaction rate may take one of the following forms, among others:
Rate = kCa first order
Rate = kC2a or kCaCb second order
Rate = kC\ or kC\Cb or kCaCbCc third order where Ca, Ch, and Cc represent the concentrations of reactants A, B, and C, respectively- Reactions that proceed according to such simple expressions are said to be reactions of the first, second, or third order as indicated, with the order of the reaction being defined as the sum of the exponents of the concentration terms in the reaction equations. Not all reactions have such simple rate equations. Some involve concentrations raised to a fractional power, while others consist of more complex algebraic expressions. Attention must be paid to the units of the reaction rate constant k. They are different for the different reaction orders and depend on the concentration units used.
Environmental engineers and scientists deal with many reactions that proceed slowly and require rate expressions so that the reactions can be dealt with on a practical basis. This is true for biotransformation reactions, microbial growth and decay, aeration, radioactive decay, disinfection, chemical hydrolysis, and oxidation and reduction. In general, first-order reactions are the most common, although reactions of other orders or of a more complex nature are sometimes involved. Additional detail is given in Sec. 5.34.
A zero-order reaction is one in which the rate of reaction is independent of concentration:
where C is the concentration of reactant and k is the rate constant in units of concentration/time. Many biologically induced reactions, particularly those involving growth on simple, soluble substrates, appear to occur in a linear manner over fairly large ranges of concentrations. That is, a plot of concentration versus time yields a straight line. These are classified as zero-order reactions. Examples of environmental interest include the oxidation of ammonia to nitrite and the oxidation of glucose by aerobic bacteria. All such biological reactions, however, become slower as the substrate concentration approaches zero, as will be discussed under Enzyme Reactions.
The decomposition of a radioactive element is the simplest example of a true first-order reaction. In such a reaction the rate of decomposition is directly proportional to the amount of undecayed material and may be expressed mathematically as
where the minus sign indicates a loss of material with time, C is its concentration, and k is the rate constant for the reaction and has units of reciprocal time.
If the initial concentration, at time t = 0, is C0, and if at some later time t the concentration has fallen to C, the integration of Eq. (3.47) gives
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