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First-Order Kinetics

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According to the law of mass action, the velocity of a chemical reaction is proportional to the product of the active masses (concentrations) of the reactants. In a monomolecular reaction, i.e., one in which only a single molecular species reacts, the velocity of the reaction is proportional to the concentration of the unreacted substance (C). The change in concentration (dC) over a time interval (dT) is the velocity of the reaction (dC/dT) and is proportional to C. For infinitely small changes of concentration over infinitely small periods of time, the reaction velocity can be written in the form of a differential equation: -dC/dt=kC. Here, dC/dt is the reaction velocity, C is concentration, and k is the constant of proportionality, or monomolecular velocity constant, which uniquely characterizes the reaction. The minus sign indicates that the velocity decreases with the passage of time, as the concentration of unreacted substance decreases; a plot of C against time would yield a curve of progressively decreasing slope. The mechanisms, the kinetics, described by the differential equation are termed first order kinetics because - although the exponent is not written - concentration (C) is raised to only the first power (C1).The differential equation above may be integrated and rearranged to yield: ln(C/C0)= k t, where ln indicates use of the natural logarithm, to the base e; C0 is the concentration of unreacted substance at the beginning of an observation period; t is the duration of the observation period; and k is the familiar proportionality or velocity constant. The units of k are independent of the units in which C is expressed; indeed, since a logarithm is dimensionless, and t has the dimension of time, the integrated equation balances, dimensionally, because k has the dimension of reciprocal time, t-1. Notice that for observation periods of equal length, the ratio C/C0 is always the same; after equal intervals, the final concentration is a constant fraction of the starting concentration, or, in equal time intervals, constant fractions of the starting concentration are lost, even though absolute decreases in concentration become progressively less as time passes and C becomes smaller and smaller. Let t1/2 represent the length of time required for C0 to be halved, so that C=0.5 C0. Then, substituting in the integrated equation above, ln 0.5 = -kt1/2, or, since -0.693 is the natural logarithm of 0.5: -0.693 = kt1/2. Multiplying both sides of the equation by -1 yields 0.693 = kt1/2 or 0.693/k = t1/2: the natural logarithm of 2 (0.693) divided by the monomolecular velocity constant yields the time required for the concentration to be halved, the "half-life" or "half-time" of the reaction. Since ln (C/C0) may be rewritten (lnC - lnC0), the integrated equation may be rewritten and given the form of a linear equation: ln C = ln C0 -kt. The existence of a monomolecular reaction can be established by plotting ln C, for unreacted material, against t and finding the relationship to be linear; the slope of the line is the original proportionality or velocity constant, and the intercept of the line with the ordinate is the natural logarithm of the original concentration of unreacted material. Since natural logarithms have a fixed relationship to common logarithms, i.e., logarithms to the base 10 (lnX =2.303 log X), one may write: 2.303 log C =2.303 log C - kt. When common logarithms of C are plotted against t, a first order reaction yields a straight line with a slope of k/2.303, and anintercept which is the common logarithm of C0.More...

 

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