Integrated Rate Law Expresses the reactant concentrations as a function of time. aA → products Kinetics are first order in [A], and the rate law is Rate = k[A] Integrated first-order rate law is ln[A] = -kt + ln[A]0 Equation shows how concentration of A depends on time. If the initial concentration of A and the rate constant k are known, the concentration of A can be calculated at any time.

ln[A] = -kt + ln[A]0 Equation is of the form y = mx + b, where a plot of y versus x is a straight line with slope m and intercept b. y = ln[A] x = t m = -k b = ln[A]0 Thus, for a first-order reaction, plotting the natural logarithm of concentration vs. time always gives a straight line. For the reaction, aA → products the reaction is first order in A if a plot of ln[A] vs. t is a straight line.

ln[A] = -kt + ln[A]0 The integrated rate law for a first-order reaction also can be expressed in terms of a ratio of [A] and [A]0 as follows:

2N2O5(g) → 4NO2(g) + O2(g) Since the plot of ln[N2O5] vs. time is a straight line, it confirms that the reaction is first order in N2O5, since it follows the equation ln[N2O5] = -kt + ln[N2O5]0.

Half-Life of a First-Order Reaction Half-life = the time required for a reactant to reach half its original concentration. Designated by the symbol t1/2. General equation for the half-life of a first order reaction is (derivation in textbook (p. 542): Note for a first-order reaction, the half-life does not depend on concentration.

Second-Order Rate Laws General reaction: aA → products That is second order in A, the rate law is: Rate = k[A]2 The integrated second-order rate law has the form A plot of 1/[A] vs. t will produce a straight line with a slope equal to k.

Equation shows how [A] depends on time and can be used to calculate [A] at any time t, provided k and [A]0 are known. 2C4H6(g) →C8H12(g) – second-order since a plot of 1/[C4H6] vs. t produces a straight line. Expression for the half-life of a second order reaction:

Zero-Order Rate Laws The rate law for a zero-order reaction is: Rate = k[A]0 = k(1) = k For a zero-order reaction, the rate is constant. It does not change with concentration as it does for first-order or second-order reactions. Integrated rate law for a zero-order reaction is: [A] = -kt + [A]0 Plot of [A] vs. t gives a straight line. Half-Life equation: