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Integrated Rate Law

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

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Integrated Rate Law

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  1. 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.

  2. 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 = tm = -kb = 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.

  3. 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:

  4. 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.

  5. 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.

  6. 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.

  7. 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:

  8. 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:

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