Integrated Rate Laws Quiz

PDF page 683: The rate laws discussed thus far relate the rate and the concentrations of reactants. We can also determine a second form of each rate law that relates the concentrations of reactants and time. These are called integrated rate laws. We can use an integrated rate law to determine the amount of reactant or product present after a period of time or to estimate the time required for a reaction to proceed to a certain extent. For example, an integrated rate law is used to determine the length of time a radioactive material must be stored for its radioactivity to decay to a safe level. Using calculus, the differential rate law for a chemical reaction can be integrated with respect to time to give an equation that relates the amount of reactant or product present in a reaction mixture to the elapsed time of the reaction. This process can either be very straightforward or very complex, depending on the complexity of the differential rate law. For purposes of discussion, we will focus on the resulting integrated rate laws for first-, second-, and zero-order reactions.

PDF page 683: The rate laws discussed thus far relate the rate and the concentrations of reactants. We can also determine a second form of each rate law that relates the concentrations of reactants and time. These are called integrated rate laws. We can use an integrated rate law to determine the amount of reactant or product present after a period of time or to estimate the time required for a reaction to proceed to a certain extent. For example, an integrated rate law is used to determine the length of time a radioactive material must be stored for its radioactivity to decay to a safe level. Using calculus, the differential rate law for a chemical reaction can be integrated with respect to time to give an equation that relates the amount of reactant or product present in a reaction mixture to the elapsed time of the reaction. This process can either be very straightforward or very complex, depending on the complexity of the differential rate law. For purposes of discussion, we will focus on the resulting integrated rate laws for first-, second-, and zero-order reactions.

PDF page 684: In the next example exercise, a linear format for the integrated rate law will be convenient: ln[A] = (−k)(t) + ln[A] 0 t y = mx + b A plot of ln[A]t versus t for a first-order reaction is a straight line with a slope of −k and a y-intercept of ln[A]0. If a set of rate data are plotted in this fashion but do not result in a straight line, the reaction is not first order in A.

PDF page 684: In the next example exercise, a linear format for the integrated rate law will be convenient: ln[A] = (−k)(t) + ln[A] 0 t y = mx + b A plot of ln[A]t versus t for a first-order reaction is a straight line with a slope of −k and a y-intercept of ln[A]0. If a set of rate data are plotted in this fashion but do not result in a straight line, the reaction is not first order in A.

PDF page 685: The plot of ln[H2O2] versus time is linear, indicating that the reaction may be described by a first-order rate law. According to the linear format of the first-order integrated rate law, the rate constant is given by the negative of this plot’s slope.

PDF page 684: In the next example exercise, a linear format for the integrated rate law will be convenient: ln[A] = (−k)(t) + ln[A] 0 t y = mx + b A plot of ln[A]t versus t for a first-order reaction is a straight line with a slope of −k and a y-intercept of ln[A]0. If a set of rate data are plotted in this fashion but do not result in a straight line, the reaction is not first order in A.

PDF page 684: In the next example exercise, a linear format for the integrated rate law will be convenient: ln[A] = (−k)(t) + ln[A] 0 t y = mx + b A plot of ln[A]t versus t for a first-order reaction is a straight line with a slope of −k and a y-intercept of ln[A]0. If a set of rate data are plotted in this fashion but do not result in a straight line, the reaction is not first order in A.

PDF page 686: The equations that relate the concentrations of reactants and the rate constant of second-order reactions can be fairly complicated. To illustrate the point with minimal complexity, only the simplest second-order reactions will be described here, namely, those whose rates depend on the concentration of just one reactant. For these types of reactions, the differential rate law is written as: 2 rate = k[A] For these second-order reactions, the integrated rate law is: 1 1 = kt + [A] [A] 0 t where the terms in the equation have their usual meanings as defined earlier.

PDF page 686: The equations that relate the concentrations of reactants and the rate constant of second-order reactions can be fairly complicated. To illustrate the point with minimal complexity, only the simplest second-order reactions will be described here, namely, those whose rates depend on the concentration of just one reactant. For these types of reactions, the differential rate law is written as: 2 rate = k[A] For these second-order reactions, the integrated rate law is: 1 1 = kt + [A] [A] 0 t where the terms in the equation have their usual meanings as defined earlier.

PDF page 687: The integrated rate law for second-order reactions has the form of the equation of a straight line: 1 1 = kt + [A] [A] 0 t y = mx + b 1 1 A plot of versus t for a second-order reaction is a straight line with a slope of k and a y-intercept of . If the [A] [A] 0 t plot is not a straight line, then the reaction is not second order.

PDF page 693: 1 ⎛ 1 ⎞ [A] = −kt + [A] ln[A] = −kt + ln[A] = kt + integrated rate law 0 0

PDF page 687: The integrated rate law for second-order reactions has the form of the equation of a straight line: 1 1 = kt + [A] [A] 0 t y = mx + b 1 1 A plot of versus t for a second-order reaction is a straight line with a slope of k and a y-intercept of . If the [A] [A] 0 t plot is not a straight line, then the reaction is not second order.

PDF page 688: 1 versus According to the second-order integrated rate law, the rate constant is equal to the slope of the [A] t t plot. Using the data for t = 0 s and t = 6200 s, the rate constant is estimated as follows: −1 −1 (481 M − 100 M ) −1 −1

PDF page 689: For zero-order reactions, the differential rate law is: rate = k A zero-order reaction thus exhibits a constant reaction rate, regardless of the concentration of its reactant(s). This may seem counterintuitive, since the reaction rate certainly can’t be finite when the reactant concentration is zero. For purposes of this introductory text, it will suffice to note that zero-order kinetics are observed for some reactions only under certain specific conditions. These same reactions exhibit different kinetic behaviors when the specific conditions aren’t met, and for this reason the more prudent term pseudo-zero-order is sometimes used. The integrated rate law for a zero-order reaction is a linear function: [A] = −kt + [A] 0 t y = mx + b A plot of [A] versus t for a zero-order reaction is a straight line with a slope of −k and a y-intercept of [A]0. Figure 12.11 shows a plot of [NH3] versus t for the thermal decomposition of ammonia at the surface of two different heated solids. The decomposition reaction exhibits first-order behavior at a quartz (SiO2) surface, as suggested by the exponentially decaying plot of concentration versus time. On a tungsten surface, however, the plot is linear, indicating zero-order kinetics.

PDF page 689: For zero-order reactions, the differential rate law is: rate = k A zero-order reaction thus exhibits a constant reaction rate, regardless of the concentration of its reactant(s). This may seem counterintuitive, since the reaction rate certainly can’t be finite when the reactant concentration is zero. For purposes of this introductory text, it will suffice to note that zero-order kinetics are observed for some reactions only under certain specific conditions. These same reactions exhibit different kinetic behaviors when the specific conditions aren’t met, and for this reason the more prudent term pseudo-zero-order is sometimes used. The integrated rate law for a zero-order reaction is a linear function: [A] = −kt + [A] 0 t y = mx + b A plot of [A] versus t for a zero-order reaction is a straight line with a slope of −k and a y-intercept of [A]0. Figure 12.11 shows a plot of [NH3] versus t for the thermal decomposition of ammonia at the surface of two different heated solids. The decomposition reaction exhibits first-order behavior at a quartz (SiO2) surface, as suggested by the exponentially decaying plot of concentration versus time. On a tungsten surface, however, the plot is linear, indicating zero-order kinetics.