Chapter 14

Complete the following statements relating to the production of ammonia by the Haber process, for which the overall reaction is \(\mathrm{N}_{2}(\mathrm{~g})+3 \mathrm{H}_{2}(\mathrm{~g}) \longrightarrow 2 \mathrm{NH}_{3}(\mathrm{~g}) \cdot\) (a) The rate of consumption of \(\mathrm{N}_{2}\) is ______ times the rate of consumption of \(\mathrm{H}_{2}\). (b) The rate of formation of \(\mathrm{NH}_{3}\) is _____ times the _______times the rate of consumption of \(\mathrm{N}_{2}\).

Complete the following statements for the reaction \(6 \mathrm{Li}(\mathrm{s})+\mathrm{N}_{2}(\mathrm{~g}) \rightarrow 2 \mathrm{Li}_{3} \mathrm{~N}(\mathrm{~s})\). The rate of consumption of \(\mathrm{N}_{2}\) is _______ times the rate of formation of \(\mathrm{Li}_{3} \mathrm{~N}\). (b) The rate of formation of \(\mathrm{Li}_{3} \mathrm{~N}\) is _____times the rate of consumption of Li. (c) The rate of consumption of \(\mathrm{N}_{2}\) is ________ times the rate of consumption of Li.

(a) In the reaction \(3 \mathrm{ClO}^{-}\)(aq) \(\longrightarrow 2 \mathrm{Cl}^{-}\)(aq) \(+\mathrm{ClO}_{3}{\underline{\phantom{xx}}}^{-}\)(aq), the rate of formation of \(\mathrm{Cl}^{-}\)is \(3.6 \mathrm{~mol} \cdot \mathrm{L}^{-1} \cdot \mathrm{min}^{-1}\). What is the rate of reaction of \(\mathrm{ClO}^{-}\)? (b) What is the unique rate of the reaction?

(a) Nitrogen dioxide, \(\mathrm{NO}_{2}\), decomposes at \(6.5\) \(\mathrm{mmol} \cdot \mathrm{L}^{-1} \cdot \mathrm{s}^{-1}\) by the reaction \(2 \mathrm{NO}_{2}(\mathrm{~g}) \longrightarrow 2 \mathrm{NO}(\mathrm{g})+\mathrm{O}_{2}(\mathrm{~g})\). Determine the rate of formation of \(\mathrm{O}_{2}\). (b) What is the unique rate of the reaction?

Manganate ions, \(\mathrm{MnO}_{4}^{2-}\), react at \(2.0 \mathrm{~mol} \cdot \mathrm{L}^{-1} \cdot \mathrm{min}^{-1}\) in acidic solution to form permanganate ions and manganese(IV) oxide: \(3 \mathrm{MnO}_{4}{\underline{\phantom{xx}}}^{2-}(\mathrm{aq})+4 \mathrm{H}^{+}(\mathrm{aq}) \rightarrow 2 \mathrm{MnO}_{4}{\underline{\phantom{xx}}}^{-}(\mathrm{aq})+\mathrm{MnO}_{2}(\mathrm{~s})+\) \(2 \mathrm{H}_{2} \mathrm{O}(\mathrm{l})\). (a) What is the rate of formation of permanganate ions? (b) What is the rate of reaction of \(\mathrm{H}^{+}(\mathrm{aq})\) ? (c) What is the unique rate of the reaction?

The decomposition of gaseous hydrogen iodide, \(2 \mathrm{HI}(\mathrm{g}) \rightarrow \mathrm{H}_{2}(\mathrm{~g})+\mathrm{I}_{2}(\mathrm{~g})\), gives the data shown here for \(700 . \mathrm{K}\). $$ \begin{array}{lcccccc} \text { Time }(\mathrm{s}) & 0 . & 1000 . & 2000 . & 3000 . & 4000 . & 5000 . \\\ {[\mathrm{HI}]\left(\mathrm{mmol} \cdot \mathrm{L}^{-1}\right)} & 10.0 & 4.4 & 2.8 & 2.1 & 1.6 & 1.3 \end{array} $$ (a) Use a graphing calculator or standard graphing software, such as that on the Web site for this book, to plot the concentration of HI as a function of time. (b) Estimate the rate of decomposition of HI at each time. (c) Plot the concentrations of \(\mathrm{H}_{2}\) and \(\mathrm{I}_{2}\) as a function of time on the same graph.

Express the units for rate constants when the concentrations are in moles per liter and time is in seconds for (a) zero-order reactions; (b) first-order reactions; (c) second-order reactions.

Because partial pressures are proportional to concentrations, rate laws for gas-phase reactions can also be expressed in terms of partial pressures, for instance, as Rate \(=\) \(k P_{\mathrm{X}}\) for a first-order reaction of a gas \(\mathrm{X}\). What are the units for the rate constants when partial pressures are expressed in torr and time is expressed in seconds for (a) zero-order reactions; (b) first-order reactions; (c) second-order reactions?

In the reaction \(\mathrm{CH}_{3} \mathrm{Br}(\mathrm{aq})+\mathrm{OH}^{-}(\mathrm{aq}) \rightarrow \mathrm{CH}_{3} \mathrm{OH}(\mathrm{aq})+\) \(\mathrm{Br}^{-}(\mathrm{aq})\), when the \(\mathrm{OH}^{-}\)concentration alone was doubled, the rate doubled; when the \(\mathrm{CH}_{3} \mathrm{Br}\) concentration alone was increased by a factor of \(1.2\), the rate increased by a factor of \(1.2\). Write the rate law for the reaction.

In the brewing of beer, ethanal, which smells like green apples, is an intermediate in the formation of ethanol. Ethanal decomposes in the following first-order reaction: \(\mathrm{CH}_{3} \mathrm{CHO}(\mathrm{g}) \longrightarrow \mathrm{CH}_{4}(\mathrm{~g})+\) \(\mathrm{CO}(\mathrm{g})\). At an elevated temperature the rate constant for the decomposition is \(1.5 \times 10^{-3} \mathrm{~s}^{-1}\). What concentration of ethanal, which had an initial concentration of \(0.120 \mathrm{~mol} \cdot \mathrm{L}^{-1}\), remains \(20.0 \mathrm{~min}\) after the start of its decomposition at this temperature?

Determine the rate constant for each of the following firstorder reactions, in each case expressed for the rate of loss of \(A\) : (a) \(\mathrm{A} \longrightarrow \mathrm{B}\), given that the concentration of \(\mathrm{A}\) decreases to one-half its initial value in \(1000 . \mathrm{s} ;\) (b) \(\mathrm{A} \longrightarrow \mathrm{B}\), given that the concentration of A decreases from \(0.67 \mathrm{~mol} \cdot \mathrm{L}^{-1}\) to \(0.53 \mathrm{~mol} \cdot \mathrm{L}^{-1}\) in \(25 \mathrm{~s}\); (c) \(2 \mathrm{~A} \longrightarrow \mathrm{B}+\mathrm{C}\), given that \([\mathrm{A}]_{0}=0.153 \mathrm{~mol} \cdot \mathrm{L}^{-1}\) and that after \(115 \mathrm{~s}\) the concentration of \(B\) rises to \(0.034 \mathrm{~mol} \cdot \mathrm{L}^{-1}\).

Determine the rate constant for each of the following firstorder reactions: (a) \(2 \mathrm{~A} \rightarrow \mathrm{B}+\mathrm{C}\), given that the concentration of A decreases to one-fourth its initial value in \(61 \mathrm{~min}\); (b) \(2 \mathrm{~A} \longrightarrow\) \(\mathrm{B}+\mathrm{C}\), given that \([\mathrm{A}]_{0}=0.021 \mathrm{~mol} \cdot \mathrm{L}^{-1}\) and that after \(45 \mathrm{~s}\) the concentration of \(\mathrm{B}\) increases to \(0.00125 \mathrm{~mol} \cdot \mathrm{L}^{-1}\); (c) \(2 \mathrm{~A} \rightarrow 3 \mathrm{~B}+\) \(\mathrm{C}\), given that \([\mathrm{A}]_{0}=0.060 \mathrm{~mol} \cdot \mathrm{L}^{-1}\) and that after \(10.8\) min the concentration of \(\mathrm{B}\) rises to \(0.040 \mathrm{~mol} \cdot \mathrm{L}^{-1}\). In each case, write the rate law for the rate of loss of \(\mathrm{A}\).

Dinitrogen pentoxide, \(\mathrm{N}_{2} \mathrm{O}_{5}\), decomposes by first- order kinetics with a rate constant of \(3.7 \times 10^{-5} \mathrm{~s}^{-1}\) at \(298 \mathrm{~K}\). (a) What is the half-life (in hours) for the decomposition of \(\mathrm{N}_{2} \mathrm{O}_{5}\) at \(298 \mathrm{~K}\) ? (b) If \(\left[\mathrm{N}_{2} \mathrm{O}_{5}\right]_{0}=0.0567 \mathrm{~mol} \cdot \mathrm{L}^{-1}\), what will be the concentration of \(\mathrm{N}_{2} \mathrm{O}_{5}\) after \(3.5 \mathrm{~h}\) ? (c) How much time (in minutes) will elapse before the \(\mathrm{N}_{2} \mathrm{O}_{5}\) concentration decreases from \(0.0567 \mathrm{~mol} \cdot \mathrm{L}^{-1}\) to \(0.0135 \mathrm{~mol} \cdot \mathrm{L}^{-1}\) ?

Dinitrogen pentoxide, \(\mathrm{N}_{2} \mathrm{O}_{5}\), decomposes by first- order kinetics with a rate constant of \(0.15 \mathrm{~s}^{-1}\) at \(353 \mathrm{~K}\). (a) What is the half-life (in seconds) for the decomposition of \(\mathrm{N}_{2} \mathrm{O}_{5}\) at \(353 \mathrm{~K}\) ? (b) If \(\left[\mathrm{N}_{2} \mathrm{O}_{5}\right]_{0}=0.0567 \mathrm{~mol} \cdot \mathrm{L}^{-1}\), what will be the concentration of \(\mathrm{N}_{2} \mathrm{O}_{5}\) after \(2.0 \mathrm{~s}\) ? (c) How much time (in minutes) will elapse before the \(\mathrm{N}_{2} \mathrm{O}_{5}\) concentration decreases from \(0.0567 \mathrm{~mol} \cdot \mathrm{L}^{-1}\) to \(0.0135 \mathrm{~mol} \cdot \mathrm{L}^{-1}\) ?

The half-life for the first-order decomposition of A is \(355 \mathrm{~s}\). How much time must elapse for the concentration of A to decrease to (a) \(\frac{1}{8}[\mathrm{~A}]_{0} ;\) (b) one-fourth of its initial concentration; (c) \(15 \%\) of its initial concentration; (d) one-ninth of its initial concentration?

The first-order rate constant for the photodissociation of \(\mathrm{A}\) is \(6.85 \times 10^{-2} \mathrm{~min}^{-1}\). Calculate the time needed for the concentration of A to decrease to (a) \(\frac{1}{8}[\mathrm{~A}]_{0} ;\) (b) \(10 . \%\) of its initial concentration; (c) one-third of its initial concentration.

The data below were collected for the reaction \(2 \mathrm{HI}(\mathrm{g}) \longrightarrow \mathrm{H}_{2}(\mathrm{~g})+\mathrm{I}_{2}(\mathrm{~g})\) at \(580 \mathrm{~K}\). $$ \begin{array}{lccccc} \text { Time }(\mathrm{s}) & 0 & 1000 . & 2000 . & 3000 . & 4000 . \\ {[\mathrm{HI}]\left(\mathrm{mol} \cdot \mathrm{L}^{-1}\right)} & 1.0 & 0.11 & 0.061 & 0.041 & 0.031 \end{array} $$ (a) Using a graphing calculator or graphing software, such as that on the Web site for this book, plot the data in an appropriate fashion to determine the order of the reaction. (b) From the graph, determine the rate constant for (i) the rate law for the loss of HI and (ii) the unique rate law.

The data below were collected for the reaction \(\mathrm{H}_{2}(\mathrm{~g})+\mathrm{I}_{2}(\mathrm{~g}) \longrightarrow 2 \mathrm{HI}(\mathrm{g})\) at \(780 \mathrm{~K}\). (a) Using a graphing calculator or graphing software, such as that on the Web site for this book, plot the data in an appropriate fashion to determine the order of the reaction. (b) From the graph, determine the rate constant for the rate of consumption of \(\mathrm{I}_{2}\). $$ \begin{array}{lccccc} \text { Time }(\mathrm{s}) & 0 & 1.0 & 2.0 & 3.0 & 4.0 \\ {\left[\mathrm{I}_{2}\right]\left(\mathrm{mmol} \cdot \mathrm{L}^{-1}\right)} & 1.00 & 0.43 & 0.27 & 0.20 & 0.16 \end{array} $$

The half-life for the second-order reaction of a substance A is \(50.5 \mathrm{~s}\) when \([\mathrm{A}]_{0}=0.84 \mathrm{~mol} \cdot \mathrm{L}^{-1}\). Calculate the time needed for the concentration of A to decrease to (a) one- sixteenth; (b) onefourth; (c) one-fifth of its original value.

Sulfuryl chloride, \(\mathrm{SO}_{2} \mathrm{Cl}_{2}\), decomposes by first- order kinetics, and \(k=2.81 \times 10^{-3} \mathrm{~min}^{-1}\) at a certain temperature. (a) Determine the half-life for the reaction. (b) Determine the time needed for the concentration of \(\mathrm{SO}_{2} \mathrm{Cl}_{2}\) to decrease to \(10 \%\) of its initial concentration. (c) If \(14.0 \mathrm{~g}\) of \(\mathrm{SO}_{2} \mathrm{Cl}_{2}\) is sealed in a \(2500 .-\mathrm{L}\) reaction vessel and heated to the specified temperature, what mass will remain after \(1.5 \mathrm{~h}\) ?

Ethane, \(\mathrm{C}_{2} \mathrm{H}_{6}\), forms \(\cdot \mathrm{CH}_{3}\) radicals at \(700 .^{\circ} \mathrm{C}\) in a firstorder reaction, for which \(k=1.98 \mathrm{~h}^{-1}\). (a) What is the half-life for the reaction? (b) Calculate the time needed for the amount of ethane to fall from \(1.15 \times 10^{-3} \mathrm{~mol}\) to \(2.35 \times 10^{-4} \mathrm{~mol}\) in a 500.-mL reaction vessel at \(700 .{ }^{\circ} \mathrm{C}\). (c) How much of a 6.88-mg sample of ethane in a \(500 .-\mathrm{mL}\) reaction vessel at \(700 .{ }^{\circ} \mathrm{C}\) will remain after \(45 \mathrm{~min}\) ?

The second-order rate constant for the decomposition of \(\mathrm{NO}_{2}\) (to \(\mathrm{NO}\) and \(\mathrm{O}_{2}\) ) at \(573 \mathrm{~K}\) is \(0.54 \mathrm{~L} \cdot \mathrm{mol}^{-1} \cdot \mathrm{s}^{-1}\). Calculate the time for an initial \(\mathrm{NO}_{2}\) concentration of \(0.20 \mathrm{~mol} \cdot \mathrm{L}^{-1}\) to decrease to (a) one-half; (b) one-sixteenth; (c) one-ninth of its initial concentration.

Derive an expression for the half-life of the reactant A that decays by a third-order reaction with rate constant \(k\).

Derive an expression for the half-life of the reactant A that decays by an \(n\) th-order reaction (with \(n>1\) ) with rate constant \(k\). Reaction Mechanisms

Write the overall reaction for the mechanism proposed below and identify any reaction intermediates. $$ \begin{aligned} &\text { Step } 1 \mathrm{AC}+\mathrm{B} \longrightarrow \mathrm{AB}+\mathrm{C} \\ &\text { Step } 2 \mathrm{AC}+\mathrm{AB} \longrightarrow \mathrm{A}_{2} \mathrm{~B}+\mathrm{C} \end{aligned} $$

Write the overall reaction for the mechanism proposed below and identify any reaction intermediates. Step \(1 \mathrm{C}_{4} \mathrm{H}_{9} \mathrm{Br} \longrightarrow \mathrm{C}_{4} \mathrm{H}_{9}{\underline{\phantom{xx}}}^{+}+\mathrm{Br}^{-}\) Step \(2 \mathrm{C}_{4} \mathrm{H}_{9}{\underline{\phantom{xx}}}^{+}+\mathrm{H}_{2} \mathrm{O} \longrightarrow \mathrm{C}_{4} \mathrm{H}_{9} \mathrm{OH}_{2}{\underline{\phantom{xx}}}^{+}\) Step \(3 \mathrm{C}_{4} \mathrm{H}_{9} \mathrm{OH}_{2}^{+}+\mathrm{H}_{2} \mathrm{O} \longrightarrow \mathrm{C}_{4} \mathrm{H}_{9} \mathrm{OH}+\mathrm{H}_{3} \mathrm{O}^{+}\)

A reaction was believed to occur by the following mechanism. Step \(1 \mathrm{~A}_{2} \longrightarrow \mathrm{A}+\mathrm{A}\) Step \(2 \mathrm{~A}+\mathrm{A}+\mathrm{B} \longrightarrow \mathrm{A}_{2} \mathrm{~B}\) Step \(3 \mathrm{~A}_{2} \mathrm{~B}+\mathrm{C} \longrightarrow \mathrm{A}_{2}+\mathrm{BC}\) (a) Write the overall reaction. (b) Write the rate law for each step and indicate its molecularity. (c) What are the reaction intermediates? (d) A catalyst is a substance that accelerates the rate of a reaction and is regenerated in the process. What is the catalyst in the reaction?

The following mechanism has been proposed for the reaction between nitric oxide and bromine: Step \(1 \mathrm{NO}+\mathrm{Br}_{2} \longrightarrow \mathrm{NOBr}_{2}\) (slow) Step \(2 \mathrm{NOBr}_{2}+\mathrm{NO} \longrightarrow \mathrm{NOBr}+\mathrm{NOBr}\) (fast) Write the rate law for the formation of NOBr implied by this mechanism.

The mechanism proposed for the oxidation of iodide ion by the hypochlorite ion in aqueous solution is as follows: Step \(1 \mathrm{ClO}^{-}+\mathrm{H}_{2} \mathrm{O} \longrightarrow \mathrm{HClO}+\mathrm{OH}^{-}\)and its reverse (both fast, equilibrium) Step \(2 \mathrm{I}^{-}+\mathrm{HClO} \longrightarrow \mathrm{HIO}+\mathrm{Cl}^{-}\)(slow) Step \(3 \mathrm{HIO}+\mathrm{OH}^{-} \longrightarrow \mathrm{IO}^{-}+\mathrm{H}_{2} \mathrm{O}\) (fast) Write the rate law for the formation of HIO implied by this mechanism.

When the rate of the reaction \(2 \mathrm{NO}(\mathrm{g})+\mathrm{O}_{2}(\mathrm{~g}) \rightarrow\) \(2 \mathrm{NO}_{2}(\mathrm{~g})\) was studied, the rate was found to double when the \(\mathrm{O}_{2}\) concentration alone was doubled but to quadruple when the NO concentration alone was doubled. Which of the following mechanisms accounts for these observations? Explain your reasoning. (a) Step \(1 \mathrm{NO}+\mathrm{O}_{2} \longrightarrow \mathrm{NO}_{3}\) and its reverse (both fast, equilibrium) Step \(2 \mathrm{NO}+\mathrm{NO}_{3} \rightarrow \mathrm{NO}_{2}+\mathrm{NO}_{2}\) (slow) (b) Step \(1 \mathrm{NO}+\mathrm{NO} \rightarrow \mathrm{N}_{2} \mathrm{O}_{2}\) (slow) Step \(2 \mathrm{O}_{2}+\mathrm{N}_{2} \mathrm{O}_{2} \rightarrow \mathrm{N}_{2} \mathrm{O}_{4}\) (fast) Step \(3 \mathrm{~N}_{2} \mathrm{O}_{4} \rightarrow \mathrm{NO}_{2}+\mathrm{NO}_{2}\) (fast)

Determine whether each of the following statements is true or false. If a statement is false, explain why. (a) For a reaction with a very large equilibrium constant, the rate constant of the forward reaction is much larger than the rate constant of the reverse reaction. (b) At equilibrium, the rate constants of the forward and reverse reactions are equal. (c) Increasing the concentration of a reactant increases the rate of a reaction by increasing the rate constant in the forward direction.

Determine whether each of the following statements is true or false. If a statement is false, explain why. (a) The equilibrium constant for a reaction equals the rate constant for the forward reaction divided by the rate constant for the reverse reaction. (b) In a reaction that is a series of equilibrium steps, the overall equilibrium constant is equal to the product of all the forward rate constants divided by the product of all the reverse rate constants. (c) Increasing the concentration of a product increases the rate of the reverse reaction, and so the rate of the forward reaction must then increase, too.

Consider the reaction \(\mathrm{A} \rightleftarrows \mathrm{B}\), which is first order in each direction with rate constants \(k\) and \(k^{\prime}\). Initially, only A is present. Show that the concentrations approach their equilibrium values at a rate that depends on \(k\) and \(k^{\prime}\).

(a) Using a graphing calculator or standard graphing software, such as that on the Web site for this book, make an appropriate Arrhenius plot of the data shown here for the conversion of cyclopropane into propene and calculate the activation energy for the reaction. (b) What is the value of the rate constant at \(600^{\circ} \mathrm{C}\) ? $$ \begin{array}{lcccc} T(\mathrm{~K}) & 750 . & 800 . & 850 . & 900 . \\ k\left(\mathrm{~s}^{-1}\right) & 1.8 \times 10^{-4} & 2.7 \times 10^{-3} & 3.0 \times 10^{-2} & 0.26 \end{array} $$

(a) Using a graphing calculator or standard graphing software, such as that on the Web site for this book, make an appropriate Arrhenius plot of the data shown here for the decomposition of iodoethane into ethene and hydrogen iodide, \(\mathrm{C}_{2} \mathrm{H} \mathrm{H}_{5} \mathrm{I}(\mathrm{g}) \longrightarrow \mathrm{C}_{2} \mathrm{H}_{4}(\mathrm{~g})+\mathrm{HI}(\mathrm{g})\), and determine the activation energy for the reaction. (b) What is the value of the rate constant at \(400^{\circ} \mathrm{C}\) ? $$ \begin{array}{lcccc} T(\mathbf{K}) & 660 & 680 & 720 & 760 \\ k\left(\mathbf{s}^{-1}\right) & 7.2 \times 10^{-4} & 2.2 \times 10^{-3} & 1.7 \times 10^{-2} & 0.11 \end{array} $$

For the reversible, one-step reaction \(\mathrm{A}+\mathrm{B} \rightleftarrows \mathrm{C}+\mathrm{D}\), the forward rate constant is \(52.4 \mathrm{~L} \cdot \mathrm{mol}^{-1} \cdot \mathrm{h}^{-1}\) and the rate constant for the reverse reaction is \(32.1 \mathrm{~L} \cdot \mathrm{mol}^{-1} \cdot \mathrm{h}^{-1}\). The activation energy was found to be \(35.2 \mathrm{~kJ} \cdot \mathrm{mol}^{-1}\) for the forward reaction and \(44.0 \mathrm{~kJ} \cdot \mathrm{mol}^{-1}\) for the reverse reaction. (a) What is the equilibrium constant for the reaction? (b) Is the reaction exothermic or endothermic? (c) What will be the effect of raising the temperature on the rate constants and the equilibrium constant?

Determine which of the following statements about catalysts are true. If the statement is false, explain why. (a) A heterogeneous catalyst works by binding one or more of the molecules undergoing reaction to the surface of the catalyst. (b) Enzymes are naturally occurring proteins that serve as catalysts in biological systems. (c) The equilibrium constant for a reaction is greater in the presence of a catalyst. (d) A catalyst changes the pathway of a reaction in such a way that the reaction becomes more exothermic.

The Michaelis constant \(\left(K_{M}\right)\) is an index of the stability of an enzyme-substrate complex. Does a high Michaelis constant indicate a stable or an unstable enzyme-substrate complex? Explain your reasoning.

Each of the following steps is an elementary reaction. Write its rate law and indicate its molecularity: (a) \(\mathrm{NO}+\mathrm{NO} \rightarrow \mathrm{N}_{2} \mathrm{O}_{2}\) (b) \(\mathrm{Cl}_{2} \rightarrow \mathrm{Cl}+\mathrm{Cl}\); (c) \(\mathrm{NO}_{2}+\mathrm{NO}_{2} \rightarrow \mathrm{NO}+\mathrm{NO}_{3}\); (d) Which o these reactions might be radical chain initiating?

Each of the following is an elementary reaction. Write its rate law and indicate its molecularity. (a) \(\mathrm{O}+\mathrm{CF}_{2} \mathrm{Cl}_{2} \rightarrow\) \(\mathrm{ClO}+\mathrm{CF}_{2} \mathrm{Cl} ;\) (b) \(\mathrm{OH}+\mathrm{NO}_{2}+\mathrm{N}_{2} \longrightarrow \mathrm{HNO}_{3}+\mathrm{N}_{2} ;\) (c) \(\mathrm{ClO}^{-}+\) \(\mathrm{H}_{2} \mathrm{O} \rightarrow \mathrm{HClO}+\mathrm{OH}^{-}\); (d) Which of these reactions might be radical chain propagating?

An organic compound A can decompose by either of two kinetically controlled pathways to form products B or C (see Exercis 14.79). The activation energy for the formation of \(B\) is greater than that for the formation of \(\mathrm{C}\). Will the ratio \([\mathrm{B}] /[\mathrm{C}]\) increase or decrease as the temperature is increased? Explain your answer.

All radioactive decay processes follow first-order kinetics. The half-life of the radioactive isotope tritium \(\left({ }^{3} \mathrm{H}\right.\), or \(\left.\mathrm{T}\right)\) is \(12.3\) years. How much of a \(25.0\)-mg sample of tritium would remain after \(10.9\) years?

The rate law of the reaction \(2 \mathrm{NO}(\mathrm{g})+2 \mathrm{H}_{2}(\mathrm{~g}) \longrightarrow \mathrm{N}_{2}(\mathrm{~g})+\) \(2 \mathrm{H}_{2} \mathrm{O}(\mathrm{g})\) is Rate \(=k[\mathrm{NO}]^{2}\left[\mathrm{H}_{2}\right]\), and the mechanism that has been proposed is Step \(1 \mathrm{NO}+\mathrm{NO} \longrightarrow \mathrm{N}_{2} \mathrm{O}_{2}\) Step \(2 \mathrm{~N}_{2} \mathrm{O}_{2}+\mathrm{H}_{2} \longrightarrow \mathrm{N}_{2} \mathrm{O}+\mathrm{H}_{2} \mathrm{O}\) Step \(3 \mathrm{~N}_{2} \mathrm{O}+\mathrm{H}_{2} \longrightarrow \mathrm{N}_{2}+\mathrm{H}_{2} \mathrm{O}\) (a) Which step in the mechanism is likely to be rate determining? Explain your answer. (b) Sketch a reaction profile for the overall reaction, which is known to be exothermic. Label the activation energies of each step and the overall reaction enthalpy.

(a) Using a graphing calculator or graphing software, such as that on the Web site for this book, calculate the activation energy for the acid hydrolysis of sucrose to give glucose and fructose from an Arrhenius plot of the data shown here. (b) Calculate the rate constant at \(37^{\circ} \mathrm{C}\) (body temperature). (c) From data in Appendix 2A, calculate the enthalpy change for this reaction, assuming that the solvation enthalpies of the sugars are negligible. Draw an energy profile for the overall process. $$ \begin{array}{cc} \text { Temperature }\left({ }^{\circ} \mathrm{C}\right) & k\left(\mathrm{~s}^{-1}\right) \\ \hline 24 & 4.8 \times 10^{-3} \\ 28 & 7.8 \times 10^{-3} \\ 32 & 13 \times 10^{-3} \\ 36 & 20 . \times 10^{-3} \\ 40 . & 32 \times 10^{-3} \\ \hline \end{array} $$

(a) From the following mechanism, derive Eq. 19a, which Michaelis and Menten proposed to represent the rate of formation of products in an enzyme-catalyzed reaction. (b) Show that the rate is independent of substrate concentration at high concentrations of substrate. $$ \begin{aligned} &\mathrm{E}+\mathrm{S} \rightleftarrows \mathrm{ES} \quad k_{1}, k_{1}^{\prime} \\ &\mathrm{ES} \longrightarrow \mathrm{E}+\mathrm{P} \quad k_{2} \end{aligned} $$ where \(E\) is the free enzyme, \(S\) is the substrate, ES is the enzyme-substrate complex, and \(P\) is the product. Note that the steady-state concentration of free enzyme will be equal to the initial concentration of the enzyme less the amount of enzyme that is present in the enzyme-substrate complex: \([\mathrm{E}]=[\mathrm{E}]_{0}-[\mathrm{ES}]\)

Models of population growth are analogous to chemical reaction rate equations. In the model developed by Malthus in 1798 , the rate of change of the population \(N\) of Earth is \(\mathrm{d} N / \mathrm{d} t=\) births - deaths. The numbers of births and deaths are proportional to the population, with proportionality constants \(b\) and \(d\). Derive the integrated rate law for population change. How well does it fit the approximate data for the population of Earth over time given below? $$ \begin{array}{lccccccc} \text { Year } & 1750 & 1825 & 1922 & 1960 & 1974 & 1987 & 2000 \\ N / 10^{9} & 0.5 & 1 & 2 & 3 & 4 & 5 & 6 \end{array} $$

The half-life of a substance taking part in a third-order reaction \(\mathrm{A} \rightarrow\) products is inversely proportional to the square of the initial concentration of A. How can this half-life be used to predict the time needed for the concentration to fall to (a) onehalf; (b) one-fourth; (c) one- sixteenth of its initial value?

Which of the following plots will be linear? (a) [A] against time for a reaction first order in \(A\); (b) [A] against time for a reaction zero order in \(A ;\) (c) \(\ln [A]\) against time for a reaction first order in \(A ;\) (d) \(1 /[A]\) against time for a reaction second order in \(\mathrm{A}\); (e) \(k\) against temperature; (f) initial rate against [A] for a reaction first order in \(A ;\) (g) half-life against [A] for a reaction zero order in A; (h) half- life against [A] for a reaction second order in A.

The pre-equilibrium and the steady-state approximations are two different approaches to deriving a rate law from a proposed mechanism. For the following mechanism, determine the rate law (a) by the pre-equilibrium approximation and (b) by the steady-state approximation. (c) Under what conditions do the two methods give the same answer? (d) What will the rate laws become at high concentrations of \(\mathrm{Br}^{-}\)? $$ \begin{aligned} &\mathrm{CH}_{3} \mathrm{OH}+\mathrm{H}^{+} \rightleftharpoons \mathrm{CH}_{3} \mathrm{OH}_{2}^{+} \text {(fast equilibrium) } \\ &\mathrm{CH}_{3} \mathrm{OH}_{2}^{+}+\mathrm{Br}^{-} \longrightarrow \mathrm{CH}_{3} \mathrm{Br}+\mathrm{H}_{2} \mathrm{O} \text { (slow) } \end{aligned} $$

The decomposition of A has the rate law Rate \(=k[\mathrm{~A}]^{a}\). Show that for this reaction the ratio \(t_{1 / 2} / t_{3 / 4}\), where \(t_{1 / 2}\) is the halflife and \(t_{3 / 4}\) is the time for the concentration of A to decrease to \(\frac{3}{4}\) of its initial concentration, can be written as a function of \(a\) alone and can therefore be used to make a quick assessment of the order of the reaction in A.