Chapter 15

Why are reaction rates important (both practically and theoretically)?

Using the idea that reactions occur as a result of collisions between particles, explain why reaction rates depend on the concentration of the reactants.

Using the idea that reactions occur as a result of collisions between particles, explain why reaction rates depend on the temperature of the reaction mixture.

What units are typically used to express the rate of a reaction?

Why is the reaction rate for reactants defined as the negative of the change in reactant concentration with respect to time, whereas for products it is defined as the change in reactant concentration with re- spect to time (with a positive sign)?

Explain the difference between the average rate of reaction and the in- stantancous rate of reaction.

Consider a simple reaction in which a reactant A forms products: $$\mathrm{A} \longrightarrow products$$ What is the rate law if the reaction is zero order with respect to \(A\) ? First order? Second order? For each case, explain how a doubling of the concentration of A would affect the rate of reaction.

How is the order of a reaction generally determined?

For a reaction with multiple reactants, how is the overall order of the reaction defined?

Explain the difference between the rate law for a reaction and the in- tegrated rate law for a reaction. What relationship does each kind of rate law express?

Write integrated rate laws for zero-order, first-order, and second-order reactions of the form \(A \longrightarrow\) products.

How do reaction rates typically depend on temperature? What part of the rate law is temperature dependent?

Explain the meaning of each term within the Arrhenius equation: activa- tion energy, frequency factor, and exponential factor. Use these terms and the Arrhenius equation to explain why small changes in temperature can result in large changes in reaction rates.

What is an Arrhenius plot? Explain the significance of the slope and intercept of an Arrhenius plot.

Explain the meaning of the orientation factor in the collision model.

Explain the difference between a normal chemical equation for a chemical reaction and the mechanism of that reaction.

In a reaction mechanism, what is an elementary step? Write down the three most common elementary steps and the corresponding rate law for each one.

What is an intermediate within a reaction mechanism?

What is a catalyst? How does a catalyst increase the rate of a chemical reaction?

Explain the difference between homogeneous catalysis and heteroge- neous catalysis.

What are the four basic steps involved in heterogeneous catalysis?

What are enzymes? What is the active site of an enzyme? What is a substrate?

What is the general two-step mechanism by which most enzymes work?

\(\begin{aligned} \text { Consider the reaction. } \\ & 2 \operatorname{HBr}(g) \longrightarrow \mathrm{H}_{2}(g)+\mathrm{Br}_{2}(g) \end{aligned}\) \begin{equation} \begin{array}{l}{\text { a. Express the rate of the reaction in terms of the change in concen- }} \\ {\text { tration of each of the reactants and products. }}\end{array} \end{equation} \begin{array}{l}{\text { b. In the first } 25.0 \text { s of this reaction, the concentration of HBr drops }} \\ {\text { from } 0.600 \mathrm{M} \text { to } 0.512 \mathrm{M} \text { . Calculate the average rate of the reac- }} \\\ {\text { tion during this time interval. }} \\ {\text { c. If the volume of the reaction vessel in part b is } 1.50 \mathrm{L}, \text { what }} \\ {\text { amount of } \mathrm{Br}_{2}(\text { in moles) forms during the first } 15.0 \mathrm{s} \text { of the }} \\ {\text { reaction? }}\end{array}

\(\begin{aligned} \text { Consider the reaction. } \\ & 2 \mathrm{N}_{2} \mathrm{O}(g) \longrightarrow 2 \mathrm{N}_{2}(g)+\mathrm{O}_{2}(g) \end{aligned}\) \begin{equation} \begin{array}{l}{\text { a. Express the rate of the reaction in terms of the change in }} \\ {\text { concentration of each of the reactants and products. }} \\ {\text { b. In the first } 15.0 \text { s of the reaction, 0.015 mol of } \mathrm{O}_{2} \text { is produced in a }} \\ {\text { reaction vessel with a volume of } 0.500 \text { L. What is the average rate }} \\ {\text { of the reaction during this time interval? }}\end{array} \end{equation} \begin{equation} \begin{array}{l}{\text { c. Predict the rate of change in the concentration of } \mathrm{N}_{2} \mathrm{O} \text { during this }} \\ {\text { time interval. In other words, what is } \Delta\left[\mathrm{N}_{2} \mathrm{O}\right] / \Delta t?}\end{array} \end{equation}

For the reaction \(2 \mathrm{A}(g)+\mathrm{B}(g) \longrightarrow 3 \mathrm{C}(g),\) \begin{equation} \begin{array}{l}{\text { a. determine the expression for the rate of the reaction in terms of }} \\ {\text { the change in concentration of each of the reactants and products. }} \\ {\text { b. when A is decreasing at a rate of } 0.100 \mathrm{M} / \mathrm{s} \text { , how fast is } \mathrm{B} \text { decreas- }} \\ {\text { ing? How fast is Cincreasing? }}\end{array} \end{equation}

For the reaction \(\mathrm{A}(g)+\frac{1}{2} \mathrm{B}(g) \longrightarrow 2 \mathrm{C}(g),\) \begin{equation} \begin{array}{l}{\text { a. determine the expression for the rate of the reaction in terms of }} \\ {\text { the change in concentration of cach of the reactants and products. }} \\ {\text { b. when } C \text { is increasing at a rate of } 0.0025 \mathrm{M} / \mathrm{s} \text { , how fast is } \mathrm{B}} \\ {\text { decreasing? How fast is A decreasing? }}\end{array} \end{equation}

What are the units of \(k\) for each type of reaction? \begin{equation} \begin{array}{l}{\text { a. first-order reaction }} \\ {\text { b. second- order reaction }} \\ {\text { c. } \text { zero-order reaction }}\end{array} \end{equation}

This reaction is first order in \(\mathrm{N}_{2} \mathrm{O}_{5}:\) $$\mathrm{N}_{2} \mathrm{O}_{5}(g) \longrightarrow \mathrm{NO}_{3}(g)+\mathrm{NO}_{2}(g)$$ The rate constant for the reaction at a certain temperature is 0.053\(/ \mathrm{s}.\) \begin{equation} \begin{array}{l}{\text { a. Calculate the rate of the reaction when }\left[\mathrm{N}_{2} \mathrm{O}_{5}\right]=0.055 \mathrm{M} \text { . }} \\\ {\text { b. What is the rate of the reaction at the concentration indicated in part }} \\ {\text { a if the reaction is second order? Zero order? (Assume the same } n u-} \\ {\text { merical value for the rate constant with the appropriate units.) }}\end{array} \end{equation}

A reaction in which \(\mathrm{A}, \mathrm{B},\) and C react to form products is first order in \(\mathrm{A},\) second order in \(\mathrm{B},\) and \(\mathrm{zero}\) order in \(\mathrm{C}\) . \begin{equation} \begin{array}{l}{\text { a. Write a rate law for the reaction. }} \\ {\text { b. What is the overall order of the reaction? }} \\ {\text { c. By what factor does the reaction rate change if }[A] \text { is doubled (and the }} \\\ {\text { other reactant concentrations are held constant)? }}\end{array} \end{equation} \begin{equation} \begin{array}{l}{\text { d. By what factor does the reaction rate change if }[\mathrm{B}] \text { is doubled (and the }} \\ {\text { other reactant concentrations are held constant)? }} \\ {\text { e. By what factor does the reaction rate change if }[\mathrm{C}] \text { is doubled (and the }} \\\ {\text { other reactant concentrations are held constant)? }}\\\\{\text { f. By what factor does the reaction rate change if the concentrations of }} \\\ {\text { all three reactants are doubled? }}\end{array} \end{equation}

A reaction in which \(\mathrm{A}, \mathrm{B},\) and \(\mathrm{C}\) react to form products is zero order in \(\mathrm{A},\) one-half order in \(\mathrm{B},\) and second order in \(\mathrm{C} .\) \begin{equation} \begin{array}{l}{\text { a. Write a rate law for the reaction. }} \\ {\text { b. What is the overall order of the reaction? }} \\ {\text { c. By what factor does the reaction rate change if }[\mathrm{A}] \text { is doubled (and the }} \\ {\text { other reactant concentrations are held constant)? }}\end{array} \end{equation} \begin{equation} \begin{array}{l}{\text { d. By what factor does the reaction rate change if }[\mathrm{B}] \text { is doubled (and the }} \\ {\text { other reactant concentrations are held constant)? }} \\ {\text { e. By what factor does the reaction rate change if }[\mathrm{C}] \text { is doubled (and the }} \\\ {\text { other reactant concentrations are held constant)? }}\\\\{\text { f. By what factor does the reaction rate change if the concentrations of }} \\\ {\text { all three reactants are doubled? }}\end{array} \end{equation}

Indicate the order of reaction consistent with each observation. \begin{equation} \begin{array}{l}{\text { a. A plot of the concentration of the reactant versus time yields a }} \\ {\text { straight line. }} \\ {\text { b. The reaction has a half-life that is independent of initial }} \\ {\text { c. A plot of the inverse of the concentration versus time yiclds a }} \\ {\text { straight line. }}\end{array} \end{equation}

This reaction was monitored as a function of time: $$\mathrm{A} \longrightarrow \mathrm{B}+\mathrm{C}$$ A plot of \(\ln [A]\) versus time yields a straight line with slope \(-0.0045 / \mathrm{s}\) \begin{equation} \begin{array}{l}{\text { a. What is the value of the rate constant }(k) \text { for this reaction at this }} \\ {\text { temperature? }} \\ {\text { b. Write the rate law for the reaction. }} \\ {\text { c. What is the half-life? }} \\ {\text { d. If the initial concentration of A is } 0.250 \mathrm{M}, \text { what is the concentra- }} \\ {\text { tion after } 225 \mathrm{s} \text { ? }}\end{array} \end{equation}

This reaction was monitored as a function of time: $$\mathrm{AB} \longrightarrow \mathrm{A}+\mathrm{B}$$ A plot of 1\(/[\mathrm{AB}]\) versus time yields a straight line with slope \(-0.055 / \mathrm{M} \cdot \mathrm{s} .\) \begin{equation} \begin{array}{l}{\text { a. What is the value of the rate constant }(k) \text { for this reaction at this }} \\ {\text { temperature? }}\\\\{\text { b. Write the rate law for the reaction. }} \\ {\text { c. What is the half-life when the initial concentration is } 0.55 \mathrm{M} \text { ? }}\\\\{\text { d. If the initial concentration of } \mathrm{AB} \text { is } 0.250 \mathrm{M}, \text { and the reaction mixture }} \\ {\text { initially contains no products, what are the concentrations of } \mathrm{A} \text { and } \mathrm{B}} \\ {\text { after } 75 \mathrm{s} \text { ? }}\end{array} \end{equation}

The decomposition of \(\mathrm{SO}_{2} \mathrm{Cl}_{2}\) is first order in \(\mathrm{SO}_{2} \mathrm{Cl}_{2}\) and has a rate constant of \(1.42 \times 10^{-4} \mathrm{s}^{-1}\) at a certain temperature. \begin{equation} \begin{array}{l}{\text { a. What is the half-life for this reaction? }} \\\ {\text { b. How long will it take for the concentration of } \mathrm{SO}_{2} \mathrm{Cl}_{2} \text { to decrease to }} \\ {25 \% \text { of its initial concentration? }}\end{array} \end{equation} \begin{equation} \begin{array}{l}{\text { c. If the initial concentration of } \mathrm{SO}_{2} \mathrm{Cl}_{2} \text { is } 1.00 \mathrm{M}, \text { how long will it take }} \\\ {\text { for the concentration to decrease to } 0.78 \mathrm{M} \text { ? }}\end{array} \end{equation} \begin{equation} \begin{array}{l}{\text { d. If the initial concentration of } \mathrm{SO}_{2} \mathrm{Cl}_{2} \text { is } 0.150 \mathrm{M}, \text { what is the concen- }} \\\ {\text { tration of } \mathrm{SO}_{2} \mathrm{Cl}_{2} \text { after } 2.00 \times 10^{2} \text { s? After } 5.00 \times 10^{2} \text { s? }}\end{array} \end{equation}

The half-life for the radioactive decay of U- 238 is 4.5 billion years and is independent of initial concentration. How long will it take for 10\(\%\) of the U- 238 atoms in a sample of \(U-238\) to decay? If a sample of \(U-238\) initially contained \(1.5 \times 10^{18}\) atoms when the universe was formed 13.8 billion years ago, how many \(\mathrm{U}-238\) atoms does it contain today?

The half-life for the radioactive decay of \(\mathrm{C}-14\) is 5730 years and is inde- pendent of the initial concentration. How long does it take for 25\(\%\) of the C-14 atoms in a sample of \(C-14\) to decay? If a sample of C-14 initially contains 1.5 mmol of C-14, how many millimoles are left after 2255 years?

A chemical reaction is endothermic and has an activation energy that is twice the value of the enthalpy change of the reaction. Draw a dia- gram depicting the energy of the reaction as it progresses. Label the position of the reactants and products and indicate the activation ener- gy and enthalpy of reaction.

The rate constant \((k)\) for a reaction is measured as a function of tem- perature. A plot of ln \(k\) versus 1\(/ T(\) in \(\mathrm{K})\) is linear and has a slope of \(-7445 \mathrm{K}\) . Calculate the activation energy for the reaction.

The rate constant \((k)\) for a reaction is measured as a function of tem- perature. A plot of ln \(k\) versus 1\(/ T(\) in \(\mathrm{K})\) is linear and has a slope of \(-1.01 \times 10^{4} \mathrm{K}\) . Calculare the activation energy for the reaction.

The tabulated data shown here were collected for the first-order reaction: $$ \mathrm{N}_{2} \mathrm{O}(g) \longrightarrow \mathrm{N}_{2}(g)+\mathrm{O}(g) $$ Use an Arrhenius plot to determine the activation barrier and frequency factor for the reaction. $$ \begin{array}{|c|c|} \hline \text { Temperature (K) } & \text { Rate Constant (s }^{-1} \text {) } \\\ \hline 800 & 3.24 \times 10^{-5} \\ \hline 900 & 0.00214 \\ \hline 1000 & 0.0614 \\ \hline 1100 & 0.955 \\ \hline \end{array} $$

Consider this overall reaction, which is experimentally observed to be second order in \(\mathrm{AB}\) and zero order in \(\mathrm{C}\) . $$\mathrm{AB}+\mathrm{C} \longrightarrow \mathrm{A}+\mathrm{BC}$$ Is the following mechanism valid for this reaction? $$\mathrm{AB}+\mathrm{AB} \longrightarrow \mathrm{AB}_{2}+\mathrm{A} \quad Slow$$ $$\mathrm{AB}_{2}+\mathrm{C} \longrightarrow \mathrm{AB}+\mathrm{BC}\quad Fast$$

Many heterogencous catalysts are deposited on high surface-area sup- ports. Why?

Suppose that the reaction \(\mathrm{A} \longrightarrow\) products is exothermic and has an ac- tivation barrier of 75 \(\mathrm{kJ} / \mathrm{mol} .\) Sketch an energy diagram showing the en- ergy of the reaction as a function of the progress of the reaction. Draw a second energy curve showing the effect of a catalyst.

The activation barrier for the hydrolysis of sucrose into glucose and fruc- tose is 108 \(\mathrm{k} / \mathrm{mol}\) . If an enzyme increases the rate of the hydrolysis reac- tion by a factor of 1 million, how much lower must the activation barrier be when sucrose is in the active site of the enzyme? (Assume that the frequency factors for the catalyzed and uncatalyzed reactions are identi- cal and a temperature of \(25^{\circ} \mathrm{C.})\)

Cyclopropane \(\left(\mathrm{C}_{3} \mathrm{H}_{6}\right)\) reacts to form propene \(\left(\mathrm{C}_{3} \mathrm{H}_{6}\right)\) in the gas phase. The reaction is first order in cyclopropane and has a rate constant of \(5.87 \times 10^{-4 / \mathrm{s}}\) at \(485^{\circ} \mathrm{C}\) . If a 2.5 L reaction vessel initially contains 722 torr of cyclopropane at \(485^{\circ} \mathrm{C}\) , how long will it take for the par- tial pressure of cyclopropane to drop to below \(1.00 \times 10^{2}\) torr?

Iodine atoms combine to form \(\mathrm{I}_{2}\) in liquid hexane solvent with a rate constant of \(1.5 \times 10^{10} \mathrm{L} / \mathrm{mol} \cdot \mathrm{s}\) . The reaction is second order in I. Since the reaction occurs so quickly, the only way to study the reaction is to create iodine atoms almost instantaneously, usually by photo- chemical decomposition of \(\mathrm{I}_{2}\) . Suppose a flash of light creates an ini- tial \([I]\) concentration of 0.0100 \(\mathrm{M}\) . How long will it take for 95\(\%\) of the newly created iodine atoms to recombine to form \(\mathrm{I}_{2} ?\)

The reaction \(2 \mathrm{H}_{2} \mathrm{O}_{2}(a q) \longrightarrow 2 \mathrm{H}_{2} \mathrm{O}(l)+\mathrm{O}_{2}(g)\) is first order in \(\mathrm{H}_{2} \mathrm{O}_{2}\) and under certain conditions has a rate constant of 0.00752 \(\mathrm{s}^{-1}\) at \(20.0^{\circ} \mathrm{C} .\) A reaction vessel initially contains 150.0 \(\mathrm{mL}\) of 30.0\(\%\) \(\mathrm{H}_{2} \mathrm{O}_{2}\) by mass solution (the density of the solution is 1.11 \(\mathrm{g} / \mathrm{mL} )\) . The gaseous oxygen is collected over water at \(20.0^{\circ} \mathrm{C}\) as it forms. What volume of \(\mathrm{O}_{2}\) forms in 85.0 seconds at a barometric pressure of 742.5 \(\mathrm{mmHg}\) ? (The vapor pressure of water at this temperature is 17.5 \(\mathrm{mm} \mathrm{g} . )\)

The desorption of a single molecular layer of \(n\) -butane from a single crystal of aluminum oxide is found to be first order with a rate constant of 0.128\(/ \mathrm{s}\) at 150 \(\mathrm{K}\) . \begin{equation} \begin{array}{l}{\text { a. What is the haff-life of the desorption reaction? }} \\ {\text { b. If the surface is initially completely covered with } n \text { -butane at }} \\ {150 \mathrm{K}, \text { how long will it take for } 25 \% \text { of the molecules to desorb? For }} \\ {50 \% \text { to desorb? }}\\\\{\text { c. If the surface is initially completely covered, what fraction will remain }} \\ {\text { covered after } 10 \text { s? After } 20 \mathrm{s?}}\end{array} \end{equation}

The evaporation of a 120 nm film of \(n\) -pentane from a single crystal of aluminum oxide is zero order with a rate constant of \(1.92 \times 10^{13}\) molecules/cm \(^{2} \cdot \mathrm{s}\) at 120 \(\mathrm{K}.\) \begin{equation} \begin{array}{l}{\text { a. If the initial surface coverage is } 8.9 \times 10^{16} \text { molecules/cm', how long }} \\ {\text { will it take for one- half of the film to evaporate? }}\end{array} \end{equation} \begin{equation} \begin{array}{l}{\text { b. What fraction of the film is left after } 10 \text { s? Assume the same initial }} \\ {\text { coverage as in part a. }}\end{array} \end{equation}