Problem 1

What variable of a chemical reaction is measured over time to obtain the reaction rate?

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Problem 2

How does an increase in pressure affect the rate of a gas-phase reaction? Explain.

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Problem 3

A reaction is carried out with water as the solvent. How does the addition of more water to the reaction vessel affect the rate of the reaction? Explain.

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Problem 4

A gas reacts with a solid that is present in large chunks. Then the reaction is run again with the solid pulverized. How does the increase in the surface area of the solid affect the rate of its reaction with the gas? Explain.

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Problem 5

How does an increase in temperature affect the rate of a reaction? Explain the two factors involved.

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Problem 6

In a kinetics experiment, a chemist places crystals of iodine in a closed reaction vessel, introduces a given quantity of $\mathrm{H}_{2}$ gas, and obtains data to calculate the rate of $\mathrm{HI}$ formation. In a second experiment, she uses the same amounts of iodine and hydrogen, but first warms the flask to $130^{\circ} \mathrm{C}$ , a temperature above the sublimation point of iodine. In which of these experiments does the reaction proceed at a higher rate? Explain.

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Problem 7

Define reaction rate. Assuming constant temperature and a closed reaction vessel, why does the rate change with time?

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Problem 8

(a) What is the difference between an average rate and an instantaneous rate? (b) What is the difference between an initial rate and an instantaneous rate?

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Problem 10

For the reaction $\mathrm{A}(g) \rightarrow \mathrm{B}(g),$ sketch two curves on the same set of axes that show

(a) The formation of product as a function of time

(b) The consumption of reactant as a function of time

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Problem 11

For the reaction $\mathrm{C}(g) \rightarrow \mathrm{D}(g),[\mathrm{C}]$ vs. time is plotted:

How do you determine each of the following?

(a) The average rate over the entire experiment

(b) The reaction rate at time $x$

(c) The initial reaction rate

(d) Would the values in parts (a), (b), and (c) be different if you plotted [D] vs.t time? Explain.

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Problem 12

The compound $\mathrm{AX}_{2}$ decomposes according to the equation $2 \mathrm{AX}_{2}(g) \longrightarrow 2 \mathrm{AX}(g)+\mathrm{X}_{2}(g) .$ In one experiment, $\left[\mathrm{AX}_{2}\right]$ was measured at various times and these data were obtained:

(a) Find the average rate over the entire experiment.

(b) Is the initial rate higher or lower than the rate in part (a)? Use graphical methods to estimate the initial rate.

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Problem 13

(a) Use the data from Problem 16.12 to calculate the average rate from 8.0 to 20.0 s.

(b) Is the rate at exactly 5.0 s higher or lower than the rate in part (a)? Use graphical methods to estimate the rate at 5.0 $\mathrm{s}$ .

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Problem 14

Express the rate of this reaction in terms of the change in concentration of each of the reactants and products:

$$2 \mathrm{A}(g) \longrightarrow \mathrm{B}(g)+\mathrm{C}(g)$$

When $[\mathrm{C}]$ is increasing at $2 \mathrm{mol} / \mathrm{L} \cdot \mathrm{s},$ how fast is $[\mathrm{A}]$ decreasing?

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Problem 15

Express the rate of this reaction in terms of the change in concentration of each of the reactants and products:

$$\mathrm{D}(g) \longrightarrow \frac{3}{2} \mathrm{E}(g)+\frac{5}{2} \mathrm{F}(g)$$

When $[\mathrm{E}]$ is increasing at 0.25 $\mathrm{mol} / \mathrm{L}$ .s, how fast is $[\mathrm{F}]$ increasing?

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Problem 16

Express the rate of this reaction in terms of the change in concentration of each of the reactants and products:

$$\mathrm{A}(g)+2 \mathrm{B}(g) \longrightarrow \mathrm{C}(g)$$

When $[\mathrm{B}]$ is decreasing at $0.5 \mathrm{mol} / \mathrm{L} \cdot \mathrm{s},$ how fast is $[\mathrm{A}]$ decreasing?

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Problem 17

Express the rate of this reaction in terms of the change in concentration of each of the reactants and products:

$$2 \mathrm{D}(g)+3 \mathrm{E}(g)+\mathrm{F}(g) \longrightarrow 2 \mathrm{G}(g)+\mathrm{H}(g)$$

When $[\mathrm{D}]$ is decreasing at 0.1 $\mathrm{mol} / \mathrm{L}$ .s, how fast is $[\mathrm{H}]$ increasing?

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Problem 18

Reaction rate is expressed in terms of changes in concentration of reactants and products. Write a balanced equation for

$$

\text {(Rate)} =-\frac{1}{2} \frac{\Delta\left[\mathrm{N}_{2} \mathrm{O}_{5}\right]}{\Delta t}=\frac{1}{4} \frac{\Delta\left[\mathrm{NO}_{2}\right]}{\Delta t}=\frac{\Delta\left[\mathrm{O}_{2}\right]}{\Delta t}

$$

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Problem 19

Reaction rate is expressed in terms of changes in concentration of reactants and products. Write a balanced equation for

$$

\text {(Rate)} =-\frac{\Delta\left[\mathrm{CH}_{4}\right]}{\Delta t}=-\frac{1}{2} \frac{\Delta\left[\mathrm{O}_{2}\right]}{\Delta t}=\frac{1}{2} \frac{\Delta\left[\mathrm{H}_{2} \mathrm{O}\right]}{\Delta t}=\frac{\Delta\left[\mathrm{CO}_{2}\right]}{\Delta t}

$$

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Problem 20

The decomposition of NOBr is studied manometrically because the number of moles of gas changes; it cannot be studied colorimetrically because both NOBr and $\mathrm{Br}_{2}$ are reddish brown:

$$

2 \mathrm{NOBr}(g) \longrightarrow 2 \mathrm{NO}(g)+\mathrm{Br}_{2}(g)

$$

Use the data below to answer the following:

(a) Determine the average rate over the entire experiment.

(b) Determine the average rate between 2.00 and 4.00 $\mathrm{s}$ .

(c) Use graphical methods to estimate the initial reaction rate.

(d) Use graphical methods to estimate the initial reaction rate.

(d) Use graphical methods to estimate the rate at 7.00 s.

(e) At what time does the instantaneous rate equal the average rate over the entire experiment?

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Problem 21

The formation of ammonia is one of the most important processes in the chemical industry:

$$

\mathrm{N}_{2}(g)+3 \mathrm{H}_{2}(g) \longrightarrow 2 \mathrm{NH}_{3}(g)

$$

Express the rate in terms of changes in $\left[\mathrm{N}_{2}\right],\left[\mathrm{H}_{2}\right],$ and $\left[\mathrm{NH}_{3}\right]$

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Problem 22

Although the depletion of stratospheric ozone threatens life on Earth today, its accumulation was one of the crucial processes that allowed life to develop in prehistoric times:

$$

3 \mathrm{O}_{2}(g) \longrightarrow 2 \mathrm{O}_{3}(g)

$$

(a) Express the reaction rate in terms of $\left[\mathrm{O}_{2}\right]$ and $\left[\mathrm{O}_{3}\right] .$

(b) At a given instant, the reaction rate in terms of $\left[\mathrm{O}_{2}\right]$ is

$2.17 \times 10^{-5} \mathrm{mol} / \mathrm{L}$ 's. What is it in terms of $\left[\mathrm{O}_{3}\right] ?$

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Problem 23

The rate law for the general reaction

$$

a \mathrm{A}+b \mathrm{B}+\cdots \longrightarrow c \mathrm{C}+d \mathrm{D}+\cdots

$$

is rate $=k[\mathrm{A}]^{m}[\mathrm{B}]^{n} \ldots .$ (a) Explain the meaning of $k .(\mathrm{b})$ Explain the meanings of $m$ and $n .$ Does $m=a$ and $n=b ?$ Explain. $(\mathrm{c})$ If the reaction is first order in $\mathrm{A}$ and second order in $\mathrm{B}$ , and time is measured in minutes (min), what are the units for $k ?$

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Problem 24

You are studying the reaction

$$\mathrm{A}_{2}(g)+\mathrm{B}_{2}(g) \longrightarrow 2 \mathrm{AB}(g)$$

to determine its rate law. Assuming that you have a valid

experimental procedure for obtaining $\left[\mathrm{A}_{2}\right]$ and $\left[\mathrm{B}_{2}\right]$ at various times, explain how you determine (a) the initial rate, (b) the reaction orders, and (c) the rate constant.

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Problem 25

By what factor does the rate change in each of the following cases (assuming constant temperature)?

(a) A reaction is first order in reactant $\mathrm{A}$ , and $[\mathrm{B}]$ is doubled.

(b) A reaction is second order in reactant $\mathrm{B},$ and $[\mathrm{B}]$ is halved.

(c) A reaction is second order in reactant $\mathrm{C}$ , and $[\mathrm{C}]$ is tripled.

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Problem 26

Give the individual reaction orders for all substances and the overall reaction order from the following rate law:

$$

\text {(Rate)} =k\left[\mathrm{BrO}_{3}^{-}\right]\left[\mathrm{Br}^{-}\right]\left[\mathrm{H}^{+}\right]^{2}

$$

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Problem 27

Give the individual reaction orders for all substances and the overall reaction order from the following rate law:

$$

\text {rate} =k \frac{\left[\mathrm{O}_{3}\right]^{2}}{\left[\mathrm{O}_{2}\right]}

$$

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Problem 28

$\mathrm{By}$ what factor does the rate in Problem 16.26 change if each of the following changes occurs: (a) $\left[\mathrm{BrO}_{3}-\right]$ is doubled; (b) $\left[\mathrm{Br}^{-}\right]$ is halved; $(\mathrm{c})\left[\mathrm{H}^{+}\right]$ is quadrupled?

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Problem 29

By what factor does the rate in Problem 16.27 change if each of the following changes occurs: (a) $\left[\mathrm{O}_{3}\right]$ is doubled; (b) $\left[\mathrm{O}_{2}\right]$ is doubled; (c) $\left[\mathrm{O}_{2}\right]$ is halved?

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Problem 30

Give the individual reaction orders for all substances and the overall reaction order from this rate law:

$$

\text {rate} =k\left[\mathrm{NO}_{2}\right]^{2}\left[\mathrm{Cl}_{2}\right]

$$

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Problem 31

Give the individual reaction orders for all substances and the overall reaction order from this rate law:

$$

\text {rate} =k \frac{\left[\mathrm{HNO}_{2}\right]^{4}}{[\mathrm{NO}]^{2}}

$$

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Problem 32

By what factor does the rate in Problem 16.30 change if each of the following changes occurs: (a) $\left[\mathrm{NO}_{2}\right]$ is tripled; (b) $\left[\mathrm{NO}_{2}\right]$ and $\left[\mathrm{Cl}_{2}\right]$ are doubled; (c) $\left[\mathrm{Cl}_{2}\right]$ is halved?

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Problem 33

By what factor does the rate in Problem 16.31 change if each of the following changes occurs: (a) $\left[\mathrm{HNO}_{2}\right]$ is doubled; (b) [NO] is doubled; (c) [HNO, ] is halved?

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Problem 34

For the reaction

$$4 \mathrm{A}(g)+3 \mathrm{B}(g) \rightarrow 2 \mathrm{C}(g)$$

(a) What is the order with respect to each reactant? (b) Write the

rate law. (c) Calculate k (using the data from Expt 1).

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Problem 35

For the reaction

$$\mathrm{A}(g)+\mathrm{B}(g)+\mathrm{C}(g) \longrightarrow \mathrm{D}(g)$$

the following data were obtained at constant temperature:

(a) What is the order with respect to each reactant? (b) Write the rate law. ( $(\text { c) Calculate } k \text { (using the data from Expt 1). }$

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Problem 36

Without consulting Table $16.3,$ give the units of the rate constants for reactions with the following overall orders: (a) first order; $(b)$ second order; $(c)$ third order; $(d) \frac{5}{2}$ order.

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Problem 37

Give the overall reaction order that corresponds to a rate constant with each of the following units:

(a) $\mathrm{mol} / \mathrm{L} \cdot \mathrm{s} ;$ (b) $\mathrm{yr}^{-1}$ $(\mathrm{c})(\mathrm{mol} / \mathrm{L})^{1 / 2} \mathrm{s}^{-1} ;(\mathrm{d})(\mathrm{mol} / \mathrm{L})^{-5 / 2} \cdot \mathrm{min}^{-1}$

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Problem 38

Phosgene is a toxic gas prepared by the reaction of carbon monoxide with chlorine:

$$

\mathrm{CO}(g)+\mathrm{Cl}_{2}(g) \rightarrow \mathrm{COCl}_{2}(g)

$$

These data were obtained in a kinetics study of its formation:

(a) Write the rate law for the formation of phosgene.

(b) Calculate the average value of the rate constant.

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Problem 39

How are integrated rate laws used to determine reaction order? What is the order in reactant if a plot of

(a) The natural logarithm of [reactant] vs. time is linear?

(b) The inverse of [reactant] vs. time is linear?

(c) [Reactant] vs. time is linear?

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Problem 40

Define the half-life of a reaction. Explain on the molecular level why the half-life of a first-order reaction is constant.

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Problem 41

For the simple decomposition reaction

$$

\mathrm{AB}(g) \longrightarrow \mathrm{A}(g)+\mathrm{B}(g)

$$

rate $=k[\mathrm{AB}]^{2}$ and $k=0.2 \mathrm{L} / \mathrm{mol}$ -s. How long will it take for $[\mathrm{AB}]$ to reach one-third of its initial concentration of 1.50 $\mathrm{M} ?$

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Problem 43

In a first-order decomposition reaction, 50.0$\%$ of a compound decomposes in 10.5 min. ( a ) What is the rate constant of the reaction? (b) How long does it take for 75.0$\%$ of the compound to decompose?

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Problem 44

A decomposition reaction has a rate constant of 0.0012 $\mathrm{yr}^{-1}$ . (a) What is the half-life of the reaction? (b) How long does it take for [reactant] to reach 12.5$\%$ of its original value?

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Problem 45

In a study of ammonia production, an industrial chemist discovers that the compound decomposes to its elements $\mathrm{N}_{2}$ and $\mathrm{H}_{2}$ in a first-order process. She collects the following data:

$$

\begin{array}{llll}{\text { Time (s) }} & {0} & {1.000} & {2.000} \\ {\left[\mathrm{NH}_{3}\right](\mathrm{mol} / \mathrm{L})} & {4.000} & {3.986} & {3.974}\end{array}

$$

(a) Use graphical methods to determine the rate constant.

(b) What is the half-life for ammonia decomposition?

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Problem 46

hat is the central idea of collision theory? How does this model explain the effect of concentration on reaction rate?

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Problem 48

rhenius proposed that each reaction has an energy threshold that must be reached for the particles to react. The kinetic theory of gases proposes that the average kinetic energy of the particles is proportional to the absolute temperature. How do these concepts relate to the effect of temperature on rate?

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Problem 49

Use the exponential term in the Arrhenius equation to explain how temperature affects reaction rate.

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Problem 51

(a) Graph the relationship between $k$ (vertical axis) and $T$ (horizontal axis). (b) Graph the relationship between $\ln k$ (vertical axis) and 1$/ T$ (horizontal axis). How is the activation energy determined from this graph?

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Problem 52

(a) For a reaction with a given $E_{a},$ how does an increase in $T$ affect the rate? (b) For a reaction at a given $T$ , how does a decrease in $E_{\mathrm{a}}$ affect the rate?

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Problem 53

In the reaction $A B+C D=E F, 4 \times 10^{-5}$ mol of $A B$ molecules collide with $4 \times 10^{-5}$ mol of $C D$ molecules. Will $4 \times 10^{-5}$ mol of $E F$ form? Explain.

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Problem 54

Assuming the activation energies are equal, which of the following reactions will proceed at a higher rate at $50^{\circ} \mathrm{C} ?$ Explain.

$$

\begin{array}{c}{\mathrm{NH}_{3}(g)+\mathrm{HCl}(g) \rightarrow \mathrm{NH}_{4} \mathrm{Cl}(s)} \\ {\mathrm{N}\left(\mathrm{CH}_{3}\right)_{3}(g)+\mathrm{HCl}(g) \longrightarrow\left(\mathrm{CH}_{3}\right)_{3} \mathrm{NHCl}(s)}\end{array}

$$

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Problem 55

For the reaction $\mathrm{A}(g)+\mathrm{B}(g) \longrightarrow \mathrm{AB}(g),$ how many unique

collisions between $\mathrm{A}$ and $\mathrm{B}$ are possible if there are four particles of $\mathrm{A}$ and three particles of $\mathrm{B}$ present in the vessel?

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Problem 56

For the reaction $\mathrm{A}(g)+\mathrm{B}(g) \rightarrow \mathrm{AB}(g),$ how many unique collisions between $\mathrm{A}$ and $\mathrm{B}$ are possible if 1.01 $\mathrm{mol}$ of $\mathrm{A}(g)$ and 2.12 $\mathrm{mol}$ of $\mathrm{B}(g)$ are present in the vessel?

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Problem 57

At $25^{\circ} \mathrm{C}$ , what is the fraction of collisions with energy equal to or greater than an activation energy of $100 . \mathrm{kJ} / \mathrm{mol} ?$

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Problem 58

If the temperature in Problem 16.57 is increased to $50 .^{\circ} \mathrm{C},$ by what factor does the fraction of collisions with energy equal to or greater than the activation energy change?

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Problem 59

The rate constant of a reaction is $4.7 \times 10^{-3} \mathrm{s}^{-1}$ at $25^{\circ} \mathrm{C},$ and

the activation energy is 33.6 $\mathrm{kJ} / \mathrm{mol}$ . What is $k$ at $75^{\circ} \mathrm{C} ?$

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Problem 60

The rate constant of a reaction is $4.50 \times 10^{-5} \mathrm{L} / \mathrm{mol}$ s at $195^{\circ} \mathrm{C}$ and $3.20 \times 10^{-3} \mathrm{L} / \mathrm{mol}$ s at $258^{\circ} \mathrm{C}$ . What is the activation energy of the reaction?

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Problem 61

For the reaction $\mathrm{ABC}+\mathrm{D} \rightleftharpoons \mathrm{AB}+\mathrm{CD}, \Delta H_{\mathrm{rxn}}^{\circ}=$ $-55 \mathrm{kJ} / \mathrm{mol}$ and $E_{\mathrm{affwd}}=215 \mathrm{kJ} / \mathrm{mol}$ . Assuming a one-step reaction, (a) draw a reaction energy diagram; (b) calculate $E_{\mathrm{arev}}$ nd (c) sketch a possible transition state if ABC is V shaped.

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Problem 62

For the reaction $\mathrm{A}_{2}+\mathrm{B}_{2} \longrightarrow 2 \mathrm{AB}, E_{\mathrm{af(rwd} )}=125 \mathrm{kJ} / \mathrm{mol}$ and $E_{\text { a(rev } )}=85 \mathrm{kJ} / \mathrm{mol}$ . Assuming the reaction occurs in one step, (a) draw a reaction energy diagram; (b) calculate $\Delta H_{\mathrm{rxn}}^{\circ} ;$ and (c) sketch a possible transition state.

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Problem 63

Understanding the high-temperature formation and breakdown of the nitrogen oxides is essential for controlling the pollutants generated from power plants and cars. The first-order breakdown of dinitrogen monoxide to its elements has rate constants of 0.76$/ \mathrm{s}$ at $727^{\circ} \mathrm{C}$ and 0.87$/ \mathrm{s}$ at $757^{\circ} \mathrm{C} .$ What is the activation energy of this reaction?

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Problem 64

Aqua regia, a mixture of $\mathrm{HCl}$ and $\mathrm{HNO}_{3},$ has been used since alchemical times to dissolve many metals, including gold. Its orange color is due to the presence of nitrosyl chloride. Consider this one-step reaction for the formation of this compound:

$$

\mathrm{NO}(g)+\mathrm{Cl}_{2}(g) \longrightarrow \mathrm{NOCl}(g)+\mathrm{Cl}(g) \quad \Delta H^{\circ}=83 \mathrm{kJ}

$$

(a) Draw a reaction energy diagram, given $E_{\text { af(wd) }}=86 \mathrm{kJ} / \mathrm{mol}$ .

(b) Calculate $E_{\text { a(rev. }} )$ .

(c) Sketch a possible transition state for the reaction. (Note: The

atom sequence of nitrosyl chloride is $\mathrm{Cl}-\mathrm{N}-$ O.)

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Problem 65

Is the rate of an overall reaction lower, higher, or equal to the average rate of the individual steps? Explain.

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Problem 66

Explain why the coefficients of an elementary step equal the reaction orders of its rate law but those of an overall reaction do not.

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Problem 67

Is it possible for more than one mechanism to be consistent with the rate law of a given reaction? Explain.

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Problem 68

What is the difference between a reaction intermediate and a transition state?

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Problem 69

Why is a bimolecular step more reasonable physically than a termolecular step?

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Problem 70

If a slow step precedes a fast step in a two-step mechanism, do the substances in the fast step appear in the rate law? Explain.

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Problem 71

If a fast step precedes a slow step in a two-step mechanism, how is the fast step affected? How is this effect used to determine the validity of the mechanism?

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Problem 72

The proposed mechanism for a reaction is

$$

\begin{array}{l}{\text { (1) } \mathrm{A}(g)+\mathrm{B}(g) \rightleftharpoons \mathrm{X}(g)} \\ {\text { (2) } \mathrm{X}(g)+\mathrm{C}(g) \longrightarrow \mathrm{Y}(g)} \\ {\text { (3) } \mathrm{Y}(g) \longrightarrow \mathrm{D}(g)}\end{array}

$$

(a) What is the overall equation?

(b) Identify the intermediate(s), if any.

(c) What are the molecularity and the rate law for each step?

(d) Is the mechanism consistent with the actual rate law: Rate $=k[\mathrm{A}][\mathrm{B}][\mathrm{C}] ?$

(e) Is the following one-step mechanism equally valid: $\mathrm{A}(g)+\mathrm{B}(g)+\mathrm{C}(g) \longrightarrow \mathrm{D}(g) ?$

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Problem 73

Consider the following mechanism:

$$

\begin{array}{l}{\text { (1) } \mathrm{ClO}^{-}(a q)+\mathrm{H}_{2} \mathrm{O}(l) \rightleftharpoons \mathrm{HClO}(a q)+\mathrm{OH}^{-}(a q)} \\ {\text { (2) } \mathrm{I}^{-}(a q)+\mathrm{HClO}(a q) \longrightarrow \mathrm{HIO}(a q)+\mathrm{Cl}^{-}(a q)} \\ {\text { (3) } \mathrm{OH}^{-}(a q)+\mathrm{HIO}(a q) \longrightarrow \mathrm{H}_{2} \mathrm{O}(l)+\mathrm{IO}^{-}(a q)}\end{array}

$$

(a) What is the overall equation?

(b) Identify the intermediate(s), if any.

(c) What are the molecularity and the rate law for each step?

(d) Is the mechanism consistent with the actual rate law: Rate $=k\left[\mathrm{ClO}^{-}\right]\left[\mathrm{I}^{-}\right] ?$

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Problem 74

n a study of nitrosyl halides, a chemist proposes the following mechanism for the synthesis of nitrosyl bromide:

$$

\begin{array}{c}{\mathrm{NO}(g)+\mathrm{Br}_{2}(g) \rightleftharpoons \mathrm{NOBr}_{2}(g)} \\ {\mathrm{NOBr}_{2}(g)+\mathrm{NO}(g) \longrightarrow 2 \mathrm{NOBr}(g)}\end{array}

$$

If the rate law is rate $=k[\mathrm{NO}]^{2}\left[\mathrm{Br}_{2}\right],$ is the proposed mechanism

valid? If so, show that it satisfies the three criteria for validity.

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Problem 75

The rate law for $2 \mathrm{NO}(g)+\mathrm{O}_{2}(g) \longrightarrow 2 \mathrm{NO}_{2}(g)$ is rate $=$ $k[\mathrm{NO}]^{2}\left[\mathrm{O}_{2}\right] .$ In addition to the mechanism in the text (p. $709 ),$ the following ones have been proposed:

$$

\begin{array}{l}{2 \mathrm{NO}(g)+\mathrm{O}_{2}(g) \longrightarrow 2 \mathrm{NO}_{2}(g)} \\ {2 \mathrm{NO}(g) \rightleftharpoons \mathrm{N}_{2} \mathrm{O}_{2}(g)} \\ {\mathrm{N}_{2} \mathrm{O}_{2}(g)+\mathrm{O}_{2}(g) \rightarrow 2 \mathrm{NO}_{2}(g)} \\ {2 \mathrm{NO}(g) \rightleftharpoons \mathrm{N}_{2}(g)+\mathrm{O}_{2}(g)} \\ {\mathrm{N}_{2}(g)+2 \mathrm{O}_{2}(g) \rightarrow 2 \mathrm{NO}_{2}(g)}\end{array}

$$

(a) Which of these mechanisms is consistent with the rate law?

(b) Which of these mechanisms is most reasonable? Why?

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Problem 76

Consider the reaction $\mathrm{N}_{2} \mathrm{O}(g)-\mathrm{Au}, \mathrm{N}_{2}(g)+\frac{1}{2} \mathrm{O}_{2}(g)$

(a) Does the gold catalyst (Au, above the arrow) act as a homogeneous or a heterogeneous catalyst?

(b) On the same set of axes, sketch the reaction energy diagrams for the catalyzed and the uncatalyzed reaction.

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Problem 77

Does a catalyst increase reaction rate by the same means as a rise in temperature does? Explain.

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Problem 78

In a classroom demonstration, hydrogen gas and oxygen gas are mixed in a balloon. The mixture is stable under normal conditions, but if a spark is applied to it or some powdered metal is added, the mixture explodes. (a) Is the spark acting as a catalyst? Explain. (b) Is the metal acting as a catalyst? Explain.

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Problem 79

A principle of green chemistry is that the energy needs of industrial processes should have minimal environmental impact. How can the use of catalysts lead to “greener” technologies?

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Problem 80

Enzymes are remarkably efficient catalysts that can increase reaction rates by as many as 20 orders of magnitude.

(a) How does an enzyme affect the transition state of a reaction, and how does this effect increase the reaction rate?

(b) What characteristics of enzymes give them this effectiveness as catalysts?

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Problem 81

Experiments show that each of the following redox reactions is second order overall:

Reaction $1 : \mathrm{NO}_{2}(g)+\mathrm{CO}(g) \longrightarrow \mathrm{NO}(g)+\mathrm{CO}_{2}(g)$

Reaction $2 : \mathrm{NO}(g)+\mathrm{O}_{3}(g) \longrightarrow \mathrm{NO}_{2}(g)+\mathrm{O}_{2}(g)$

(a) When $\left[\mathrm{NO}_{2}\right]$ in reaction 1 is doubled, the rate quadruples. Write the rate law for this reaction.

(b) When [NO] in reaction 2 is doubled, the rate doubles. Write the rate law for this reaction.

(c) In each reaction, the initial concentrations of the reactants are equal. For each reaction, what is the ratio of the initial rate to the rate when the reaction is 50$\%$ complete?

(d) In reaction $1,$ the initial $\left[\mathrm{NO}_{2}\right]$ is twice the initial $[\mathrm{CO}] .$ What is the ratio of the initial rate to the rate at 50$\%$ completion?

(e) In reaction $2,$ the initial [NO] is twice the initial $\left[\mathrm{O}_{3}\right] .$ What is the ratio of the initial rate to the rate at 50$\%$ completion?

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Problem 82

Consider the following reaction energy diagram:

(a) How many elementary steps are in the reaction mechanism?

(b) Which step is rate limiting?

(c) Is the overall reaction exothermic or endothermic?

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Problem 83

Reactions between certain organic (alkyl) halides and water produce alcohols. Consider the overall reaction for $t$ -butyl bromide (2-bromo-2-methylpropane):

$\left(\mathrm{CH}_{3}\right)_{3} \mathrm{CBr}(a q)+\mathrm{H}_{2} \mathrm{O}(l) \longrightarrow {\left(\mathrm{CH}_{3}\right)_{3}} \mathrm{COH}(a q)+\mathrm{H}^{+}(a q)+\mathrm{Br}^{-}(a q)$

The experimental rate law is rate $=k\left[\left(\mathrm{CH}_{3}\right)_{3} \mathrm{CBr}\right] .$ The accepted mechanism for the reaction is

$$

\begin{array}{l}{(1)\left(\mathrm{CH}_{3}\right)_{3} \mathrm{C}-\mathrm{Br}(a q) \longrightarrow\left(\mathrm{CH}_{3}\right)_{3} \mathrm{C}^{+}(a q)+\mathrm{Br}^{-}(a q)} \\ {\text { (2) }\left(\mathrm{CH}_{3}\right)_{3} \mathrm{C}^{+}(a q)+\mathrm{H}_{2} \mathrm{O}(l) \longrightarrow\left(\mathrm{CH}_{3}\right)_{3} \mathrm{C}-\mathrm{OH}_{2}^{+}(a q)}\end{array}

$$

$$

(3)\left(\mathrm{CH}_{3}\right)_{3} \mathrm{C}-\mathrm{OH}_{2}^{+}(a q) \rightarrow \mathrm{H}^{+}(a q)+\left(\mathrm{CH}_{3}\right)_{3} \mathrm{C}-\mathrm{OH}(a q)

$$

(a) Why doesn't $\mathrm{H}_{2} \mathrm{O}$ appear in the rate law?

(b) Write rate laws for the elementary steps.

(c) What reaction intermediates appear in the mechanism?

(d) Show that the mechanism is consistent with the experimental rate law.

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Problem 84

Archeologists can determine the age of an artifact made of wood or bone by measuring the amount of the radioactive isotope 14 $\mathrm{C}$ present in the object. The amount of this isotope decreases in a first-order process. If 15.5$\%$ of the original amount of $^{14} \mathrm{C}$ is present in a wooden tool at the time of analysis, what is the age of the tool? The half-life of $^{14} \mathrm{C}$ is 5730 yr.

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Problem 85

A slightly bruised apple will rot extensively in about 4 days at room temperature $\left(20^{\circ} \mathrm{C}\right) .$ If it is kept in the refrigerator at $0^{\circ} \mathrm{C},$ the same extent of rotting takes about 16 days. What is the activation energy for the rotting reaction?

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Problem 86

Benzoyl peroxide, the substance most widely used against acne, has a half-life of $9.8 \times 10^{3}$ days when refrigerated. How long will it take to lose 5$\%$ of its potency $(95 \% \text { remaining)? }$

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Problem 87

The rate law for the reaction

$$

\mathrm{NO}_{2}(g)+\mathrm{CO}(g) \rightarrow \mathrm{NO}(g)+\mathrm{CO}_{2}(g)

$$

is rate $=k\left[\mathrm{NO}_{2}\right]^{2} ;$ one possible mechanism is shown on p. 708 .

(a) Draw a reaction energy diagram for that mechanism, given that $\Delta H_{\text { overall }}^{\circ}=-226 \mathrm{kJ} / \mathrm{mol}$

(b) Consider the following alternative mechanism:

$$

\begin{array}{l}{\text { (1) } 2 \mathrm{NO}_{2}(g) \rightarrow \mathrm{N}_{2}(g)+2 \mathrm{O}_{2}(g)} \\ {\text { (2) } 2 \mathrm{CO}(g)+\mathrm{O}_{2}(g) \longrightarrow 2 \mathrm{CO}_{2}(g)} \\ {\text { (3) } \mathrm{N}_{2}(g)+\mathrm{O}_{2}(g) \longrightarrow 2 \mathrm{NO}(g)}\end{array}

$$

Is the alternative mechanism consistent with the rate law? Is one mechanism more reasonable physically? Explain.

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Problem 88

Consider the following general reaction and data:

$$

2 \mathrm{A}+2 \mathrm{B}+\mathrm{C} \longrightarrow \mathrm{D}+3 \mathrm{E}

$$

(a) What is the reaction order with respect to each reactant?

(b) Calculate the rate constant.

(c) Write the rate law for this reaction.

(d) Express the rate in terms of changes in concentration with time for each of the components.

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Problem 89

In acidic solution, the breakdown of sucrose into glucose and fructose has this rate law: Rate $=k\left[\mathrm{H}^{+}\right][\text { sucrose }] .$ The initial rate of sucrose breakdown is measured in a solution that is 0.01$M \mathrm{H}^{+}$ , 1.0 $\mathrm{M}$ sucrose, 0.1 $\mathrm{M}$ fructose, and 0.1 $\mathrm{M}$ glucose. How does the rate change if

(a) [Sucrose $]$ is changed to 2.5$M ?$

(b) [Sucrose $],$ [fructose $],$ and [glucose $]$ are all changed to 0.5$M ?$

(c) $\left[\mathrm{H}^{+}\right]$ is changed to 0.0001$M ?$

(d) [Sucrose $]$ and $\left[\mathrm{H}^{+}\right]$ are both changed to 0.1 $\mathrm{M}$ ?

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Problem 90

The citric acid cycle is the central reaction sequence in the cellular metabolism of humans and many other organisms. One of the key steps is catalyzed by the enzyme isocitrate dehydrogenase and the oxidizing agent NAD $^{+} .$ In yeast, the reaction is eleventh order:

$$

\text {rate} =k[\text { enzyme }][\text { isocitrate }]^{4}[\mathrm{AMP}]^{2}\left[\mathrm{NAD}^{+}\right]^{m}\left[\mathrm{Mg}^{2+}\right]^{2}

$$

What is the order with respect to $\mathrm{NAD}^{+} ?$

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Problem 91

The following molecular scenes represent starting mixtures I and II for the reaction of A (black) with B (orange): Each sphere represents 0.010 mol, and the volume is 0.50 L. If the reaction is first order in A and first order in B and the initial rate for I is $8.3 \times 10^{-4} \mathrm{mol} / \mathrm{L} \cdot \mathrm{min}$ , what is the initial rate for II?

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Problem 92

Experiment shows that the rate of formation of carbon tetrachloride from chloroform,

$$

\mathrm{CHCl}_{3}(g)+\mathrm{Cl}_{2}(g) \rightarrow \mathrm{CCl}_{4}(g)+\mathrm{HCl}(g)

$$

is first order in $\mathrm{CHCl}_{3}, \frac{1}{2}$ order in $\mathrm{Cl}_{2},$ and $\frac{3}{2}$ order overall. Show that the following mechanism is consistent with the rate law:

$$

\begin{array}{l}{\text { (1) } \mathrm{Cl}_{2}(g)=2 \mathrm{Cl}(g)} \\ {\text { (2) } \mathrm{Cl}(g)+\mathrm{CHCl}_{3}(g) \longrightarrow \mathrm{HCl}(g)+\mathrm{CCl}_{3}(g)} \\ {\text { (3) } \mathrm{CCl}_{3}(g)+\mathrm{Cl}(g) \longrightarrow \mathrm{CCl}_{4}(g)}\end{array}

$$

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Problem 93

A biochemist studying the breakdown of the insecticide DDT finds that it decomposes by a first-order reaction with a half-life of 12 yr. How long does it take DDT in a soil sample to decrease from 275 ppbm to 10. ppbm (parts per billion by mass)?

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Problem 94

Insulin is a polypeptide hormone that is released into the blood from the pancreas and stimulates fat and muscle to take up glucose; the insulin is used up in a first-order process. In a certain patient, it has a half-life of 3.5 min. To maintain an adequate blood concentration of insulin, it must be replenished in a time interval equal to 1/k. How long is the time interval for this patient?

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Problem 95

For the reaction $\mathrm{A}(g)+\mathrm{B}(g) \longrightarrow \mathrm{AB}(g),$ the rate is $0.20 \mathrm{mol} / \mathrm{L} \cdot \mathrm{s},$ when $[\mathrm{A}]_{0}=[\mathrm{B}]_{0}=1.0 \mathrm{mol} / \mathrm{L} .$ If the reaction is first order in $\mathrm{B}$ and second order in A, what is the rate when $[\mathrm{A}]_{0}=2.0 \mathrm{mol} / \mathrm{L}$ and $[\mathrm{B}]_{0}=3.0 \mathrm{mol} / \mathrm{L} ?$

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Problem 96

The acid-catalyzed hydrolysis of sucrose occurs by the following overall reaction, whose kinetic data are given below:

$$

\mathrm{C}_{12} \mathrm{H}_{22} \mathrm{O}_{11}(s)+\mathrm{H}_{2} \mathrm{O}(l) \longrightarrow \mathrm{C}_{6} \mathrm{H}_{12} \mathrm{O}_{6}(a q)+\mathrm{C}_{6} \mathrm{H}_{12} \mathrm{O}_{6}(a q)

$$

(a) Determine the rate constant and the half-life of the reaction.

(b) How long does it take to hydrolyze 75$\%$ of the sucrose?

(c) Other studies have shown that this reaction is actually second order overall but appears to follow first-order kinetics. (Such a reaction is called a pseudo-first-order reaction.) Suggest a reason for this apparent first-order behavior.

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Problem 97

At body temperature $\left(37^{\circ} \mathrm{C}\right),$ the rate constant of an enzyme- catalyzed decomposition is $2.3 \times 10^{14}$ times that of the uncatalyzed reaction. If the frequency factor, $A$ , is the same for both processes, by how much does the enzyme lower the $E_{\mathrm{a}} ?$

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Problem 98

Is each of these statements true? If not, explain why.

(a) At a given T, all molecules have the same kinetic energy.

(b) Halving the P of a gaseous reaction doubles the rate.

(c) A higher activation energy gives a lower reaction rate

(d) A temperature rise of $10^{\circ} \mathrm{C}$ doubles the rate of any reaction.

(e) If reactant molecules collide with greater energy than the activation energy, they change into product molecules.

(f) The activation energy of a reaction depends on temperature.

(g) The rate of a reaction increases as the reaction proceeds.

(h) Activation energy depends on collision frequency.

(i) A catalyst increases the rate by increasing collision frequency.

(j) Exothermic reactions are faster than endothermic reactions.

(k) Temperature has no effect on the frequency factor (A).

(l) The activation energy of a reaction is lowered by a catalyst.

(m) For most reactions, $\Delta H_{\text { ran }}$ is lowered by a catalyst.

(o) The initial rate of a reaction is its maximum rate.

(p) A bimolecular reaction is generally twice as fast as a unimolecular reaction.

(q) The molecularity of an elementary reaction is proportional to the molecular complexity of the reactant(s).

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Problem 99

For the decomposition of gaseous dinitrogen pentoxide,

$$

2 \mathrm{N}_{2} \mathrm{O}_{5}(g) \longrightarrow 4 \mathrm{NO}_{2}(g)+\mathrm{O}_{2}(g)

$$

the rate constant is $k=2.8 \times 10^{-3} \mathrm{s}^{-1}$ at $60^{\circ} \mathrm{C} .$ The initial

concentration of $\mathrm{N}_{2} \mathrm{O}_{5}$ is 1.58 $\mathrm{mol} / \mathrm{L}$ . (a) What is $\left[\mathrm{N}_{2} \mathrm{O}_{5}\right]$ after 5.00 $\min ?$ (b) What fraction of the $\mathrm{N}_{2} \mathrm{O}_{5}$ has decomposed after 5.00 $\mathrm{min} ?$

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Problem 100

Even when a mechanism is consistent with the rate law, later work may show it to be incorrect. For example, the reaction between hydrogen and iodine has this rate law: Rate $=k\left[\mathrm{H}_{2}\right]\left[\mathrm{I}_{2}\right]$ The long-accepted mechanism had a single bimolecular step; that is, the overall reaction was thought to be elementary:

$$

\mathrm{H}_{2}(g)+\mathrm{I}_{2}(g) \longrightarrow 2 \mathrm{HI}(g)

$$

In the 1960s, however, spectroscopic evidence showed the presence of free I atoms during the reaction. Kineticists have since proposed a three-step mechanism:

$$

\begin{array}{l}{\text { (1) } \mathrm{I}_{2}(g) \rightleftharpoons 2 \mathrm{I}(g)} \\ {\text { (2) } \mathrm{H}_{2}(g)+\mathrm{I}(g) \rightleftharpoons \mathrm{H}_{2} \mathrm{I}(g)} \\ {\text { (3) } \mathrm{H}_{2} \mathrm{I}(g)+\mathrm{I}(g) \longrightarrow 2 \mathrm{HI}(g)}\end{array}

$$

show that this mechanism is consistent with the rate law

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Problem 101

Suggest an experimental method for measuring the change in concentration with time for each of the following reactions:

(a) $\mathrm{CH}_{3} \mathrm{CH}_{2} \mathrm{Br}(l)+\mathrm{H}_{2} \mathrm{O}(l) \rightarrow \mathrm{CH}_{3} \mathrm{CH}_{2} \mathrm{OH}(l)+\mathrm{HBr}(a q)$

(b) $2 \mathrm{NO}(g)+\mathrm{Cl}_{2}(g) \longrightarrow 2 \mathrm{NOCl}(g)$

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Problem 102

An atmospheric chemist fills a container with gaseous $\mathrm{N}_{2} \mathrm{O}_{5}$ to a pressure of 125 $\mathrm{kPa}$ , and the gas decomposes to $\mathrm{NO}_{2}$ and $\mathrm{O}_{2} .$ What is the partial pressure of $\mathrm{NO}_{2}, P_{\mathrm{NO}_{2}}(\text { in } \mathrm{kPa}),$ when the total pressure is 178 $\mathrm{kPa}$ ?

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Problem 103

Many drugs decompose in blood by a first-order process.

(a) Two tablets of aspirin supply 0.60 g of the active compound. After 30 min, this compound reaches a maximum concentration of 2 mg/100 mL of blood. If the half-life for its breakdown is 90 min, what is its concentration (in mg/100 mL) 2.5 h after it reaches its maximum concentration?

(b) For the decomposition of an antibiotic in a person with a nor-

mal temperature $\left(98.6^{\circ} \mathrm{F}\right), k=3.1 \times 10^{-5} \mathrm{s}^{-1} ;$ for a person with a fever (temperature of $101.9^{\circ} \mathrm{F} ), k=3.9 \times 10^{-5} \mathrm{s}^{-1}$ If the person with the fever must take another pill when $\frac{2}{3}$ of the first pill has decomposed, how many hours should she wait to take a second pill? A third pill? (Assume that the pill is effective immediately.)

(c) Calculate $E_{\mathrm{a}}$ for decomposition of the antibiotic in part (b).

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Problem 103

(a) Two tablets of aspirin supply 0.60 g of the active compound. After 30 min, this compound reaches a maximum concentration of 2 mg/100 mL of blood. If the half-life for its breakdown is 90 min, what is its concentration (in mg/100 mL) 2.5 h after it reaches its maximum concentration?

(b) For the decomposition of an antibiotic in a person with a nor-

mal temperature $\left(98.6^{\circ} \mathrm{F}\right), k=3.1 \times 10^{-5} \mathrm{s}^{-1} ;$ for a person with a fever (temperature of $101.9^{\circ} \mathrm{F} ), k=3.9 \times 10^{-5} \mathrm{s}^{-1}$ If the person with the fever must take another pill when $\frac{2}{3}$ of the first pill has decomposed, how many hours should she wait to take a second pill? A third pill? (Assume that the pill is effective immediately.)

(c) Calculate $E_{\mathrm{a}}$ for decomposition of the antibiotic in part (b).

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Problem 104

While developing a catalytic process to make ethylene glycol from synthesis gas $\left(\mathrm{CO}+\mathrm{H}_{2}\right),$ a chemical engineer finds the rate is fourth order with respect to gas pressure. The uncertainty in the pressure reading is 5$\% .$ When the catalyst is modified, the rate increases by 10$\% .$ If you were the company patent attorney, would you file for a patent on this catallyst modification? Explain.

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Problem 105

Iodide ion reacts with chloromethane to displace chloride ion in a common organic substitution reaction:

$$

\mathrm{I}^{-}+\mathrm{CH}_{3} \mathrm{Cl} \longrightarrow \mathrm{CH}_{3} \mathrm{I}+\mathrm{Cl}^{-}

$$

(a) Draw a wedge-bond structure of chloroform and indicate the most effective direction of I - attack.

(b) The analogous reaction with 2 -chlorobutane results in a major change in specific rotation as measured by polarimetry. Explain, showing a wedge-bond structure of the product.

(c) Under different conditions, 2 -chlorobutane loses $\mathrm{Cl}^{-}$ in a rate-determining step to form a planar intermediate. This cationic species reacts with HI and then loses $\mathrm{H}^{+}$ to form a product that exhibits no optical activity. Explain, showing a wedge-bond structure.

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Problem 106

Assume that water boils at $100.0^{\circ} \mathrm{C}$ in Houston (near sea level), and at $90.0^{\circ} \mathrm{C}$ in Cripple Creek, Colorado (near 9500 $\mathrm{ft} ) .$ If it takes 4.8 $\mathrm{min}$ to cook an egg in Cripple Creek and 4.5 $\mathrm{min}$ in Houston, what is $E_{\mathrm{a}}$ for this process?

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Problem 107

Sulfonation of benzene has the following mechanism:

$$

\begin{array}{l}{\text { (1) } 2 \mathrm{H}_{2} \mathrm{SO}_{4} \longrightarrow \mathrm{H}_{3} \mathrm{O}^{+}+\mathrm{HSO}_{4}^{-}+\mathrm{SO}_{3}} \\ {\text { (2) } \mathrm{SO}_{3}+\mathrm{C}_{6} \mathrm{H}_{6} \rightarrow \mathrm{H}\left(\mathrm{C}_{6} \mathrm{H}_{5}^{+}\right) \mathrm{SO}_{3}^{-}} \\ {\text { (3) } \mathrm{H}\left(\mathrm{C}_{6} \mathrm{H}_{5}^{+}\right) \mathrm{SO}_{3}^{-}+\mathrm{HSO}_{4}^{-} \longrightarrow \mathrm{C}_{6} \mathrm{H}_{5} \mathrm{SO}_{3}^{-}+\mathrm{H}_{2} \mathrm{SO}_{4}} \\ {\text { (4) } \mathrm{C}_{6} \mathrm{H}_{5} \mathrm{SO}_{3}^{-}+\mathrm{H}_{3} \mathrm{O}^{+} \longrightarrow \mathrm{C}_{6} \mathrm{H}_{5} \mathrm{SO}_{3} \mathrm{H}+\mathrm{H}_{2} \mathrm{O}}\end{array}

$$

(a) Write an overall equation for the reaction. (b) Write the overall rate law in terms of the initial rate of the reaction.

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Problem 108

In the lower troposphere, ozone is one of the components of photochemical smog. It is generated in air when nitrogen dioxide, formed by the oxidation of nitrogen monoxide from car exhaust, reacts by the following mechanism:

$$

\begin{array}{l}{\text { (1) } \mathrm{NO}_{2}(g) \frac{k_{1}}{h v} \mathrm{NO}(g)+\mathrm{O}(g)} \\ {\text { (2) } \mathrm{O}(g)+\mathrm{O}_{2}(g) \frac{k_{2}}{\ln v} \mathrm{O}_{3}(g)}\end{array}

$$

Assuming the rate of formation of atomic oxygen in step 1 equals the rate of its consumption in step 2, use the data below to calculate (a) the concentration of atomic oxygen [O]; (b) the rate of ozone formation.

$$

\begin{array}{ll}{k_{1}=6.0 \times 10^{-3} \mathrm{s}^{-1}} & {\left[\mathrm{NO}_{2}\right]=4.0 \times 10^{-9} \mathrm{M}} \\ {k_{2}=1.0 \times 10^{6} \mathrm{L} / \mathrm{mol} \cdot \mathrm{s}} & {\left[\mathrm{O}_{2}\right]=1.0 \times 10^{-2} \mathrm{M}}\end{array}

$$

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Problem 109

Chlorine is commonly used to disinfect drinking water, and inactivation of pathogens by chlorine follows first-order kinetics. The following data show E. coli inactivation:

(a) Determine the first-order inactivation constant, $k$ . [Hint:

$\%$ inactivation $=100 \times\left(1-[\mathrm{A}]_{t} /[\mathrm{A}]_{0}\right) . ]$

(b) How much contact time is required for 95$\%$ inactivation?

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Problem 110

The overall equation and rate law for the gas-phase decomposition of dinitrogen pentoxide are

$$

2 \mathrm{N}_{2} \mathrm{O}_{5}(g) \rightarrow 4 \mathrm{NO}_{2}(g)+\mathrm{O}_{2}(g) \quad \text { rate }=k\left[\mathrm{N}_{2} \mathrm{O}_{5}\right]

$$

Which of the following can be considered valid mechanisms for the reaction?

$$

\begin{array}{l}{\text { I One-step collision }} \\ {\text { II } 2 \mathrm{N}_{2} \mathrm{O}_{5}(g) \longrightarrow 2 \mathrm{NO}_{3}(g)+2 \mathrm{NO}_{2}(g)} \\ {2 \mathrm{NO}_{3}(g) \rightarrow 2 \mathrm{NO}_{2}(g)+2 \mathrm{O}(g)} \\ {2 \mathrm{O}(g) \longrightarrow \mathrm{O}_{2}(g)}\end{array}

$$

$$

\begin{array}{l}{\text { III } \mathrm{N}_{2} \mathrm{O}_{5}(g)=\mathrm{NO}_{3}(g)+\mathrm{NO}_{2}(g)} \\ {\mathrm{NO}_{2}(g)+\mathrm{N}_{2} \mathrm{O}_{5}(g) \longrightarrow 3 \mathrm{NO}_{2}(g)+\mathrm{O}(g)} \\ {\mathrm{NO}_{3}(g)+\mathrm{O}(g) \longrightarrow \mathrm{NO}_{2}(g)+\mathrm{O}_{2}(g)}\end{array}

$$

$$

\begin{array}{l}{\text { IV } 2 \mathrm{N}_{2} \mathrm{O}_{5}(g) \rightleftharpoons 2 \mathrm{NO}_{2}(g)+\mathrm{N}_{2} \mathrm{O}_{3}(g)+3 \mathrm{O}(g)} \\ {\mathrm{N}_{2} \mathrm{O}_{3}(g)+\mathrm{O}(g) \longrightarrow 2 \mathrm{NO}_{2}(g)} \\ {\quad 2 \mathrm{O}(g) \longrightarrow \mathrm{O}_{2}(g)}\end{array}

$$

$$

\begin{array}{rl}{\mathrm{V}} & {2 \mathrm{N}_{2} \mathrm{O}_{5}(g) \longrightarrow \mathrm{N}_{4} \mathrm{O}_{10}(g)} \\ {\mathrm{N}_{4} \mathrm{O}_{10}(g) \longrightarrow} & {4 \mathrm{NO}_{2}(g)+\mathrm{O}_{2}(g)}\end{array}

$$

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Problem 111

Nitrification is a biological process for removing $\mathrm{NH}_{3}$ from wastewater as $\mathrm{NH}_{4}^{+} :$

$$

\mathrm{NH}_{4}^{+}+2 \mathrm{O}_{2} \longrightarrow \mathrm{NO}_{3}^{-}+2 \mathrm{H}^{+}+\mathrm{H}_{2} \mathrm{O}

$$

The first-order rate constant is given as

$$

k_{1}=0.47 e^{0.095\left(T=15^{\circ} \mathrm{C}\right)}

$$

where $k_{1}$ is in day $^{-1}$ and $T$ is in $^{\circ} \mathrm{C}$ .

(a) If the initial concentration of $\mathrm{NH}_{3}$ is 3.0 $\mathrm{mol} / \mathrm{m}^{3}$ , how long will it take to reduce the concentration to 0.35 $\mathrm{mol} / \mathrm{m}^{3}$ in the spring $\left(T=20^{\circ} \mathrm{C}\right) ?$

(b) In the winter $\left(T=10^{\circ} \mathrm{C}\right) ?$

(c) Using your answer to part (a), what is the rate of $\mathrm{O}_{2}$ consumption?

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Problem 112

Carbon disulfide, a poisonous, flammable liquid, is an excellent solvent for phosphorus, sulfur, and some other nonmetals. A kinetic study of its gaseous decomposition reveals these data:

(a) Write the rate law for the decomposition of $\mathrm{CS}_{2}$ .

(b) Calculate the average value of the rate constant.

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Problem 112

(a) Write the rate law for the decomposition of $\mathrm{CS}_{2}$ .

(b) Calculate the average value of the rate constant.

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Problem 113

Like any catalyst, palladium, platinum, or nickel catalyzes both directions of a reaction: addition of hydrogen to (hydrogenation) and its elimination from (dehydrogenation) carbon double bonds.

(a) Which variable determines whether an alkene will be hydrogenated or dehydrogenated?

(b) Which reaction requires a higher temperature?

(c) How can all-trans fats arise during hydrogenation of fats that contain some cis- double bonds?

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Problem 114

In a clock reaction, a dramatic color change occurs at a time determined by concentration and temperature. Consider the iodine clock reaction, whose overall equation is

$$

2 \mathrm{I}^{-}(a q)+\mathrm{S}_{2} \mathrm{O}_{8}^{2-}(a q) \longrightarrow \mathrm{I}_{2}(a q)+2 \mathrm{SO}_{4}^{2-}(a q)

$$

As $\mathrm{I}_{2}$ forms, it is immediately consumed by its reaction with a fixed amount of added $\mathrm{S}_{2} \mathrm{O}_{3}^{2-}$ :

$$

\mathrm{I}_{2}(a q)+2 \mathrm{S}_{2} \mathrm{O}_{3}^{2-}(a q) \longrightarrow 2 \mathrm{I}^{-}(a q)+\mathrm{S}_{4} \mathrm{O}_{6}^{2-}(a q)

$$

Once the $\mathrm{S}_{2} \mathrm{O}_{3}^{2-}$ is consumed, the excess $\mathrm{I}_{2}$ forms a blue-black product with starch present in solution:

$$

\mathrm{I}_{2}+\text { starch } \longrightarrow \text { starch } \cdot \mathrm{I}_{2}

$$

The rate of the reaction is also influenced by the total concentration of ions, so $\mathrm{KCl}$ and $\left(\mathrm{NH}_{4}\right) \mathrm{SO}_{4}$ are added to maintain a constant value. Use the data below, obtained at $23^{\circ} \mathrm{C}$ , to determine:

(a) The average rate for each trial

(b) The order with respect trial

(c) The rate constant

(d) The rate law for the overall reaction

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Problem 115

Heat transfer to and from a reaction flask is often a critical factor in controlling reaction rate. The heat transferred $(q)$ depends on a heat transfer coefficient $(h)$ for the flask material, the temperature difference $(\Delta T)$ across the flask wall, and the commonly "wetted" area $(A)$ of the flask and bath, $q=h A \Delta T$ . When an exothermic reaction is run at a given $T$ , there is a bath temperature at which the reaction can no longer be controlled, and the reaction "runs away" suddenly. A similar problem is often seen when a reaction is "scaled up" from, say, a half-filled small flask to a half-filled large flask. Explain these behaviors.

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Problem 116

The molecular scenes below represent the first-order reaction in which cyclopropane (red) is converted to propene (green): Determine (a) the half-life and (b) the rate constant.

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Problem 117

The growth of Pseudomonas bacteria is modeled as a first-order process with $k=0.035 \mathrm{min}^{-1}$ at $37^{\circ} \mathrm{C} .$ The initial Pseudomonas population density is $1.0 \times 10^{3}$ cells/L. (a) What is the population density after 2 $\mathrm{h}$ (b) What is the time required for the population to go from 1.0 $\times 10^{3}$ to $2.0 \times 10^{3}$ cells/L?

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Problem 118

Consider the following organic reaction, in which one halogen replaces another in an alkyl halide:

$$

\mathrm{CH}_{3} \mathrm{CH}_{2} \mathrm{Br}+\mathrm{KI} \longrightarrow \mathrm{CH}_{3} \mathrm{CH}_{2} \mathrm{I}+\mathrm{KBr}

$$

In acetone, this particular reaction goes to completion because KI is soluble in acetone but $\mathrm{KBr}$ is not. In the mechanism, I- approaches the carbon opposite to the Br (see Figure 16.20 , p. $704,$ with $\mathrm{I}^{-}$ instead of $\mathrm{OH}^{-} ) .$ After $\mathrm{Br}^{-}$ has been replaced by $\mathrm{I}^{-}$ and precipitates as $\mathrm{KBr}$ , other $\mathrm{I}^{-}$ ions react with the ethyl iodide by the same mechanism.

(a) If we designate the carbon bonded to the halogen as $\mathrm{C}-1$ what is the shape around $\mathrm{C}-1$ and the hybridization of $\mathrm{C}-1$ in ethyl iodide?

(b) In the transition state, one of the two lobes of the unhybridized 2$p$ orbital of $\mathrm{C}-1$ overlaps a $p$ orbital of I, while the other lobe overlaps a $p$ orbital of Br. What is the shape around $\mathrm{C}-1$ and the hybridization of $\mathrm{C}-1$ in the transition state?

(c) The deuterated reactant, $\mathrm{CH}_{3} \mathrm{CHDBr}$ (where $\mathrm{D}$ is deuterium, 'H), has two optical isomers because $\mathrm{C}-1$ is chiral. If the reaction is run with one of the isomers, the ethyl iodide is not optically active. Explain.

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Problem 119

Another radioisotope of iodine, $131 \mathrm{I},$ is also used to study thyroid function (see Follow-up Problem 16.7 $\mathrm{A} ) .$ A patient is given a sample that is $1.7 \times 10^{-4} M^{131} \mathrm{I}$ the half-life is 8.04 days, what fraction of the radioactivity remains after $30 .$ days?

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Problem 120

The effect of substrate concentration on the first-order growth rate of a microbial population follows the Monod equation:

$$

\mu=\frac{\mu_{\max } S}{K_{s}+S}

$$

where $\mu$ is the first-order growth rate $\left(\mathrm{s}^{-1}\right), \mu_{\max }$ is the maximum

growth rate $\left(\mathrm{s}^{-1}\right), S$ is the substrate concentration $\left(\mathrm{kg} / \mathrm{m}^{3}\right),$ and $K_{\mathrm{s}}$ is the value of $S$ that gives one-half of the maximum growth rate $\left(\text { in } \mathrm{kg} / \mathrm{m}^{3}\right) .$ For $\mu_{\mathrm{max}}=1.5 \times 10^{-4} \mathrm{s}^{-1}$ and $K_{\mathrm{s}}=0.03 \mathrm{kg} / \mathrm{m}^{3} :$

(a) Plot $\mu$ vs. $S$ for $S$ between 0.0 and 1.0 $\mathrm{kg} / \mathrm{m}^{3}$

(b) The initial population density is $5.0 \times 10^{3}$ cells 3 . What is the density after $1.0 \mathrm{h},$ if the initial $S$ is 0.30 $\mathrm{kg} / \mathrm{m}^{3}$ ?

(c) What is it if the initial $S$ is 0.70 $\mathrm{kg} / \mathrm{m}^{3}$ ?

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Problem 121

The scenes depict four initial reaction mixtures for the reaction of $A$ (blue) and $B$ (yellow), with and without a solid present (gray cubes). The initial rate, $-\Delta[\mathrm{A}] / \Delta t$ (in mol/L $\cdot$ s), is shown, with each sphere representing 0.010 mol and the container volume at 0.50 $\mathrm{L}$ .

(a) What is the rate law in the absence of a catalyst?

(b) What is the overall reaction order?

(c) Find the rate constant.

(d) Do the gray cubes have a catalytic effect? Explain.

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Problem 122

The mathematics of the first-order rate law can be applied to any situation in which a quantity decreases by a constant fraction per unit of time (or unit of any other variable).

(a) As light moves through a solution, its intensity decreases per unit distance traveled in the solution. Show that

$$

\begin{array}{l}{\text { In }\left(\frac{\text { intensity of light leaving the solution }}{\text { intensity of light entering the solution }}\right)} \\ {\qquad \begin{aligned} &=-\text { fraction of light removed per unit of length } \\ & \times \text { distance traveled in solution } \end{aligned}}\end{array}

$$

(b) The value of your savings declines under conditions of constant inflation. Show that

$$

\begin{array}{l}{\text { In }\left(\frac{\text { value remaining }}{\text { initial value }}\right)} \\ {\qquad=-\text { fraction lost per unit of time } \times \text { savings time interval }}\end{array}

$$

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Problem 123

Figure 16.25$(\mathrm{p} .714)$ shows key steps in the metal-catalyzed (M) hydrogenation of ethylene:

$$

\mathrm{C}_{2} \mathrm{H}_{4}(g)+\mathrm{H}_{2}(g) \longrightarrow \mathrm{C}_{2} \mathrm{H}_{6}(g)

$$

Use the following symbols to write a mechanism that gives the overall equation:

$$

\begin{array}{ll}{\mathrm{H}_{2}(\mathrm{ads})} & {\text { adsorbed hydrogen molecules }} \\ {\mathrm{M}-\mathrm{H}} & {\text { hydrogen atoms bonded to metal atoms }} \\ {\mathrm{C}_{2} \mathrm{H}_{4}(\mathrm{ads})} & {\text { adsorbed ethylene molecules }} \\ {\mathrm{C}_{2} \mathrm{H}_{5}(\mathrm{ads})} & {\text { adsorbed ethyl radicals }}\end{array}

$$

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Problem 124

Human liver enzymes catalyze the degradation of ingested toxins. By what factor is the rate of a detoxification changed if an enzyme lowers the $E_{\mathrm{a}}$ by 5 $\mathrm{kJ} / \mathrm{mol}$ at $37^{\circ} \mathrm{C} ?$

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Problem 125

Acetone is one of the most important solvents in organic chemistry, used to dissolve everything from fats and waxes to airplane glue and nail polish. At high temperatures, it decomposes in a first-order process to methane and ketene $\left(\mathrm{CH}_{2}=\mathrm{C}=\mathrm{O}\right)$ . At

$600^{\circ} \mathrm{C},$ the rate constant is $8.7 \times 10^{-3} \mathrm{s}^{-1}$ .

(a) What is the half-life of the reaction?

(b) How long does it take for 40.% of a sample of acetone to decompose?

(c) How long does it take for 90.% of a sample of acetone to decompose?

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Problem 126

A (green), B (blue), and C (red) are structural isomers. The molecular filmstrip depicts them undergoing a chemical change as time proceeds.

(a) Write a mechanism for the reaction.

(b) What role does C play?

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