Chapter 15

The tabulated data show the rate constant of a reaction mea- sured at several different temperatures. Use an Arrhenius plot to determine the activation barrier and frequency factor for the reaction. $$ \begin{array}{cl} \text { Temperature (K) } & \text { Rate Constant (1/s) } \\ \hline 310 & 0.00434 \\ \hline 320 & 0.0140 \\ \hline 330 & 0.0421 \\ \hline 340 & 0.118 \\ \hline 350 & 0.316 \\ \hline \end{array} $$

Consider this overall reaction, which is experimentally observed to be second order in AB and zero order in C: $$ \mathrm{AB}+\mathrm{C} \longrightarrow \mathrm{A}+\mathrm{BC} $$ Is the following mechanism valid for this reaction? $$ \begin{array}{ll} \mathrm{AB}+\mathrm{AB} \longrightarrow \mathrm{AB}_{2}+\mathrm{A} & \text { Slow } \\ \mathrm{AB}_{2}+\mathrm{C} \longrightarrow \mathrm{AB}+\mathrm{BC} & \text { Fast } \end{array} $$

Consider this three-step mechanism for a reaction: \(\mathrm{Cl}_{2}(g) \underset{k_{2}}{\stackrel{k_{1}}{\rightleftarrows}} 2 \mathrm{Cl}(g)\) Fast \(\mathrm{Cl}(g)+\mathrm{CHCl}_{3}(g) \longrightarrow \mathrm{HCl}(g)+\mathrm{CCl}_{3}(g) \quad\) Slow \(\mathrm{Cl}(g)+\mathrm{CCl}_{3}(g) \longrightarrow \mathrm{CCl}_{4}(g)\) Fast a. What is the overall reaction? b. Identify the intermediates in the mechanism. c. What is the predicted rate law?

Consider this two-step mechanism for a reaction: $$ \mathrm{NO}_{2}(g)+\mathrm{Cl}_{2}(g) \longrightarrow \mathrm{ClNO}_{2}(g)+\mathrm{Cl}(g) \quad \text { Slow } $$ $$ \mathrm{NO}_{2}(g)+\mathrm{Cl}(g) \longrightarrow \mathrm{ClNO}_{2}(g) $$ Fast a. What is the overall reaction? b. Identify the intermediates in the mechanism. c. What is the predicted rate law?

Many heterogeneous catalysts are deposited on high-surfacearea supports. Why?

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

The tabulated data were collected for this reaction at \(500^{\circ} \mathrm{C}\) : $$ \mathrm{CH}_{3} \mathrm{CN}(g) \longrightarrow \mathrm{CH}_{3} \mathrm{NC}(g) $$ $$ \begin{array}{cc} \text { Time (h) } & {\left[\mathrm{CH}_{3} \mathrm{CN]}\right. \text { (M) }} \\\ 0.0 & 1.000 \\ \hline 5.0 & 0.794 \\ \hline 10.0 & 0.631 \\ \hline 15.0 & 0.501 \\ \hline 20.0 & 0.398 \\ \hline 25.0 & 0.316 \\ \hline \end{array} $$ a. Determine the order of the reaction and the value of the rate constant at this temperature. b. What is the half-life for this reaction (at the initial concentration)? c. How long will it take for \(90 \%\) of the \(\mathrm{CH}_{3} \mathrm{CN}\) to convert to \(\mathrm{CH}_{3} \mathrm{NC} ?\)

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 - \(\mathrm{L}\) reaction vessel initially contains 722 torr of cyclopropane at \(485^{\circ} \mathrm{C}\), how long will it take for the partial 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 instanta- neously, usually by photochemical decomposition of \(\mathrm{I}_{2} .\) Suppose a flash of light creates an initial [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 \(\begin{array}{lllll}85.0 & \text { seconds at a barometric pressure of } & 742.5 & \mathrm{mmHg} ?\end{array}\) (The vapor pressure of water at this temperature is \(17.5 \mathrm{mmHg}\).)

The desorption (leaving of the surface) 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 \(150 \mathrm{~K}\). a. What is the half-life of the desorption reaction? b. If the surface is initially completely covered with \(n\) -butane at \(150 \mathrm{~K},\) how long will it take for \(25 \%\) of the molecules to desorb (leave the surface)? For \(50 \%\) to desorb? c. If the surface is initially completely covered, what fraction will remain covered after 10 s? After 20 s?

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 \(1.92 \times 10^{13} \mathrm{molecules} / \mathrm{cm}^{2} \cdot \mathrm{s}\) at \(120 \mathrm{~K}\) a. If the initial surface coverage is \(8.9 \times 10^{16}\) molecules \(/ \mathrm{cm}^{2}\), how long will it take for one-half of the film to evaporate? b. What fraction of the film is left after 10 s? Assume the same initial coverage as in part a.

This reaction has an activation energy of zero in the gas phase: $$ \mathrm{CH}_{3}+\mathrm{CH}_{3} \longrightarrow \mathrm{C}_{2} \mathrm{H}_{6} $$ a. Would you expect the rate of this reaction to change very much with temperature? b. Why might the activation energy be zero? c. What other types of reactions would you expect to have little or no activation energy?

Anthropologists can estimate the age of a bone or other sample of organic matter by its carbon- 14 content. The carbon- 14 in a living organism is constant until the organism dies, after which carbon-14 decays with first- order kinetics and a half-life of 5730 years. Suppose a bone from an ancient human contains \(19.5 \%\) of the C-14 found in living organisms. How old is the bone?

Geologists can estimate the age of rocks by their uranium- 238 content. The uranium is incorporated in the rock as it hardens and then decays with first- order kinetics and a half-life of 4.5 billion years. A rock contains \(83.2 \%\) of the amount of uranium- 238 that it contained when it was formed. (The amount that the rock contained when it was formed can be deduced from the presence of the decay products of U-238.) How old is the rock?

Phosgene \(\left(\mathrm{Cl}_{2} \mathrm{CO}\right)\), a poison gas used in World War I, is formed by the reaction of \(\mathrm{Cl}_{2}\) and \(\mathrm{CO}\). The proposed mechanism for the reaction is: \(\mathrm{Cl}_{2} \rightleftharpoons 2 \mathrm{Cl} \quad\) (fast, equilibrium) \(\mathrm{Cl}+\mathrm{CO} \rightleftharpoons \mathrm{ClCO} \quad\) (fast, equilibrium) \(\mathrm{ClCO}+\mathrm{Cl}_{2} \longrightarrow \mathrm{Cl}_{2} \mathrm{CO}+\mathrm{Cl} \quad(\) slow \()\) What rate law is consistent with this mechanism?

The proposed mechanism for the formation of hydrogen bromide can be written in a simplified form as: \(\begin{array}{ll}\operatorname{Br}_{2}(g) \stackrel{k_{1}}{k_{1}} 2 \operatorname{Br}(g) & \text { Fast } \\\ \operatorname{Br}(g)+\mathrm{H}_{2}(g) \stackrel{k_{2}}{\longrightarrow} \operatorname{HBr}(g)+\mathrm{H}(g) & \text { Slow } \\\ \mathrm{H}(g)+\mathrm{Br}_{2}(g) \stackrel{k_{3}}{\longrightarrow} \mathrm{HBr}(g)+\operatorname{Br}(g) & \text { Fast }\end{array}\)

A certain substance X decomposes. Fifty percent of X remains after 100 minutes. How much \(X\) remains after 200 minutes if the reaction order with respect to \(X\) is (a) zero order, (b) first order, (c) second order?

The energy of activation for the decomposition of \(2 \mathrm{~mol}\) of \(\mathrm{HI}\) to \(\mathrm{H}_{2}\) and \(\mathrm{I}_{2}\) in the gas phase is \(185 \mathrm{~kJ}\). The heat of formation of \(\mathrm{HI}(g)\) from \(\mathrm{H}_{2}(g)\) and \(\mathrm{I}_{2}(g)\) is \(-5.65 \mathrm{~kJ} / \mathrm{mol} .\) Find the energy of activation for the reaction of \(1 \mathrm{~mol}\) of \(\mathrm{H}_{2}\) and \(1 \mathrm{~mol}\) of \(\mathrm{I}_{2}\) to form 2 mol of HI in the gas phase.

Ethyl chloride vapor decomposes by the first-order reaction: $$ \mathrm{C}_{2} \mathrm{H}_{5} \mathrm{Cl} \longrightarrow \mathrm{C}_{2} \mathrm{H}_{4}+\mathrm{HCl} $$ The activation energy is \(249 \mathrm{~kJ} / \mathrm{mol}\), and the frequency factor is \(1.6 \times 10^{14} \mathrm{~s}^{-1} .\) Find the value of the rate constant at \(710 \mathrm{~K}\) What fraction of the ethyl chloride decomposes in 15 minutes at this temperature? Find the temperature at which the rate of the reaction would be twice as fast.

In this chapter, we have seen a number of reactions in which a single reactant forms products. For example, consider the following first-order reaction: \(\mathrm{CH}_{3} \mathrm{NC}(g) \longrightarrow \mathrm{CH}_{3} \mathrm{CN}(g)\) However, we also learned that gas-phase reactions occur through collisions. a. One possible explanation for how this reaction occurs is that two molecules of \(\mathrm{CH}_{3} \mathrm{NC}\) collide with each other and form two molecules of the product in a single elementary step. If that were the case, what reaction order would you expect? b. Another possibility is that the reaction occurs through more than one step. For example, a possible mechanism involves one step in which the two \(\mathrm{CH}_{3} \mathrm{NC}\) molecules collide, resulting in the "activation" of one of them. In a second step, the activated molecule goes on to form the product. Write down this mechanism and determine which step must be rate determining in order for the kinetics of the reaction to be first order. Show explicitly how the mechanism predicts first-order kinetics.

The previous exercise shows how the first-order integrated rate law is derived from the first-order differential rate law. Begin with the second-order differential rate law and derive the sec- ond-order integrated rate law.

Three different reactions involve a single reactant converting to products. Reaction A has a half-life that is independent of the initial concentration of the reactant, reaction \(\mathrm{B}\) has a half-life that doubles when the initial concentration of the reactant doubles, and reaction \(\mathrm{C}\) has a half-life that doubles when the initial concentration of the reactant is halved. Which state- ment is most consistent with these observations? a. Reaction A is first order; reaction \(\mathrm{B}\) is second order; and reaction C is zero order. b. Reaction A is first order; reaction \(\mathrm{B}\) is zero order; and reaction C is zero order. c. Reaction A is zero order; reaction B is first order; and reaction C is second order. d. Reaction \(\mathrm{A}\) is second order; reaction \(\mathrm{B}\) is first order; and reaction C is zero order.