Chapter 11

For the reaction $$2 \mathrm{~N}_{2} \mathrm{O}(g) \longrightarrow 2 \mathrm{~N}_{2}(g)+\mathrm{O}_{2}(g)$$ the rate constant is \(0.066 \mathrm{~L} / \mathrm{mol} \cdot \mathrm{min}\) at \(565^{\circ} \mathrm{C}\) and \(22.8 \mathrm{~L} / \mathrm{mol} \cdot \mathrm{min}\) at \(728^{\circ} \mathrm{C} .\) (a) What is the activation energy of the reaction? (b) What is \(k\) at \(485^{\circ} \mathrm{C}\) ? (c) At what temperature is \(k\), the rate constant, equal to \(11.6 \mathrm{~L} / \mathrm{mol} \cdot \mathrm{min} ?\)

For the decomposition of HI, the activation energy is \(182 \mathrm{~kJ} / \mathrm{mol}\). The rate constant at \(850^{\circ} \mathrm{C}\) is \(0.0174 \mathrm{~L} / \mathrm{mol} \cdot \mathrm{h}\). (a) What is the rate constant at \(700^{\circ} \mathrm{C} ?\) (b) At what temperature will the rate constant be a fourth of what it is at \(850^{\circ} \mathrm{C} ?\)

At high temperatures, the decomposition of cyclobutane is a first-order reaction. Its activation energy is \(262 \mathrm{~kJ} / \mathrm{mol}\). At \(477^{\circ} \mathrm{C}\), its half-life is \(5.00 \mathrm{~min}\). What is its half-life (in seconds) at \(527^{\circ} \mathrm{C}\) ?

The decomposition of \(\mathrm{N}_{2} \mathrm{O}_{5}\) to \(\mathrm{NO}_{2}\) and \(\mathrm{NO}_{3}\) is a first-order gas-phase reaction. At \(25^{\circ} \mathrm{C}\), the reaction has a half-life of \(2.81\) s. At \(45^{\circ} \mathrm{C}\), the reaction has a half-life of \(0.313 \mathrm{~s}\). What is the activation energy of the reaction?

For a certain reaction, \(E_{a}\) is \(135 \mathrm{~kJ}\) and \(\Delta H=45 \mathrm{~kJ}\). In the presence of a catalyst, the activation energy is \(39 \%\) of that for the uncatalyzed reaction. Draw a diagram similar to Figure \(11.11\) but instead of showing two activated complexes (two humps) show only one activated complex (i.e., only one hump) for the reaction. What is the activation energy of the uncatalyzed reverse reaction?

A catalyst lowers the activation energy of a reaction from \(215 \mathrm{~kJ}\) to \(206 \mathrm{~kJ} .\) By what factor would you expect the reaction-rate constant to increase at \(25^{\circ} \mathrm{C} ?\) Assume that the frequency factors \((\mathrm{A})\) are the same for both reactions. (Hint: Use the formula \(\ln k=\ln \mathrm{A}-E_{2} / R T .\) )

Write the rate expression for each of the following elementary steps: (a) \(\mathrm{NO}_{3}+\mathrm{CO} \longrightarrow \mathrm{NO}_{2}+\mathrm{CO}_{2}\) (b) \(\mathrm{I}_{2} \longrightarrow 2 \mathrm{I}\) (c) \(\mathrm{NO}+\mathrm{O}_{2} \longrightarrow \mathrm{NO}_{3}\)

Write the rate expression for each of the following elementary steps: (a) \(\mathrm{NO}+\mathrm{O}_{3} \longrightarrow \mathrm{NO}_{2}+\mathrm{O}_{2}\) (b) \(2 \mathrm{NO}_{2} \longrightarrow 2 \mathrm{NO}+\mathrm{O}_{2}\) (c) \(\mathrm{K}+\mathrm{HCl} \longrightarrow \mathrm{KCl}+\mathrm{H}\)

For the reaction between hydrogen and iodine, $$\mathrm{H}_{2}(\mathrm{~g})+\mathrm{I}_{2}(\mathrm{~g}) \longrightarrow 2 \mathrm{HI}(\mathrm{g})$$ the experimental rate expression is rate \(=k\left[\mathrm{H}_{2}\right] \times\left[\mathrm{I}_{2}\right] .\) Show that this expression is consistent with the mechanism $$\begin{array}{cc}\mathrm{I}_{2}(g) \rightleftharpoons 2 \mathrm{I}(g) & \text { (fast) } \\ \mathrm{H}_{2}(g)+\mathrm{I}(g)+\mathrm{I}(g) \longrightarrow 2 \mathrm{HI}(g) & \text { (slow) }\end{array}$$

For the reaction $$2 \mathrm{H}_{2}(g)+2 \mathrm{NO}(g) \longrightarrow \mathrm{N}_{2}(g)+2 \mathrm{H}_{2} \mathrm{O}(g)$$ the experimental rate expression is rate \(=k[\mathrm{NO}]^{2} \times\left[\mathrm{H}_{2}\right] .\) The following mechanism is proposed: $$\begin{array}{cc}2 \mathrm{NO} \rightleftharpoons \mathrm{N}_{2} \mathrm{O}_{2} & \text { (fast) } \\ \mathrm{N}_{2} \mathrm{O}_{2}+\mathrm{H}_{2} \longrightarrow \mathrm{H}_{2} \mathrm{O}+\mathrm{N}_{2} \mathrm{O} & \text { (slow) } \\ \mathrm{N}_{2} \mathrm{O}+\mathrm{H}_{2} \longrightarrow \mathrm{N}_{2}+\mathrm{H}_{2} \mathrm{O} & \text { (fast) } \end{array}$$ Is this mechanism consistent with the rate expression?

At low temperatures, the rate law for the reaction $$\mathrm{CO}(\mathrm{g})+\mathrm{NO}_{2}(g) \longrightarrow \mathrm{CO}_{2}(g)+\mathrm{NO}(g)$$ is as follows: rate \(=\) constant \(\times\left[\mathrm{NO}_{2}\right]^{2}\). Which of the following mechanisms is consistent with the rate law? (a) \(\mathrm{CO}+\mathrm{NO}_{2} \longrightarrow \mathrm{CO}_{2}+\mathrm{NO}\) (b) \(2 \mathrm{NO}_{2} \rightleftharpoons \mathrm{N}_{2} \mathrm{O}_{4} \quad\) (fast) \(\mathrm{N}_{2} \mathrm{O}_{4}+2 \mathrm{CO} \longrightarrow 2 \mathrm{CO}_{2}+2 \mathrm{NO} \quad\) (slow) (c) \(2 \mathrm{NO}_{2} \longrightarrow \mathrm{NO}_{3}+\) NO \(\quad\) (slow) \(\mathrm{NO}_{3}+\mathrm{CO} \longrightarrow \mathrm{NO}_{2}+\mathrm{CO}_{2} \quad\) (fast) (d) \(2 \mathrm{NO}_{2} \longrightarrow 2 \mathrm{NO}+\mathrm{O}_{2} \quad\) (slow) \(\mathrm{O}_{2}+2 \mathrm{CO} \longrightarrow 2 \mathrm{CO}_{2} \quad\) (fast)

Two mechanisms are proposed for the reaction $$\begin{array}{cl}2 \mathrm{NO}(g)+\mathrm{O}_{2}(g) \longrightarrow 2 \mathrm{NO}_{2}(g) & \\ \text { Mechanism 1: } \mathrm{NO}+\mathrm{O}_{2} \rightleftharpoons \mathrm{NO}_{3} & \text { (fast) } \\ \mathrm{NO}_{3}+\mathrm{NO} \longrightarrow 2 \mathrm{NO}_{2} & \text { (slow) } \\ \text { Mechanism 2: } \mathrm{NO}+\mathrm{NO} \rightleftharpoons \mathrm{N}_{2} \mathrm{O}_{2} & \text { (fast) } \\ \mathrm{N}_{2} \mathrm{O}_{2}+\mathrm{O}_{2} \longrightarrow 2 \mathrm{NO}_{2} & \text { (slow) } \end{array}$$ Show that each of these mechanisms is consistent with the observed rate law: rate \(=k[\mathrm{NO}]^{2} \times\left[\mathrm{O}_{2}\right]\).

The decomposition of \(\mathrm{A}_{2} \mathrm{~B}_{2}\) to \(\mathrm{A}_{2}\) and \(\mathrm{B}_{2}\) at \(38^{\circ} \mathrm{C}\) was monitored as a function of time. A plot of \(1 /\left[\mathrm{A}_{2} \mathrm{~B}_{2}\right]\) vs. time is linear, with slope \(0.137 / M \cdot \mathrm{min}\) (a) Write the rate expression for the reaction. (b) What is the rate constant for the decomposition at \(38^{\circ} \mathrm{C} ?\) (c) What is the half-life of the decomposition when \(\left[\mathrm{A}_{2} \mathrm{~B}_{2}\right]\) is \(0.631 \mathrm{M} ?\) (d) What is the rate of the decomposition when \(\left[\mathrm{A}_{2} \mathrm{~B}_{2}\right]\) is \(0.219 \mathrm{M}\) ? (e) If the initial concentration of \(\mathrm{A}_{2} \mathrm{~B}_{2}\) is \(0.822 \mathrm{M}\) with no products present, then what is the concentration of \(\mathrm{A}_{2}\) after \(8.6\) minutes?

When a base is added to an aqueous solution of chlorine dioxide gas, the following reaction occurs: $$2 \mathrm{ClO}_{2}(a q)+2 \mathrm{OH}^{-}(a q) \longrightarrow \mathrm{ClO}_{3}^{-}(a q)+\mathrm{ClO}_{2}^{-}(a q)+\mathrm{H}_{2} \mathrm{O}$$ The reaction is first-order in \(\mathrm{OH}^{-}\) and second-order for \(\mathrm{ClO}_{2}\). Initially, when \(\left[\mathrm{ClO}_{2}\right]=0.010 \mathrm{M}\) and \(\left[\mathrm{OH}^{-}\right]=0.030 \mathrm{M}\), the rate of the reaction is \(6.00 \times 10^{-4} \mathrm{~mol} / \mathrm{L} \cdot \mathrm{s}\). What is the rate of the reaction when \(50.0 \mathrm{~mL}\) of \(0.200 \mathrm{M} \mathrm{ClO}_{2}\) and \(95.0 \mathrm{~mL}\) of \(0.155 \mathrm{M} \mathrm{NaOH}\) are added?

The decomposition of sulfuryl chloride, \(\mathrm{SO}_{2} \mathrm{Cl}_{2}\), to sulfur dioxide and chlorine gases is a first-order reaction. $$\mathrm{SO}_{2} \mathrm{Cl}_{2}(g) \longrightarrow \mathrm{SO}_{2}(g)+\mathrm{Cl}_{2}(g)$$ At a certain temperature, the half-life of \(\mathrm{SO}_{2} \mathrm{Cl}_{2}\) is \(7.5 \times 10^{2} \mathrm{~min}\). Consider a sealed flask with \(122.0 \mathrm{~g}\) of \(\mathrm{SO}_{2} \mathrm{Cl}_{2}\) (a) How long will it take to reduce the amount of \(\mathrm{SO}_{2} \mathrm{Cl}_{2}\) in the sealed flask to \(45.0 \mathrm{~g}\) ? (b) If the decomposition is stopped after \(29.0 \mathrm{~h}\), what volume of \(\mathrm{Cl}_{2}\) at \(27^{\circ} \mathrm{C}\) and \(1.00\) atm is produced?

For the first-order thermal decomposition of ozone $$\mathrm{O}_{3}(g) \longrightarrow \mathrm{O}_{2}(g)+\mathrm{O}(g)$$ \(k=3 \times 10^{-26} \mathrm{~s}^{-1}\) at \(25^{\circ} \mathrm{C}\). What is the half-life for this reaction in years? Comment on the likelihood that this reaction contributes to the depletion of the ozone layer.

Derive the integrated rate law, \([\mathrm{A}]=[\mathrm{A}]_{0}-k t\), for a zero-order reaction. (Hint: Start with the relation \(-\Delta[\mathrm{A}]=k \Delta \mathrm{t}\).)

The greatest increase in the reaction rate for the reaction between \(\mathrm{A}\) and \(\mathrm{C}\), where rate \(=k[\mathrm{~A}]^{1 / 2}[\mathrm{C}]\), is caused by (a) doubling [A] (b) halving [C] (c) halving [A] (d) doubling [A] and [C]

The following reaction is second-order in A and first-order in B. $$\mathrm{A}+\mathrm{B} \longrightarrow \text { products }$$ (a) Write the rate expression. (b) Consider the following one-liter vessel in which each square represents a mole of \(\mathrm{A}\) and each circle represents a mole of \(\mathrm{B}\). What is the rate of the reaction in terms of \(k ?\) (c) Assuming the same rate and \(k\) as (b), fill the similar one-liter vessel shown in the figure with an appropriate number of circles (representing B).

For the reaction $$\mathrm{A}+\mathrm{B} \longrightarrow \mathrm{C}$$ the rate expression is rate \(=k[\mathrm{~A}][\mathrm{B}]\) (a) Given three test tubes, with different concentrations of \(\mathrm{A}\) and \(\mathrm{B}\), which test tube has the smallest rate? (1) \(0.10 M \mathrm{~A}_{\mathrm{i}} 0.10 \mathrm{M} \mathrm{B}\) (2) \(0.15 \mathrm{MA}_{;} 0.15 \mathrm{M} \mathrm{B}\) (3) \(0.06 M \mathrm{~A} ; 1.0 M \mathrm{~B}\) (b) If the temperature is increased, describe (using the words increases, decreases, or remains the same) what happens to the rate, the value of \(k\), and \(E_{x}\)

The gas-phase reaction between hydrogen and iodine $$\mathrm{H}_{2}(g)+\mathrm{I}_{2}(g) \rightleftharpoons 2 \mathrm{HI}(g)$$ proceeds with a rate constant for the forward reaction at \(700^{\circ} \mathrm{C}\) of \(138 \mathrm{~L} / \mathrm{mol} \cdot \mathrm{s}\) and an activation energy of \(165 \mathrm{~kJ} / \mathrm{mol}\). (a) Calculate the activation energy of the reverse reaction given that \(\Delta H_{i}^{\circ}\) for HI is \(26.48 \mathrm{~kJ} / \mathrm{mol}\) and \(\Delta H_{i}^{\circ}\) for \(\mathrm{I}_{2}(\mathrm{~g})\) is \(62.44 \mathrm{~kJ} / \mathrm{mol}\). (b) Calculate the rate constant for the reverse reaction at \(700^{\circ} \mathrm{C}\). (Assume \(\mathrm{A}\) in the equation \(k=\mathrm{Ae}^{-\mathcal{P}_{2} / \mathrm{RT}^{\prime}}\) is the same for both forward and reverse reactions.) (c) Calculate the rate of the reverse reaction if the concentration of HI is \(0.200 M\). The reverse reaction is second-order in HI.

For a first-order reaction \(a \mathrm{~A} \longrightarrow\) products, where \(a \neq 1\), the rate is \(-\Delta[\mathrm{A}] / a \Delta t\), or in derivative notation, \(-\frac{1}{a} \frac{d[\mathrm{~A}]}{d t} .\) Derive the integrated rate law for the first-order decomposition of \(a\) moles of reactant.

In a first-order reaction, suppose that a quantity \(X\) of a reactant is added at regular intervals of time, \(\Delta t\). At first the amount of reactant in the system builds up; eventually, however, it levels off at a saturation value given by the expression $$\text { saturation value }=\frac{X}{1-10^{-a}} \quad \text { where } a=0.30 \frac{\Delta t}{t_{1 / 2}}$$ This analysis applies to prescription drugs, of which you take a certain amount each day. Suppose that you take \(0.100 \mathrm{~g}\) of a drug three times a day and that the half-life for elimination is \(2.0\) days. Using this equation, calculate the mass of the drug in the body at saturation. Suppose further that side effects show up when \(0.500 \mathrm{~g}\) of the drug accumulates in the body. As a pharmacist, what is the maximum dosage you could assign to a patient for an 8 -h period without causing side effects?