Chapter 15

Give the relative rates of disappearance of reactants and formation of products for each of the following reactions. (a) \(2 \mathrm{O}_{3}(\mathrm{g}) \longrightarrow 3 \mathrm{O}_{2}(\mathrm{g})\) (b) \(2 \mathrm{HOF}(\mathrm{g}) \longrightarrow 2 \mathrm{HF}(\mathrm{g})+\mathrm{O}_{2}(\mathrm{g})\)

Give the relative rates of disappearance of reactants and formation of products for each of the following reactions. (a) \(2 \mathrm{NO}(\mathrm{g})+\mathrm{Br}_{2}(\mathrm{g}) \longrightarrow 2 \mathrm{NOBr}(\mathrm{g})\) (b) \(\mathrm{N}_{2}(\mathrm{g})+3 \mathrm{H}_{2}(\mathrm{g}) \longrightarrow 2 \mathrm{NH}_{3}(\mathrm{g})\)

In the reaction \(2 \mathrm{O}_{3}(\mathrm{g}) \longrightarrow 3 \mathrm{O}_{2}(\mathrm{g}),\) the rate of formation of \(\mathrm{O}_{2}\) is \(1.5 \times 10^{-3} \mathrm{mol} / \mathrm{L} \cdot\) s. What is the rate of decomposition of \(\mathrm{O}_{3} ?\)

In the synthesis of ammonia, if \(-\Delta\left[\mathrm{H}_{2}\right] / \Delta t=4.5 \times 10^{-4}\) \(\mathrm{mol} / \mathrm{L} \cdot \mathrm{min},\) what is \(\Delta\left[\mathrm{NH}_{3}\right] / \Delta t ?\) \(\mathrm{N}_{2}(\mathrm{g})+3 \mathrm{H}_{2}(\mathrm{g}) \longrightarrow 2 \mathrm{NH}_{3}(\mathrm{g})\)

Experimental data are listed here for the reaction \(\mathrm{A} \longrightarrow 2 \mathrm{B}.\) $$\begin{array}{ll}\hline \begin{array}{l}\text { Time } \\\\(\mathrm{s})\end{array} & \begin{array}{l}{[\mathrm{B}]} \\\\(\mathrm{mol} / \mathrm{L})\end{array} \\\\\hline 0.00 & 0.000 \\\10.0 & 0.326 \\\20.0 & 0.572 \\\30.0 & 0.750 \\\40.0 & 0.890 \\\\\hline\end{array}$$ (a) Prepare a graph from these data, connect the points with a smooth line, and calculate the rate of change of [B] for each \(10-\) s interval from 0.0 to 40.0 s. Does the rate of change decrease from one time interval to the next? Suggest a reason for this result. (b) How is the rate of change of \([\mathrm{A}]\) related to the rate of change of \([\mathrm{B}]\) in each time interval? Calculate the rate of change of \([\mathrm{A}]\) for the time interval from 10.0 to \(20.0 \mathrm{s}\) (c) What is the instantaneous rate when \([\mathrm{B}]=0.750\) \(\mathrm{mol} / \mathrm{L} ?\)

Using the rate equation "Rate \(=k[\mathrm{A}]^{2}[\mathrm{B}],\) define the order of the reaction with respect to A and B. What is the total order of the reaction?

A reaction has the experimental rate equation "Rate = \(k[\mathrm{A}]^{2} .\) How will the rate change if the concentration of \(\mathrm{A}\) is tripled? If the concentration of A is halved?

The reaction between ozone and nitrogen dioxide at \(231 \mathrm{K}\) is first order in both \(\left[\mathrm{NO}_{2}\right]\) and \(\left[\mathrm{O}_{3}\right]\) $$2 \mathrm{NO}_{2}(\mathrm{g})+\mathrm{O}_{3}(\mathrm{g}) \longrightarrow \mathrm{N}_{2} \mathrm{O}_{5}(\mathrm{s})+\mathrm{O}_{2}(\mathrm{g})$$ (a) Write the rate equation for the reaction. (b) If the concentration of \(\mathrm{NO}_{2}\) is tripled, what is the change in the reaction rate? (c) What is the effect on reaction rate if the concentration of \(\mathrm{O}_{3}\) is halved?

Nitrosyl bromide, NOBr, is formed from NO and Bre: $$2 \mathrm{NO}(\mathrm{g})+\mathrm{Br}_{2}(\mathrm{g}) \longrightarrow 2 \mathrm{NOBr}(\mathrm{g})$$ Experiments show that this reaction is second order in NO and first order in \(\mathrm{Br}_{2}.\) (a) Write the rate equation for the reaction. (b) How does the initial reaction rate change if the concentration of \(\mathrm{Br}_{2}\) is changed from \(0.0022 \mathrm{mol} / \mathrm{L}\) to \(0.0066 \mathrm{mol} / \mathrm{L} ?\) (c) What is the change in the initial rate if the concentration of NO is changed from \(0.0024 \mathrm{mol} / \mathrm{L}\) to \(0.0012 \mathrm{mol} / \mathrm{L} ?\)

The reaction $$2 \mathrm{NO}(\mathrm{g})+2 \mathrm{H}_{2}(\mathrm{g}) \longrightarrow \mathrm{N}_{2}(\mathrm{g})+2 \mathrm{H}_{2} \mathrm{O}(\mathrm{g})$$ was studied at \(904^{\circ} \mathrm{C},\) and the data in the table were collected. $$\begin{array}{lll}\hline \begin{array}{l}\text { Reactant Concentration } \\\\(\mathrm{mol} / \mathrm{L})\end{array} & & \\\\\hline[\mathrm{N} 0] & {\left[\mathrm{H}_{2}\right]} & \begin{array}{l}\text { Rate of Appearance of } \mathrm{N}_{2} \\\\(\mathrm{mol} / \mathrm{L} \cdot \mathrm{s})\end{array} \\\\\hline 0.420 & 0.122 & 0.136 \\\0.210 & 0.122 & 0.0339 \\\0.210 & 0.244 & 0.0678 \\\0.105 & 0.488 & 0.0339 \\\\\hline\end{array}$$ (a) Determine the order of the reaction for each reactant. (b) Write the rate equation for the reaction. (c) Calculate the rate constant for the reaction. (d) Find the rate of appearance of \(\mathrm{N}_{2}\) at the instant when \([\mathrm{NO}]=0.350 \mathrm{mol} / \mathrm{L}\) and \(\left[\mathrm{H}_{2}\right]=0.205 \mathrm{mol} / \mathrm{L}.\)

Carbon monoxide reacts with \(\mathrm{O}_{2}\) to form \(\mathrm{CO}_{2}\) : $$2 \mathrm{CO}(\mathrm{g})+\mathrm{O}_{2}(\mathrm{g}) \longrightarrow 2 \mathrm{CO}_{2}(\mathrm{g})$$ Information on this reaction is given in the table below. $$\begin{array}{lll}\hline[\mathrm{CO}](\mathrm{mol} / \mathrm{L}) & {\left[\mathrm{O}_{2}\right](\mathrm{mol} /\mathrm{L})} & \text { Rate }(\mathrm{mol} / \mathrm{L} \cdot \mathrm{min}) \\\\\hline 0.02 & 0.02 & 3.68 \times 10^{-5} \\\0.04 & 0.02 & 1.47 \times 10^{-4} \\\0.02 & 0.04 & 7.36 \times 10^{-5} \\\\\hline\end{array}$$ (a) What is the rate law for this reaction? (b) What is the order of the reaction with respect to CO? What is the order with respect \(\mathrm{O}_{2} ?\) What is the overall order of the reaction? (c) What is the value for the rate constant, \(k ?\)

Data for the reaction $$\mathrm{H}_{2} \mathrm{PO}_{4}^{-}(\mathrm{aq})+\mathrm{OH}^{-}(\mathrm{aq}) \longrightarrow \mathrm{HPO}_{4}^{2-}(\mathrm{aq})+\mathrm{H}_{2} \mathrm{O}(\ell)$$ are provided in the table. $$\begin{array}{llll}\hline & & & \text { Initial Rate } \\\\\text { Experiment } & {\left[\mathrm{H}_{2} \mathrm{PO}_{4}^{-}\right](\mathrm{M})} &{\left[\mathrm{OH}^{-}\right](\mathrm{M})} & (\mathrm{mol} / \mathrm{L} \cdot \mathrm{min}) \\\\\hline 1 & 0.0030 & 0.00040 & 0.0020 \\\2 & 0.0030 & 0.00080 & 0.0080 \\\3 & 0.0090 & 0.00040 & 0.0060 \\\4 & ? & 0.00033 & 0.0020 \\\\\hline\end{array}$$ (a) What is the rate law for this reaction? (b) What is the value of \(k ?\) (c) What is the concentration of \(\mathrm{H}_{2} \mathrm{PO}_{4}^{-}\) in experiment 4?

The rate equation for the hydrolysis of sucrose to fructose and glucose $$\mathrm{C}_{12} \mathrm{H}_{22} \mathrm{O}_{11}(\mathrm{aq})+\mathrm{H}_{2} \mathrm{O}(\ell) \longrightarrow 2 \mathrm{C}_{6} \mathrm{H}_{12} \mathrm{O}_{6}(\mathrm{aq})$$ is " \(-\Delta[\text { sucrose }] / \Delta t=k\left[\mathrm{C}_{12} \mathrm{H}_{22} \mathrm{O}_{11}\right] .\) After \(2.57 \mathrm{h}\) at \(27^{\circ} \mathrm{C}\) the sucrose concentration decreased from \(0.0146 \mathrm{M}\) to \(0.0132 \mathrm{M} .\) Find the rate constant, \(k.\)

The decomposition of \(\mathrm{N}_{2} \mathrm{O}_{5}\) in \(\mathrm{CCl}_{4}\) is a first-order reaction. If \(2.56 \mathrm{mg}\) of \(\mathrm{N}_{2} \mathrm{O}_{5}\) is present initially, and \(2.50 \mathrm{mg}\) is present after 4.26 min at \(55^{\circ} \mathrm{C},\) what is the value of the rate constant, \(k ?\)

The decomposition of \(\mathrm{SO}_{2} \mathrm{Cl}_{2}\) is a first-order reaction: $$\mathrm{SO}_{2} \mathrm{Cl}_{2}(\mathrm{g}) \longrightarrow \mathrm{SO}_{2}(\mathrm{g})+\mathrm{Cl}_{2}(\mathrm{g})$$ The rate constant for the reaction is \(2.8 \times 10^{-3} \mathrm{min}^{-1}\) at \(600 \mathrm{K} .\) If the initial concentration of \(\mathrm{SO}_{2} \mathrm{Cl}_{2}\) is \(1.24 \times 10^{-3}\) mol/L, how long will it take for the concentration to drop to \(0.31 \times 10^{-3} \mathrm{mol} / \mathrm{L} ?\)

The conversion of cyclopropane to propene, described in Example \(15.5,\) occurs with a first-order rate constant of \(5.4 \times 10^{-2} \mathrm{h}^{-1} .\) How long will it take for the concentration of cyclopropane to decrease from an initial concentration \(0.080 \mathrm{mol} / \mathrm{L}\) to \(0.020 \mathrm{mol} / \mathrm{L} ?\)

Ammonium cyanate, NH_NCO, rearranges in water to give urea, (NH\(_{2}\)) \(_{2}\) CO: $$\mathrm{NH}_{4} \mathrm{NCO}(\mathrm{aq}) \longrightarrow\left(\mathrm{NH}_{2}\right)_{2} \mathrm{CO}(\mathrm{aq})$$ The rate equation for this process is "Rate \(=k\) \(\left[\mathrm{NH}_{4} \mathrm{NCO}\right]^{2}, "\) where \(k=0.0113 \mathrm{L} / \mathrm{mol} \cdot\) min. If the original concentration of \(\mathrm{NH}_{4} \mathrm{NCO}\) in solution is \(0.229 \mathrm{mol} / \mathrm{L}\) how long will it take for the concentration to decrease to \(0.180 \mathrm{mol} / \mathrm{L} ?\)

The decomposition of nitrogen dioxide at a high temperature $$\mathrm{NO}_{2}(\mathrm{g}) \longrightarrow \mathrm{NO}(\mathrm{g})+\frac{1}{2} \mathrm{O}_{2}(\mathrm{g})$$ is second order in this reactant. The rate constant for this reaction is \(3.40 \mathrm{L} / \mathrm{mol} \cdot\) min. Determine the time needed for the concentration of \(\mathrm{NO}_{2}\) to decrease from \(2.00 \mathrm{mol} / \mathrm{L}\) to \(1.50 \mathrm{mol} / \mathrm{L}.\)

Hydrogen peroxide, \(\mathrm{H}_{2} \mathrm{O}_{2}(\mathrm{aq}),\) decomposes to \(\mathrm{H}_{2} \mathrm{O}(\ell)\) and \(\mathrm{O}_{2}(\mathrm{g})\) in a reaction that is first order in \(\mathrm{H}_{2} \mathrm{O}_{2}\) and has a rate constant \(k=1.06 \times 10^{-3} \mathrm{min}^{-1}\) (a) How long will it take for \(15 \%\) of a sample of \(\mathrm{H}_{2} \mathrm{O}_{2}\) to decompose? (b) How long will it take for \(85 \%\) of the sample to decompose?

The thermal decomposition of \(\mathrm{HCO}_{2} \mathrm{H}\) is a first-order reaction with a rate constant of \(2.4 \times 10^{-3} \mathrm{s}^{-1}\) at a given temperature. How long will it take for three fourths of a sample of HCO, H to decompose?

The rate equation for the decomposition of \(\mathrm{N}_{2} \mathrm{O}_{5}\) (giving \(\mathrm{NO}_{2}\) and \(\mathrm{O}_{2}\) ) is \(^{*}-\Delta\left[\mathrm{N}_{2} \mathrm{O}_{5}\right] / \Delta t=k\left[\mathrm{N}_{2} \mathrm{O}_{5}\right] .\) "The value of \(k\) is \(5.0 \times 10^{-4} \mathrm{s}^{-1}\) for the reaction at a particular temperature. (a) Calculate the half-life of \(\mathrm{N}_{2} \mathrm{O}_{5}\) (b) How long does it take for the \(\mathrm{N}_{2} \mathrm{O}_{5}\) concentration to drop to one tenth of its original value?

The decomposition of \(\mathrm{SO}_{2} \mathrm{Cl}_{2}\) $$\mathrm{SO}_{2} \mathrm{Cl}_{2}(\mathrm{g}) \longrightarrow \mathrm{SO}_{2}(\mathrm{g})+\mathrm{Cl}_{2}(\mathrm{g})$$ is first order in \(\mathrm{SO}_{2} \mathrm{Cl}_{2}\), and the reaction has a half-life of 245 min at \(600 \mathrm{K}\). If you begin with \(3.6 \times 10^{-3} \mathrm{mol}\) of \(\mathrm{SO}_{2} \mathrm{Cl}_{2}\) in a \(1.0-\mathrm{L}\) flask, how long will it take for the amount of \(\mathrm{SO}_{2} \mathrm{Cl}_{2}\) to decrease to \(2.00 \times 10^{-4}\) mol?

Gaseous azomethane, \(\mathrm{CH}_{3} \mathrm{N}=\mathrm{NCH}_{3},\) decomposes in a first-order reaction when heated: $$\mathrm{CH}_{3} \mathrm{N}=\mathrm{NCH}_{3}(\mathrm{g}) \longrightarrow \mathrm{N}_{2}(\mathrm{g})+\mathrm{C}_{2} \mathrm{H}_{6}(\mathrm{g})$$ The rate constant for this reaction at \(425^{\circ} \mathrm{C}\) is \(40.8 \mathrm{min}^{-1}\) If the initial quantity of azomethane in the flask is \(2.00 \mathrm{g}\) how much remains after 0.0500 min? What quantity of \(\mathrm{N}_{2}\) is formed in this time?

The compound \(\mathrm{Xe}\left(\mathrm{CF}_{3}\right)_{2}\) decomposes in a first-order reaction to elemental Xe with a half-life of 30. min. If you place \(7.50 \mathrm{mg}\) of \(\mathrm{Xe}\left(\mathrm{CF}_{3}\right)_{2}\) in a flask, how long must you wait until only 0.25 mg of \(\mathrm{Xe}\left(\mathrm{CF}_{3}\right)_{2}\) remains?

The radioactive isotope \(^{64} \mathrm{Cu}\) is used in the form of \(\mathrm{cop}\) per(II) acetate to study Wilson's disease. The isotope has a half-life of \(12.70 \mathrm{h}\). What fraction of radioactive copper (II) acetate remains after \(64 \mathrm{h} ?\)

Radioactive gold-198 is used in the diagnosis of liver problems. The half-life of this isotope is 2.7 days. If you begin with a 5.6-mg sample of the isotope, how much of this sample remains after 1.0 day?

Formic acid decomposes at \(550^{\circ} \mathrm{C}\) according to the equation $$\mathrm{HCO}_{2} \mathrm{H}(\mathrm{g}) \longrightarrow \mathrm{CO}_{2}(\mathrm{g})+\mathrm{H}_{2}(\mathrm{g})$$ The reaction follows first-order kinetics. In an experiment, it is determined that \(75 \%\) of a sample of \(\mathrm{HCO}_{2} \mathrm{H}\) has decomposed in 72 seconds. Determine \(t_{1 / 2}\) for this reaction.

The decomposition of \(\mathrm{SO}_{2} \mathrm{Cl}_{2}\) to \(\mathrm{SO}_{2}\) and \(\mathrm{Cl}_{2}\) at high temperature is a first-order reaction with a half-life of \(2.5 \times 10^{3}\) min. What fraction of \(\mathrm{SO}_{2} \mathrm{Cl}_{2}\) will remain after 750 min?

Common sugar, sucrose, breaks down in dilute acid solution to form glucose and fructose. Both products have the same formula, \(\mathrm{C}_{6} \mathrm{H}_{12} \mathrm{O}_{6}.\) $$\mathrm{C}_{12} \mathrm{H}_{22} \mathrm{O}_{11}(\mathrm{aq})+\mathrm{H}_{2} \mathrm{O}(\ell) \longrightarrow 2 \mathrm{C}_{6} \mathrm{H}_{12} \mathrm{O}_{6}(\mathrm{aq})$$ The rate of this reaction has been studied in acid solution, and the data in the table were obtained. $$\begin{array}{cc}\hline \begin{array}{c}\text { Time } \\\\(\text { min })\end{array} & \begin{array}{c}{\left[\mathrm{C}_{12} \mathrm{H}_{22} \mathbf{0}_{11}\right]} \\\\(\mathrm{mol} / \mathrm{L})\end{array} \\\\\hline 0 & 0.316 \\\39 & 0.274 \\\80 & 0.238 \\\140 & 0.190 \\\210 & 0.146 \\\\\hline\end{array}$$ (a) Plot ln [sucrose] versus time and \(1 /\) [sucrose] versus time. What is the order of the reaction? (b) Write the rate equation for the reaction, and calculate the rate constant, \(k.\) (c) Estimate the concentration of sucrose after 175 min.

Data for the decomposition of dinitrogen oxide $$2 \mathrm{N}_{2} \mathrm{O}(\mathrm{g}) \longrightarrow 2 \mathrm{N}_{2} (\mathrm{g})+\mathrm{O}_{2}(\mathrm{g})$$ on a gold surface at \(900^{\circ} \mathrm{C}\) are given below. Verify that the reaction is first order by preparing a graph of \(\ln \left[\mathrm{N}_{2} \mathrm{O}\right]\) versus time. Derive the rate constant from the slope of the line in this graph. Using the rate law and value of \(k\), determine the decomposition rate at \(900^{\circ} \mathrm{C}\) when \(\left[\mathrm{N}_{2} \mathrm{O}\right]=\) \(0.035 \mathrm{mol} / \mathrm{L}.\) $$\begin{array}{cc}\hline \begin{array}{c}\text { Time } \\\\\text { (min) }\end{array} & \begin{array}{c}{\left[\mathrm{N}_{2} 0\right]} \\\\(\mathrm{mol} / \mathrm{L})\end{array} \\\\\hline 15.0 & 0.0835 \\\30.0 & 0.0680 \\\80.0 & 0.0350 \\\120.0 & 0.0220 \\\\\hline\end{array}$$

Ammonia decomposes when heated according to the equation $$\mathrm{NH}_{3}(\mathrm{g}) \longrightarrow \mathrm{NH}_{2}(\mathrm{g})+\mathrm{H}(\mathrm{g})$$ The data in the table for this reaction were collected at a high temperature. $$\begin{array}{ll}\hline \begin{array}{l}\text { Time } \\\\\text { (h) }\end{array} & \begin{array}{l}{\left[\mathrm{NH}_{3}\right]} \\\\(\mathrm{mol} / \mathrm{L})\end{array} \\\\\hline 0 & 8.00 \times 10^{-7} \\\25 & 6.75 \times 10^{-7} \\\50 & 5.84 \times 10^{-7} \\\75 & 5.15 \times 10^{-7} \\\\\hline\end{array}$$ Plot \(\ln \left[\mathrm{NH}_{3}\right]\) versus time and \(1 /\left[\mathrm{NH}_{3}\right]\) versus time. What is the order of this reaction with respect to NH \(_{3} ?\) Find the rate constant for the reaction from the slope.

Gaseous \(\left[\mathrm{NO}_{2}\right]\) decomposes at \(573 \mathrm{K}\) $$2 \mathrm{NO}_{2}(\mathrm{g}) \longrightarrow 2 \mathrm{NO}(\mathrm{g})+\mathrm{O}_{2}(\mathrm{g})$$ The concentration of \(\mathrm{NO}_{2}\) was measured as a function of time. A graph of \(1 /\left[\mathrm{NO}_{2}\right]\) versus time gives a straight line with a slope of \(1.1 \mathrm{L} / \mathrm{mol} \cdot\) s. What is the rate law for this reaction? What is the rate constant?

The decomposition of HOF occurs at \(25^{\circ} \mathrm{C}\) $$2 \mathrm{HOF}(\mathrm{g}) \longrightarrow 2 \mathrm{HF}(\mathrm{g})+\mathrm{O}_{2}(\mathrm{g})$$ Using the data in the table below, determine the rate law and then calculate the rate constant. $$\begin{array}{lc}\hline \begin{array}{l}{[\mathrm{HOF}]} \\\\(\mathrm{mol} / \mathrm{L})\end{array} & \begin{array}{l}\text { Time } \\\\(\mathrm{min})\end{array} \\\\\hline 0.850 & 0 \\\0.810 & 2.00 \\\0.754 & 5.00 \\\0.526 & 20.0 \\\0.243 & 50.0 \\\\\hline\end{array}$$

For the reaction \(2 \mathrm{C}_{2} \mathrm{F}_{4} \longrightarrow \mathrm{C}_{4} \mathrm{F}_{8},\) a graph of \(1 /\left[\mathrm{C}_{2} \mathrm{F}_{4}\right]\) versus time gives a straight line with a slope of \(+0.04 \mathrm{L} / \mathrm{mol} \cdot \mathrm{s} .\) What is the rate law for this reaction?

Calculate the activation energy, \(E_{\mathrm{a}},\) for the reaction $$\mathrm{N}_{2} \mathrm{O}_{5}(\mathrm{g}) \longrightarrow 2 \mathrm{NO}_{2}(\mathrm{g})+\frac{1}{2} \mathrm{O}_{2}(\mathrm{g})$$ from the observed rate constants: \(k\) at \(25^{\circ} \mathrm{C}=3.46 \times\) \(10^{-5} s^{-1}\) and \(k\) at \(55^{\circ} \mathrm{C}=1.5 \times 10^{-3} \mathrm{s}^{-1}.\)

If the rate constant for a reaction triples when the temperature rises from \(3.00 \times 10^{2} \mathrm{K}\) to \(3.10 \times 10^{2} \mathrm{K},\) what is the activation energy of the reaction?

When heated, cyclopropane is converted to propene (see Example 15.5 ). Rate constants for this reaction at \(470^{\circ} \mathrm{C}\) and \(510^{\circ} \mathrm{C}\) are \(k=1.10 \times 10^{-4} \mathrm{s}^{-1}\) and \(k=1.02 \times10^{-3} \mathrm{s}^{-1},\) respectively. Determine the activation energy, \(E_{\mathrm{a}}\) from these data.

What is the rate law for each of the following elementary reactions? (a) \(\mathrm{NO}(\mathrm{g})+\mathrm{NO}_{3}(\mathrm{g}) \longrightarrow 2 \mathrm{NO}_{2}(\mathrm{g})\) (b) \(\mathrm{Cl}(\mathrm{g})+\mathrm{H}_{2}(\mathrm{g}) \longrightarrow \mathrm{HCl}(\mathrm{g})+\mathrm{H}(\mathrm{g})\) (c) \(\left(\mathrm{CH}_{3}\right)_{3} \mathrm{CBr}(\mathrm{aq}) \longrightarrow\left(\mathrm{CH}_{3}\right)_{3} \mathrm{C}^{+}(\mathrm{aq})+\mathrm{Br}^{-}(\mathrm{aq})\)

What is the rate law for each of the following elementary reactions? (a) \(\mathrm{Cl}(\mathrm{g})+\mathrm{ICl}(\mathrm{g}) \longrightarrow \mathrm{I}(\mathrm{g})+\mathrm{Cl}_{2}(\mathrm{g})\) (b) \(\mathbf{O}(\mathrm{g})+\mathbf{O}_{3}(\mathrm{g}) \longrightarrow 2 \mathrm{O}_{2}(\mathrm{g})\) (c) \(2 \mathrm{NO}_{2}(\mathrm{g}) \longrightarrow \mathrm{N}_{2} \mathrm{O}_{4}(\mathrm{g})\)

Ozone, \(\mathrm{O}_{3},\) in the earth's upper atmosphere decomposes according to the equation $$2 \mathrm{O}_{3}(\mathrm{g}) \longrightarrow 3 \mathrm{O}_{2}(\mathrm{g})$$ The mechanism of the reaction is thought to proceed through an initial fast, reversible step followed by a slow, second step. Step 1 \(\quad\) Fast, reversible \(\mathbf{O}_{3}(\mathrm{g}) \rightleftarrows \mathrm{O}_{2}(\mathrm{g})+\mathrm{O}(\mathrm{g})\) Step 2 Slow \(\quad \mathrm{O}_{3}(\mathrm{g})+\mathrm{O}(\mathrm{g}) \longrightarrow 2 \mathrm{O}_{2}(\mathrm{g})\) (a) Which of the steps is rate-determining? (b) Write the rate equation for the rate-determining step.

The reaction of \(\mathrm{NO}_{2}(\mathrm{g})\) and \(\mathrm{CO}(\mathrm{g})\) is thought to occur in two steps: Step 1 Slow \(\quad \mathrm{NO}_{2}(\mathrm{g})+\mathrm{NO}_{2}(\mathrm{g}) \longrightarrow \mathrm{NO}(\mathrm{g})+\mathrm{NO}_{3}(\mathrm{g})\) Step 2 Fast \(\quad \mathrm{NO}_{3}(\mathrm{g})+\mathrm{CO}(\mathrm{g}) \longrightarrow \mathrm{NO}_{2}(\mathrm{g})+\mathrm{CO}_{2}(\mathrm{g})\) (a) Show that the elementary steps add up to give the overall, stoichiometric equation. (b) What is the molecularity of each step? (c) For this mechanism to be consistent with kinetic data, what must be the experimental rate equation? (d) Identify any intermediates in this reaction.

The mechanism for the reaction of \(\mathrm{CH}_{3} \mathrm{OH}\) and \(\mathrm{HBr}\) is believed to involve two steps. The overall reaction is exothermic. Step 1 \(\quad\) Fast, endothermic $$\mathrm{CH}_{3} \mathrm{OH}+\mathrm{H}^{+} \rightleftarrows \mathrm{CH}_{3} \mathrm{OH}_{2}^{+}$$ Step 2 Slow $$\mathrm{CH}_{3} \mathrm{OH}_{2}^{+}+\mathrm{Br}^{-} \longrightarrow \mathrm{CH}_{3} \mathrm{Br}+\mathrm{H}_{2} \mathrm{O}$$ (a) Write an equation for the overall reaction. (b) Draw a reaction coordinate diagram for this reaction. (c) Show that the rate law for this reaction is \(-\Delta\left[\mathrm{CH}_{3} \mathrm{OH}\right] / \Delta t=k\left[\mathrm{CH}_{3} \mathrm{OH}\right]\left[\mathrm{H}^{+}\right]\left[\mathrm{Br}^{-}\right]\)

A proposed mechanism for the reaction of \(\mathrm{NO}_{2}\) and \(\mathrm{CO}\) is Step 1 Slow, endothermic $$2 \mathrm{NO}_{2}(\mathrm{g}) \longrightarrow \mathrm{NO}(\mathrm{g})+\mathrm{NO}_{3}(\mathrm{g})$$ Step 2 \(\quad\) Fast, exothermic $$\mathrm{NO}_{3}(\mathrm{g})+\mathrm{CO}(\mathrm{g}) \longrightarrow \mathrm{NO}_{2}(\mathrm{g})+\mathrm{CO}_{2}(\mathrm{g})$$ Overall Reaction Exothermic $$\mathrm{NO}_{2}(\mathrm{g})+\mathrm{CO}(\mathrm{g}) \longrightarrow \mathrm{NO}(\mathrm{g})+\mathrm{CO}_{2}(\mathrm{g})$$ (a) Identify each of the following as a reactant, product, or intermediate: \(\mathrm{NO}_{2}(\mathrm{g}), \mathrm{CO}(\mathrm{g}), \mathrm{NO}_{3}(\mathrm{g}), \mathrm{CO}_{2}(\mathrm{g})\) \(\mathrm{NO}(\mathrm{g})\) (b) Draw a reaction coordinate diagram for this reaction. Indicate on this drawing the activation energy for each step and the overall reaction enthalpy.

A three-step mechanism for the reaction of \(\left(\mathrm{CH}_{3}\right)_{3} \mathrm{CBr}\) and \(\mathrm{H}_{2} \mathrm{O}\) is proposed: Step 1 Slow $$\left(\mathrm{CH}_{3}\right)_{3} \mathrm{CBr} \longrightarrow\left(\mathrm{CH}_{3}\right)_{3} \mathrm{C}^{+}+\mathrm{Br}^{-}$$ Step 2 Fast $$\left(\mathrm{CH}_{3}\right)_{3} \mathrm{C}^{+}+\mathrm{H}_{2} \mathrm{O} \longrightarrow\left(\mathrm{CH}_{3}\right)_{3} \mathrm{COH}_{2}^{+}$$ Step 3 Fast $$\left(\mathrm{CH}_{3}\right)_{3} \mathrm{COH}_{2}^{+}+\mathrm{Br}^{-} \longrightarrow\left(\mathrm{CH}_{3}\right)_{3} \mathrm{COH}+\mathrm{HBr}$$ (a) Write an equation for the overall reaction. (b) Which step is rate-determining? (c) What rate law is expected for this reaction?

A reaction has the following experimental rate equation: Rate \(=k[\mathrm{A}]^{2}[\mathrm{B}] .\) If the concentration of \(\mathrm{A}\) is doubled and the concentration of \(\mathrm{B}\) is halved, what happens to the reaction rate?

After five half-life periods for a first-order reaction, what fraction of reactant remains?

Gaseous ammonia is made by the reaction $$\mathrm{N}_{2}(\mathrm{g})+3 \mathrm{H}_{2}(\mathrm{g}) \longrightarrow 2 \mathrm{NH}_{3}(\mathrm{g})$$ Use the information on the formation of \(\mathrm{NH}_{3}\) given in the table to answer the questions that follow. $$\begin{array}{lll}\hline\left[\mathrm{N}_{2}\right](\mathrm{M}) & {\left[\mathrm{H}_{2}\right](\mathrm{M})} & \text { Rate }(\mathrm{mol} / \mathrm{L} \cdot \mathrm{min}) \\\\\hline 0.030 & 0.010 & 4.21 \times 10^{-5} \\\0.060 & 0.010 & 1.68 \times 10^{-4} \\\0.030 & 0.020 & 3.37 \times 10^{-4} \\\\\hline\end{array}$$ (a) Determine \(n\) and \(m\) in the rate equation: Rate \(=\) \(k\left[\mathrm{N}_{2}\right]^{n}\left[\mathrm{H}_{2}\right]^{m}\) (b) Calculate the value of the rate constant. (c) What is the order of the reaction with respect to \(\left[\mathrm{H}_{2}\right] ?\) (d) What is the overall order of the reaction?

The decomposition of ammonia is first order with respect to \(\mathrm{NH}_{3}\). (Compare with Study Question 58.) $$2 \mathrm{NH}_{3}(\mathrm{g}) \longrightarrow \mathrm{N}_{2}(\mathrm{g})+3 \mathrm{H}_{2}(\mathrm{g})$$ (a) What is the rate equation for this reaction? (b) Calculate the rate constant, \(k\), given the following data: $$\begin{array}{lc}\hline\left[\mathrm{NH}_{3}\right](\mathrm{mol} / \mathrm{L}) & \text { Time }(\mathrm{s}) \\\\\hline 0.67 & 0 \\\0.26 & 19 \\\\\hline\end{array}$$ (c) Determine the half-life of \(\mathrm{NH}_{3}\)

Data for the following reaction are given in the table. $$2 \mathrm{NO}(\mathrm{g})+\mathrm{Br}_{2}(\mathrm{g}) \longrightarrow 2 \mathrm{NOBr}(\mathrm{g})$$ $$\begin{array}{llll}\hline \text { Experiment } & \begin{array}{l}{[\mathrm{NO}]} \\\\(\mathrm{M})\end{array} & \begin{array}{l}{\left[\mathrm{Br}_{2}\right]} \\\\(\mathrm{M})\end{array} & \begin{array}{l}\text { Initial Rate } \\\\(\mathrm{mol} / \mathrm{L} \cdot \mathrm{s})\end{array} \\\\\hline 1 & 1.0 \times 10^{-2} & 2.0 \times 10^{-2} & 2.4 \times 10^{-2} \\\2 & 4.0 \times 10^{-2} & 2.0 \times 10^{-2} & 0.384 \\\3 & 1.0 \times 10^{-2} & 5.0 \times 10^{-2} & 6.0 \times 10^{-2} \\\\\hline\end{array}$$ (a) What is the order of the reaction with respect to \([\mathrm{NO}] ?\) (b) What is the order with respect to \(\left[\mathrm{Br}_{2}\right] ?\) (c) What is the overall order of the reaction?

The decomposition of \(\mathrm{CO}_{2}\) is first order with respect to the concentration of \(\mathrm{CO}_{2}.\) $$2 \mathrm{CO}_{2}(\mathrm{g}) \longrightarrow 2 \mathrm{CO}(\mathrm{g})+\mathrm{O}_{2}(\mathrm{g})$$ Data on this reaction are provided in the table below. $$\begin{array}{lc}\hline\left[\mathrm{CO}_{2}\right](\mathrm{mol} / \mathrm{L}) & \text { Time }(\mathrm{s}) \\\\\hline 0.38 & 0 \\\0.27 & 12 \\\\\hline\end{array}$$ (a) Write the rate equation for this reaction. (b) Use the data to determine the value of \(k\) (c) What is the half-life of \(\mathrm{CO}_{2}\) under these conditions?