Chapter 15

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

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

In the reaction \(2 \mathrm{O}_{3}(\mathrm{g}) \rightarrow 3 \mathrm{O}_{2}(\mathrm{g}),\) 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 \(\Delta\left[\mathrm{NH}_{3}\right] / \Delta t ?\) $$\mathrm{N}_{2}(\mathrm{g})+3 \mathrm{H}_{2}(\mathrm{g}) \rightarrow 2 \mathrm{NH}_{3}(\mathrm{g})$$

Experimental data are listed here for the reaction \(A \rightarrow 2 B.\) $$\begin{array}{cc}\text { Time (s) } & {[\mathrm{B}](\mathrm{mol} / \mathrm{L})} \\\\\hline 0.00 & 0.000 \\\10.0 & 0.326 \\\20.0 & 0.572 \\\30.0 & 0.750 \\\40.0 & 0.890 \\\\\hline\end{array}$$ (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 -second interval from 0.0 to 40.0 seconds. 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 [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 seconds. (c) What is the instantaneous rate, \(\Delta[\mathrm{B}] / \Delta \mathrm{t},\) 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 \(\mathrm{A}\) and \(\mathrm{B}\). 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 \(\left[\mathrm{O}_{3}\right].\) $$2 \mathrm{NO}_{2}(\mathrm{g})+\mathrm{O}_{3}(\mathrm{g}) \rightarrow \mathrm{N}_{2} \mathrm{O}_{5}(\mathrm{g})+\mathrm{O}_{2}(\mathrm{g})$$ (a) Write the rate equation for the reaction. (b) If the concentration of \(\mathrm{NO}_{2}\) is tripled (and \(\left[\mathrm{O}_{3}\right]\) is not changed), what is the change in the reaction rate? (c) What is the effect on reaction rate if the concentration of \(\left.\mathbf{O}_{3} \text { is halved (with no change in }\left[\mathrm{NO}_{2}\right]\right) ?\)

Nitrosyl bromide, NOBr, is formed from \(\mathrm{NO}\) and \(\mathrm{Br}_{2}\) : $$2 \mathrm{NO}(\mathrm{g})+\mathrm{Br}_{2}(\mathrm{g}) \rightarrow 2 \mathrm{NOBr}(\mathrm{g})$$ 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 rate equation for the hydrolysis of sucrose to fructose and glucose $$\mathrm{C}_{12} \mathrm{H}_{22} \mathrm{O}_{11}(\mathrm{aq})+\mathrm{H}_{2} \mathrm{O}(\ell) \rightarrow 2 \mathrm{C}_{6} \mathrm{H}_{12} \mathrm{O}_{6}(\mathrm{aq})$$ is \(-\Delta[\text { sucrose }] / \Delta t=k\left[\mathrm{C}_{12} \mathrm{H}_{22} \mathrm{O}_{11}\right] .\) After 27 minutes 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 minutes 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: $$\mathrm{SO}_{2} \mathrm{Cl}_{2}(\mathrm{g}) \rightarrow \mathrm{SO}_{2}(\mathrm{g})+\mathrm{Cl}_{2}(\mathrm{g})$$ 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 \(1.24 \times 10^{-3} \mathrm{mol} / \mathrm{L},\) 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 (see Example 15.5 ) occurs with a first-order rate constant of \(2.42 \times 10^{-2} \mathrm{h}^{-1} .\) How long will it take for the concentration of cyclopropane to decrease from an initial concentration of \(0.080 \mathrm{mol} / \mathrm{L}\) to \(0.020 \mathrm{mol} / \mathrm{L} ?\)

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 \(k=1.06 \times 10^{-3} \mathrm{min}^{-1}\) at a given temperature. (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 decomposition of nitrogen dioxide at a high temperature $$\mathrm{NO}_{2}(\mathrm{g}) \rightarrow \mathrm{NO}(\mathrm{g})+1 / 2 \mathrm{O}_{2}(\mathrm{g})$$ 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 \(1.50 \mathrm{mol} / \mathrm{L}\)

At \(573 \mathrm{K},\) gaseous \(\mathrm{NO}_{2}(\mathrm{g})\) decomposes, forming \(\mathrm{NO}(\mathrm{g})\) and \(\mathrm{O}_{2}(\mathrm{g}) .\) If a vessel containing \(\mathrm{NO}_{2}(\mathrm{g})\) has an initial concentration of \(2.8 \times 10^{-2} \mathrm{mol} / \mathrm{L},\) how long will it take for \(75 \%\) of the \(\mathrm{NO}_{2}(\mathrm{g})\) to decompose? The decomposition of \(\mathrm{NO}_{2}(\mathrm{g})\) is second order in the reactant and the rate constant for this reaction, at \(573 \mathrm{K},\) is \(1.1 \mathrm{L} / \mathrm{mol} \cdot \mathrm{s}.\)

The dimerization of butadiene, \(\mathrm{C}_{4} \mathrm{H}_{6},\) to form 1,5-cyclooctadiene is a second-order process that occurs when the diene is heated. In an experiment, a sample of 0.0087 mol of \(\mathrm{C}_{4} \mathrm{H}_{6}\) was heated in a \(1.0-\mathrm{L}\) flask. After 600 seconds, \(21 \%\) of the butadiene had dimerized. Calculate the rate constant for this reaction.

Hydrogen iodide decomposes when heated, forming \(\mathrm{H}_{2}(\mathrm{g})\) and \(\mathrm{I}_{2}(\mathrm{g}) .\) The rate law for this reaction is \(-\Delta[\mathrm{HI}] / \Delta t=k[\mathrm{HI}]^{2} .\) At \(443^{\circ} \mathrm{C}, k=30 . \mathrm{L} / \mathrm{mol} \cdot\) min. If the initial \(\mathrm{HI}(\mathrm{g})\) concentration is \(3.5 \times 10^{-2} \mathrm{mol} / \mathrm{L},\) what concentration of HI (g) will remain after \(10 .\) minutes?

The rate equation for the decomposition of \(\mathrm{N}_{2} \mathrm{O}_{5}\) (giving \(\mathrm{NO}_{2}\) and \(\mathrm{O}_{2}\) ) is Rate \(=k\left[\mathrm{N}_{2} \mathrm{O}_{5}\right] .\) The value of \(k\) is \(6.7 \times 10^{-5} \mathrm{s}^{-1}\) 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 \(\mathrm{SO}_{2} \mathrm{Cl}_{2}\) $$\mathrm{sO}_{2} \mathrm{Cl}_{2}(\mathrm{g}) \rightarrow \mathrm{SO}_{2}(\mathrm{g})+\mathrm{Cl}_{2}(\mathrm{g})$$ is first order in \(\mathrm{SO}_{2} \mathrm{Cl}_{2},\) and the reaction has a half-life of 245 minutes at \(600 \mathrm{K}\). If you begin with \(3.6 \times 10^{-3}\) mol of \(\mathrm{SO}_{2} \mathrm{Cl}_{2}\) in a 1.0 -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: $$\mathrm{CH}_{3} \mathrm{N}=\mathrm{NCH}_{3}(\mathrm{g}) \rightarrow \mathrm{N}_{2}(\mathrm{g})+\mathrm{C}_{2} \mathrm{H}_{6}(\mathrm{g})$$ The rate constant for this reaction at \(600 \mathrm{K}\) is 0.0216 \(\min ^{-1} .\) If the initial quantity of azomethane in the flask is \(2.00 \mathrm{g},\) how much remains after 0.0500 hour? 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 \(30 .\) minutes. 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 copper(II) acetate to study Wilson's disease. The isotope has a half-life of 12.70 hours. What fraction of radioactive copper(II) acetate remains after 64 hours?

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?

Ammonia decomposes when heated according to the equation $$\mathrm{NH}_{3}(\mathrm{g}) \rightarrow \mathrm{NH}_{2}(\mathrm{g})+\mathrm{H}(\mathrm{g})$$ The data in the table for this reaction were collected at a high temperature. $$\begin{array}{cc}\text { Time (h) } & \text { [NH }\left._{3}\right] \text { (mol/L) } \\\\\hline 0 & 8.00 \times 10^{-7} \\\25 & 6.75 \times 10^{-7} \\\50 & 5.84 \times 10^{-7} \\\75 & 5.15 \times 10^{-7} \\\\\hline\end{array}$$ 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 NO, decomposes at \(573 \mathrm{K}.\) $$\mathrm{NO}_{2}(\mathrm{g}) \rightarrow \mathrm{NO}(\mathrm{g})+1 / 2 \mathrm{O}_{2}(\mathrm{g})$$ 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 \(25^{\circ} \mathrm{C}\) $$\mathrm{HOF}(\mathrm{g}) \rightarrow \mathrm{HF}(\mathrm{g})+1 / 2 \mathrm{O}_{2}(\mathrm{g})$$ Using the data in the table below, determine the rate law, and then calculate the rate constant. $$\begin{array}{cc}{[\mathrm{HOF}](\mathrm{mol} / \mathrm{L})} & \mathrm{Time}(\mathrm{min}) \\ \hline 0.850 & 0 \\\0.810 & 2.00 \\\0.754 & 5.00 \\\0.526 & 20.0 \\\0.243 & 50.0 \\\\\hline\end{array}$$

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

Calculate the activation energy, \(E_{\alpha}\), for the reaction $$2 \mathrm{N}_{2} \mathrm{O}_{5}(\mathrm{g}) \rightarrow 4 \mathrm{NO}_{2}(\mathrm{g})+\mathrm{O}_{2}(\mathrm{g})$$ 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 to a high temperature, cyclobutane, \(\mathrm{C}_{4} \mathrm{H}_{8}\) decomposes to ethylene: $$\mathrm{C}_{4} \mathrm{H}_{8}(\mathrm{g}) \rightarrow 2 \mathrm{C}_{2} \mathrm{H}_{4}(\mathrm{g})$$ The activation energy, \(E_{a},\) for this reaction is \(260 \mathrm{kJ} / \mathrm{mol} .\) At \(800 \mathrm{K},\) the rate constant \(k=0.0315 \mathrm{s}^{-1} .\) Determine the value of \(k\) at \(850 \mathrm{K}.\)

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 \(k=\) \(1.02 \times 10^{-3} \mathrm{s}^{-1},\) respectively. Determine the activation energy, \(E_{\omega},\) from these data.

The reaction of \(\mathrm{H}_{2}\) molecules with \(\mathrm{F}\) atoms $$\mathrm{H}_{2}(\mathrm{g})+\mathrm{F}(\mathrm{g}) \rightarrow \mathrm{HF}(\mathrm{g})+\mathrm{H}(\mathrm{g})$$ has an activation energy of \(8 \mathrm{kJ} / \mathrm{mol}\) and an enthalpy change of \(-133 \mathrm{kJ} / \mathrm{mol} .\) Draw a diagram similar to Figure 15.13 for this process. Indicate the activation energy and enthalpy change on this diagram.

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

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

Ozone, \(\mathbf{O}_{3},\) in the earth's upper atmosphere decomposes according to the equation $$2 \mathrm{O}_{3}(\mathrm{g}) \rightarrow 3 \mathrm{O}_{2}(\mathrm{g})$$ 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: \(\quad\) Slow \(\quad \mathbf{O}_{3}(\mathrm{g})+\mathbf{O}(\mathrm{g}) \rightarrow 2 \mathrm{O}_{2}(\mathrm{g})\) (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 \(\mathrm{NO}_{2}(\mathrm{g})+\mathrm{NO}_{2}(\mathrm{g}) \rightarrow \mathrm{NO}(\mathrm{g})+\mathrm{NO}_{3}(\mathrm{g})\) Step 2: Fast \(\mathrm{NO}_{3}(\mathrm{g})+\mathrm{CO}(\mathrm{g}) \rightarrow \mathrm{NO}_{2}(\mathrm{g})+\mathrm{CO}_{2}(\mathrm{g})\) (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.

A proposed mechanism for the reaction of \(\mathrm{NO}_{2}\) and \(\mathrm{CO}\) is Step 1: Slow, endothermic $$2 \mathrm{NO}_{2}(\mathrm{g}) \rightarrow \mathrm{NO}(\mathrm{g})+\mathrm{NO}_{3}(\mathrm{g})$$ Step 2: Fast, exothermic $$\mathrm{NO}_{3}(\mathrm{g})+\mathrm{CO}(\mathrm{g}) \rightarrow \mathrm{NO}_{2}(\mathrm{g})+\mathrm{CO}_{2}(\mathrm{g})$$ Overall Reaction: Exothermic $$\mathrm{NO}_{2}(\mathrm{g})+\mathrm{CO}(\mathrm{g}) \rightarrow \mathrm{NO}(\mathrm{g})+\mathrm{CO}_{2}(\mathrm{g})$$ (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 enthalpy change.

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: Fast, endothermic $$\mathrm{CH}_{3} \mathrm{OH}+\mathrm{H}^{+} \rightleftarrows \mathrm{CH}_{3} \mathrm{OH}_{2}^{+}$$ Step 2: Slow $$\mathrm{CH}_{3} \mathrm{OH}_{2}^{+}+\mathrm{Br}^{-} \rightarrow \mathrm{CH}_{3} \mathrm{Br}+\mathrm{H}_{2} \mathrm{O}$$ (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 Rate \(=k\left[\mathrm{CH}_{3} \mathrm{OH}\right]\left[\mathrm{H}^{+}\right]\left[\mathrm{Br}^{-}\right]\)

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?

For a first-order reaction, what fraction of reactant remains after five half- lives have elapsed?

To determine the concentration dependence of the rate of the reaction $$\mathrm{H}_{2} \mathrm{PO}_{3}^{-}(\mathrm{aq})+\mathrm{OH}^{-}(\mathrm{aq}) \rightarrow \mathrm{HPO}_{3}^{2-}(\mathrm{aq})+\mathrm{H}_{2} \mathrm{O}(\ell)$$ you might measure \(\left[\mathrm{OH}^{-}\right]\) as a function of time using a pH meter. (To do so, you would set up conditions under which \(\left[\mathrm{H}_{2} \mathrm{PO}_{3}^{-}\right]\) remains constant by using a large excess of this reactant.) How would you prove a second-order rate dependence for \(\left[\mathrm{OH}^{-}\right] ?\)

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

Formic acid decomposes at \(550^{\circ} \mathrm{C}\) according to the equation $$\mathrm{HCO}_{2} \mathrm{H}(\mathrm{g}) \rightarrow \mathrm{CO}_{2}(\mathrm{g})+\mathrm{H}_{2}(\mathrm{g})$$ 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 \(t_{1 / 4}\) for this reaction.

Ammonium cyanate, NH_NCO, rearranges in water to give urea, \(\left(\mathrm{NH}_{2}\right)_{2} \mathrm{CO}.\) $$\mathrm{NH}_{4} \mathrm{NCO}(\mathrm{aq}) \rightarrow\left(\mathrm{NH}_{2}\right)_{2} \mathrm{CO}(\mathrm{aq})$$ $$\begin{array}{cc}\text { Time (min) } & {\left[\mathrm{NH}_{4} \mathrm{NCO}\right](\mathrm{mol} / \mathrm{L})} \\\\\hline 0 & 0.458 \\\4.50 \times 10^{1} & 0.370 \\\1.07 \times 10^{2} & 0.292 \\\2.30 \times 10^{2} & 0.212 \\\6.00 \times 10^{2} & 0.114 \\\\\hline\end{array}$$ Using the data in the table: (a) Decide whether the reaction is first order or second order. (b) Calculate \(k\) for this reaction. (c) Calculate the half-life of ammonium cyanate under these conditions. (d) Calculate the concentration of \(\mathrm{NH}_{4} \mathrm{NCO}\) after 12.0 hours.

Nitrogen oxides, \(\mathrm{NO}_{x}\) (a mixture of \(\mathrm{NO}\) and \(\mathrm{NO}_{2}\) collectively designated as \(\mathrm{NO}_{x}\) ), play an essential role in the production of pollutants found in photochemical smog. The \(\mathrm{NO}_{x}\) in the atmosphere is slowly broken down to \(\mathrm{N}_{2}\) and \(\mathrm{O}_{2}\) in a first-order reaction. The average half-life of \(\mathrm{NO}_{x}\) in the smokestack emissions in a large city during daylight is 3.9 hours. (a) Starting with \(1.50 \mathrm{mg}\) in an experiment, what quantity of NO, remains after 5.25 hours? (b) How many hours of daylight must have elapsed to decrease \(1.50 \mathrm{mg}\) of \(\mathrm{NO}_{x}\) to \(2.50 \times 10^{-6} \mathrm{mg} ?\)

At temperatures below \(500 \mathrm{K},\) the reaction between carbon monoxide and nitrogen dioxide $$\mathrm{CO}(\mathrm{g})+\mathrm{NO}_{2}(\mathrm{g}) \rightarrow \mathrm{CO}_{2}(\mathrm{g})+\mathrm{NO}(\mathrm{g})$$ has the following rate equation: Rate \(=k\left[\mathrm{NO}_{2}\right]^{2} .\) Which of the three mechanisms suggested here best agrees with the experimentally observed rate equation? Mechanism \(1 \quad\) single, elementary step $$\mathrm{NO}_{2}+\mathrm{CO} \rightarrow \mathrm{CO}_{2}+\mathrm{NO}$$ Mechanism \(2 \quad\) Two steps $$\begin{aligned}&\text { Slow } \quad \mathrm{NO}_{2}+\mathrm{NO}_{2} \rightarrow\mathrm{NO}_{3}+\mathrm{NO}\\\&\text { Fast } \quad \mathrm{NO}_{3}+\mathrm{CO} \rightarrow \mathrm{NO}_{2}+\mathrm{CO}_{2}\end{aligned}$$ Mechanism \(3 \quad\) Two steps $$\begin{array}{ll}\text { Slow } & \mathrm{NO}_{2} \rightarrow \mathrm{NO}+\mathrm{O} \\\\\text { Fast } & \mathrm{CO}+\mathrm{O} \rightarrow \mathrm{CO}_{2}\end{array}$$

Nitryl fluoride can be made by treating nitrogen dioxide with fluorine: $$2 \mathrm{NO}_{2}(\mathrm{g})+\mathrm{F}_{2}(\mathrm{g}) \rightarrow 2 \mathrm{NO}_{2} \mathrm{F}(\mathrm{g})$$ Use the rate data in the table to do the following: (a) Write the rate equation for the reaction. (b) Indicate the order of reaction with respect to each component of the reaction. (c) Find the numerical value of the rate constant, \(k.\) $$\begin{array}{ccccl}\hline \text { Experiment } & {\left[\mathrm{NO}_{2}\right]} & {\left[\mathrm{F}_{2}\right]} & {\left[\mathrm{NO}_{2} \mathrm{F}\right]} & \left(\mathrm{mol} \mathrm{F}_{2} / \mathrm{L} \cdot \mathrm{s}\right) \\\\\hline 1 & 0.001 & 0.005 & 0.001 & 2.0 \times 10^{-4} \\\2 & 0.002 & 0.005 & 0.001 & 4.0 \times 10^{-4} \\\3 & 0.006 & 0.002 & 0.001 & 4.8 \times 10^{-4} \\\4 & 0.006 & 0.004 & 0.001 & 9.6 \times 10^{-4} \\\5 & 0.001 & 0.001 & 0.001 & 4.0 \times 10^{-5} \\\6 & 0.001 & 0.001 & 0.002 & 4.0 \times 10^{-5} \\\\\hline\end{array}$$

The decomposition of dinitrogen pentaoxide $$\mathrm{N}_{2} \mathrm{O}_{5}(\mathrm{g}) \rightarrow 2 \mathrm{NO}_{2}(\mathrm{g})+1 / 2 \mathrm{O}_{2}(\mathrm{g})$$ has the following rate equation: Rate \(=k\left[\mathrm{N}_{2} \mathrm{O}_{5}\right] .\) It has been found experimentally that the decomposition is \(20.5 \%\) complete in 13.0 hours at \(298 \mathrm{K}\). Calculate the rate constant and the half-life at \(298 \mathrm{K}.\)

The decomposition of gaseous dimethyl ether at ordinary pressures is first order. Its half-life is 25.0 minutes at \(500^{\circ} \mathrm{C}\) $$\mathrm{CH}_{3} \mathrm{OCH}_{3}(\mathrm{g}) \rightarrow \mathrm{CH}_{4}(\mathrm{g})+\mathrm{CO}(\mathrm{g})+\mathrm{H}_{2}(\mathrm{g})$$ (a) Starting with \(8.00 \mathrm{g}\) of dimethyl ether, what mass remains (in grams) after 125 minutes and after 145 minutes? (b) Calculate the time in minutes required to decrease \(7.60 \mathrm{ng}\) (nanograms) to 2.25 ng. (c) What fraction of the original dimethyl ether remains after 150 minutes?