Chapter 14

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 for- mation of \(\mathrm{O}_{2}\) is \(1.5 \times 10^{-3} \mathrm{mol} / \mathrm{L} \cdot \mathrm{s}\). What is the rate of decomposition of \(\mathrm{O}_{3} ?\)

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 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 \(\mathbf{O}_{3}\) is halved (with no change in \(\left.\left[\mathrm{NO}_{2}\right]\right) ?\)

Nitrosyl bromide, NOBr, is formed from 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 Bra. (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 \(2.8 \times\) \(10^{-3} \min ^{-1}\) 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 (Example \(14.5)\) 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 \(3.40 \mathrm{L} / \mathrm{mol} \cdot \mathrm{min} .\) 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 \(1.9 \times 10^{-2} \mathrm{mol} / \mathrm{L}_{y}\) 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 secondorder 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-L flask. After 600. seconds, \(21 \%\) of the butadiene had dimerized. Calculate the rate constant for this reaction.

The decomposition of ammonia on a metal surface to form \(\mathrm{N}_{2}\) and \(\mathrm{H}_{2}\) is a zero-order reaction (Figure \(14.7 \mathrm{c}) .\) At \(873^{\circ} \mathrm{C},\) the value of the rate constant is \(1.5 \times 10^{-3} \mathrm{mol} / \mathrm{L} \cdot \mathrm{s}\). How long it will take to completely decompose 0.16 g of \(\mathrm{NH}_{3}\) in a \(1.0-\mathrm{L}\) flask?

Hydrogen iodide decomposes when heated, forming \(\mathrm{H}_{2}(\mathrm{g})\) and \(\mathrm{I}_{2}(\mathrm{g}) .\) The rate law for this reaction is \(-\Delta|\mathrm{HI}| / \Delta t=k|\mathrm{HI}|^{2} \cdot \mathrm{At} 443^{\circ} \mathrm{C}\) \(k=30 . \mathrm{L} / \mathrm{mol} \cdot\) min. If the initial HI(g) concentration is \(1.5 \times 10^{-2} \mathrm{mol} / \mathrm{L},\) what concentration of \(\mathrm{HI}(\mathrm{g})\) 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?

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 \(0.0216 \mathrm{min}^{-1} .\) If the initial quantity of azomethane in the flask is \(2.00 \mathrm{g}\), how much remains after 0.0500 hour? What mass of \(\mathrm{N}_{2}\) is formed in this time?

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 halflife of 245 minutes at 600 K. If you begin with \(3.6 \times 10^{-3}\) 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?

The compound \(\mathrm{Xe}\left(\mathrm{CF}_{3}\right)_{2}\) decomposes in a firstorder 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 \(0.25 \mathrm{mg}\) 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?

Gaseous \(\mathrm{NO}_{2}\) 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 L/mol \(\cdot\) s. What is the rate law for this reaction? What is the rate constant?

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 \(+0.04 \mathrm{L} / \mathrm{mol} \cdot \mathrm{s} .\) What is the rate law for this reaction?

Calculate the activation energy, \(E_{a}\) 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 \(25^{\circ} \mathrm{C}=\) \(3.46 \times 10^{-5} \mathrm{s}^{-1}\) 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, \(E_{x}\) 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 (Example \(14.5) .\) 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 \(k=1.02 \times 10^{-3} \mathrm{s}^{-1},\) respec- tively. Determine the activation energy, \(E_{a}\), from these data.

Compare the lock-and-key and induced-fit models for substrate binding to an enzyme.

To which species should an enzyme bind best: the substrate, transition state, or product of a reaction?

According to the Michaelis-Menten model, if \(1 /\) Rate is plotted versus \(1 /[\mathrm{S}],\) the intercept of the plot (when \(1 /[S]=0\) ) is \(1 /\) Rate \(_{\max }\). Using the data below at a given temperature, for a given enzyme and its substrate (S), calculate the maximum rate of the reaction, Rate_max.

What is the rate law for each of the following elementary reactions? (a) \(\mathrm{NO}(\mathrm{g})+\mathrm{NO}_{3}(\mathrm{g}) \rightarrow 2 \mathrm{NO}_{2}(\mathrm{g})\) (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, \(\mathrm{O}_{3,}\) 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: Fast, reversible $$ \mathrm{O}_{3}(\mathrm{g}) \rightleftarrows \mathrm{O}_{2}(\mathrm{g})+\mathrm{O}(\mathrm{g}) $$ Step 2: Slow $$ \mathrm{O}_{3}(\mathrm{g})+\mathrm{O}(\mathrm{g}) \rightarrow 2 \mathrm{O}_{2}(\mathrm{g}) $$ (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 to give NO and \(\mathrm{CO}_{2}\) : 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 HBr is believed to involve two steps. The overall reaction is exothermic. Step 1: Fast, endothermic $$ \mathrm{CH}_{3} \mathrm{OH}+\mathrm{H}^{+} \rightleftharpoons \mathrm{CH}_{3} \mathrm{OH}_{2}^{+} $$ 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 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 \(\left[\mathrm{OH}^{-}\right] ?\)

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 \(t_{1 / 2}\) for this reaction.

\(\mathrm{NO}_{x^{\prime}}\) a mixture of \(\mathrm{NO}\) and \(\mathrm{NO}_{2},\) plays 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 \(\mathrm{NO}_{x}\) 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 Slow $$ \mathrm{NO}_{2}+\mathrm{NO}_{2} \rightarrow \mathrm{NO}_{3}+\mathrm{NO} $$ Fast $$ \mathrm{NO}_{3}+\mathrm{CO} \rightarrow \mathrm{NO}_{2}+\mathrm{CO}_{2} $$ Mechanism 3 Two steps Slow $$ \mathrm{NO}_{2} \rightarrow \mathrm{NO}+\mathrm{O} $$ Fast $$ \mathrm{CO}+\mathrm{O} \rightarrow \mathrm{CO}_{2} $$

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 K. Calculate the rate constant and the half-life at 298 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 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 ng (nanograms) to 2.25 ng. (c) What fraction of the original dimethyl ether remains after 150 minutes?

The decomposition of phosphine, \(\mathrm{PH}_{3}\), proceeds according to the equation $$ \mathrm{PH}_{3}(\mathrm{g}) \rightarrow^{1 / 4} \mathrm{P}_{4}(\mathrm{g})+3 / 2 \mathrm{H}_{2}(\mathrm{g}) $$ It is found that the reaction has the following rate equation: Rate \(=k\left[\mathrm{PH}_{3}\right] .\) The half-life of \(\mathrm{PH}_{3}\) is 37.9 seconds at \(120^{\circ} \mathrm{C}\) (a) How much time is required for three fourths of the \(\mathrm{PH}_{3}\) to decompose? (b) What fraction of the original sample of \(\mathrm{PH}_{3}\) remains after 1.00 minute?

The ozone in the Earth's ozone layer 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 equilibrium and a slow step: Step 1: Fast, reversible \(\quad \mathrm{O}_{3}(\mathrm{g}) \rightleftharpoons \mathrm{O}_{2}(\mathrm{g})+\mathrm{O}(\mathrm{g})\) Step 2: Slow \(\quad \mathrm{O}_{3}(\mathrm{g})+\mathrm{O}(\mathrm{g}) \rightarrow 2 \mathrm{O}_{2}(\mathrm{g})\) Show that the mechanism agrees with this experimental rate law: $$ \text { Rate }=-(1 / 2) \Delta\left[\mathrm{O}_{3}\right] / \Delta t=k\left[\mathrm{O}_{3}\right]^{2} /\left[\mathrm{O}_{2}\right] $$.

The gas-phase reaction $$ 2 \mathrm{N}_{2} \mathrm{O}_{5}(\mathrm{g}) \rightarrow 4 \mathrm{NO}_{2}(\mathrm{g})+\mathrm{O}_{2}(\mathrm{g}) $$ has an activation energy of \(103 \mathrm{kJ} / \mathrm{mol},\) and the rate constant is 0.0900 min \(^{-1}\) at 328.0 K. Find the rate constant at \(318.0 \mathrm{K}\).

A reaction that occurs in our atmosphere is the oxidation of NO to the brown gas \(\mathrm{NO}_{2}\) $$ 2 \mathrm{NO}(\mathrm{g})+\mathrm{O}_{2}(\mathrm{g}) \rightarrow 2 \mathrm{NO}_{2}(\mathrm{g}) $$ The mechanism of the reaction is thought to be Step \(1: \quad 2 \mathrm{NO}(\mathrm{g}) \rightleftharpoons \mathrm{N}_{2} \mathrm{O}_{2}(\mathrm{g})\) rapidly established equilibrium Step \(2: \quad \mathrm{N}_{2} \mathrm{O}_{2}(\mathrm{g})+\mathrm{O}_{2}(\mathrm{g}) \rightarrow 2 \mathrm{NO}_{2}(\mathrm{g}) \quad\) slow Which is the rate determining step? Is there an intermediate in the reaction? If this is the correct mechanism for this reaction, what is the experimentally determined rate law?