Chapter 13

Which are more reactive: \(\mathrm{O}\) atoms or \(\mathrm{O}_{2}\) molecules? Why?

Why is gaseous OH so much more reactive than \(\mathrm{H}_{2} \mathrm{O}\) vapor?

Reaction Rates Explain the difference between the rate of a reaction at \(25^{\circ} \mathrm{C}\) and its rate constant at \(25^{\circ} \mathrm{C}.\)

Explain the difference between the average rate and the instantaneous rate of a chemical reaction.

Suggest three possible ways of monitoring the rate of the following reaction. Would the rate data (changing concentration with time) be the same for all the alternatives? $$ \mathrm{CH}_{3} \mathrm{CHO}(g) \rightarrow \mathrm{CH}_{4}(g)+\mathrm{CO}(g) $$

Suggest two ways to monitor the rate of the following reaction. Would the rate data (changing concentration with time) be the same for either method? $$ 2 \mathrm{H}_{2} \mathrm{O}_{2}(a q) \rightarrow 2 \mathrm{H}_{2} \mathrm{O}(\ell)+\mathrm{O}_{2}(g) $$

If baking soda (NaHCO \(_{3}\) ) and sidewalk deicer (CaCl_) are mixed, there is no sign of a chemical reaction. However, if these two solids are dissolved in water and mixed, they rapidly react. Explain the difference in reaction rate.

Any gas-phase reaction occurs more rapidly as the temperature of the gas increases. Why?

In the decomposition reaction \(A \rightarrow B+C\), how is the rate at which \(\mathrm{A}\) is consumed related to the rate at which \(\mathrm{B}\) is produced?

During the Haber process for synthesizing ammonia, \(\mathrm{N}_{2}(g)+3 \mathrm{H}_{2}(g) \rightarrow 2 \mathrm{NH}_{3}(g),\) the rate of formation of ammonia is twice the rate at which nitrogen is consumed. Does this mean that the mass of the reaction mixture increases as the reaction proceeds--seemingly defying the law of conservation of mass? Explain why or why not.

If the rate of change in the concentration of a reactant increases (becomes less negative) with time, does the rate of change in the concentration of a product of the same reaction increase or decrease?

During a reaction, can there be a time when the instantaneous rate of the reaction does not change? If you think so, describe such a time.

Catalytic Converters in Automobiles (i) Catalytic converters combat air pollution by converting \(\mathrm{NO}\) into \(\mathrm{N}_{2}\) and \(\mathrm{O}_{2}\). a. How is the rate of formation of \(\mathrm{O}_{2}\) related to the rate of formation of \(\mathrm{N}_{2} ?\) b. How is the rate of change in \(\left[\mathrm{N}_{2}\right]\) related to the rate of change in [NO]?

Catalytic Converters in Automobiles (II) Catalytic converters also combat air pollution by promoting the reaction between \(\mathrm{CO}\) and \(\mathrm{O}_{2}\) that produces \(\mathrm{CO}_{2}\). Aow is the rate of change in \(\left[\mathrm{CO}_{2}\right]\) related to the rate of change in \(\left[\mathrm{O}_{2}\right] ?\) b. How is the rate of change in \(\left[\mathrm{CO}_{2}\right]\) related to the rate of change in \([\mathrm{CO}] ?\)

Write an equation relating the rates of change in the concentrations of the products and reactants in each of the following reactions: a. \(F_{2}(g)+H_{2} O(\ell) \rightarrow H O F(g)+H F(g)\) b. \(\mathrm{Si}(s)+3 \mathrm{HCl}(g) \rightarrow \mathrm{SiHCl}_{3}(\ell)+\mathrm{H}_{2}(g)\) c. \(4 \mathrm{NH}_{3}(g)+3 \mathrm{O}_{2}(g) \rightarrow 2 \mathrm{N}_{2}(g)+6 \mathrm{H}_{2} \mathrm{O}(g)\)

Write an equation relating the rates of change in the concentrations of the products and reactants in each of the following reactions: a. \(\operatorname{SOF}_{2}(g)+2 \mathrm{F}_{2}(g) \rightarrow \mathrm{F}_{5} \mathrm{SOF}(g)\) b. \(\mathrm{B}_{2} \mathrm{H}_{6}(g)+6 \mathrm{Cl}_{2}(g) \rightarrow 2 \mathrm{BCl}_{3}(g)+6 \mathrm{HCl}(g)\) c. \(\mathrm{N}_{2} \mathrm{H}_{4}(g)+2 \mathrm{NH}_{2} \mathrm{Cl}(g) \rightarrow 2 \mathrm{NH}_{4} \mathrm{Cl}(s)+\mathrm{N}_{2}(g)\)

In a study of the thermal decomposition of ammonia into nitrogen and hydrogen: $$ 2 \mathrm{NH}_{3}(g) \rightarrow \mathrm{N}_{2}(g)+3 \mathrm{H}_{2}(g) $$ the average rate of change in the concentration of ammonia is \(-0.38 M / s.\) a. What is the average rate of change in \(\left[\mathrm{H}_{2}\right] ?\) b. What is the average rate of change in \(\left[\mathrm{N}_{2}\right] ?\) c. What is the average rate of the reaction?

Power Plant Emissions Sulfur dioxide emissions in stack gases at power plants may react with carbon monoxide as follows: $$ \mathrm{SO}_{2}(g)+3 \mathrm{CO}(g) \rightarrow 2 \mathrm{CO}_{2}(g)+\operatorname{COS}(g) $$ Write an equation relating the rates for each of the following: a. The rate of formation of \(\mathrm{CO}_{2}\) to the rate of consumption of CO b. The rate of formation of \(\mathrm{COS}\) to the rate of consumption of \(\mathrm{SO}_{2}\) c. The rate of consumption of \(\mathrm{CO}\) to the rate of consumption of \(\mathrm{SO}_{2}\)

Reducing Power Plant Emissions Nitrogen monoxide can be removed from gas-fired power plant emissions by reaction with methane as follows: $$ \mathrm{CH}_{4}(g)+4 \mathrm{NO}(g) \rightarrow 2 \mathrm{~N}_{2}(g)+\mathrm{CO}_{2}(g)+2 \mathrm{H}_{2} \mathrm{O}(g) $$ Write an equation relating the rates for each of the following: a. The rate of formation of \(\mathrm{N}_{2}\) to the rate of formation of \(\mathrm{CO}_{2}\) b. The rate of formation of \(\mathrm{CO}_{2}\) to the rate of consumption of NO c. The rate of consumption of \(\mathrm{CH}_{4}\) to the rate of formation of \(\mathrm{H}_{2} \mathrm{O}\)

Stratospheric Ozone Depletion Chlorine monoxide (C1O) plays a major role in the creation of the ozone holes in the stratosphere over Earth's polar regions. a. If \(\Delta[\mathrm{ClO}] / \Delta t\) at \(298 \mathrm{K}\) is \(-2.3 \times 10^{7} M / \mathrm{s},\) what is the rate of change in \(\left[\mathrm{Cl}_{2}\right]\) and \(\left[\mathrm{O}_{2}\right]\) in the following reaction? $$ 2 \mathrm{ClO}(g) \rightarrow \mathrm{Cl}_{2}(g)+\mathrm{O}_{2}(g) $$ b. If \(\Delta[\mathrm{ClO}] / \Delta t\) is \(-2.9 \times 10^{4} M / s,\) what is the rate of formation of oxygen and \(\mathrm{ClO}_{2}\) in the following reaction? $$ \mathrm{ClO}(g)+\mathrm{O}_{3}(g) \rightarrow \mathrm{O}_{2}(g)+\mathrm{ClO}_{2}(g) $$

The chemistry of smog formation includes \(\mathrm{NO}_{3}\) as an intermediate in several reactions. a. If \(\Delta\left[\mathrm{NO}_{3}\right] / \Delta t\) is \(-2.2 \times 10^{5} \mathrm{m} M / \mathrm{min}\) in the following reaction, what is the rate of formation of \(\mathrm{NO}_{2} ?\) $$ \mathrm{NO}_{3}(g)+\mathrm{NO}(g) \rightarrow 2 \mathrm{NO}_{2}(g) $$ b. What is the rate of change of \(\left[\mathrm{NO}_{2}\right]\) in the following reaction if \(\Delta\left[\mathrm{NO}_{3}\right] / \Delta t\) is \(-2.3 \mathrm{m} M / \mathrm{min}\) ? $$ 2 \mathrm{NO}_{3}(g) \rightarrow 2 \mathrm{NO}_{2}(g)+\mathrm{O}_{2}(g) $$

Nitrite ion reacts with ozone in aqueous solution, producing nitrate ion and oxygen: $$ \mathrm{NO}_{2}^{-}(a q)+\mathrm{O}_{3}(g) \rightarrow \mathrm{NO}_{3}^{-}(a q)+\mathrm{O}_{2}(g) $$ The following data were collected for this reaction at \(298 \mathrm{K}\) Calculate the average reaction rate between 0.0 and \(100.0 \mu \mathrm{s}\) (microseconds) and between 200.0 and \(300.0 \mu \mathrm{s}\) $$\begin{array}{cc} \text { Time }(\mu \mathrm{s}) & {\left[\mathrm{O}_{3}\right](\mathrm{M})} \\ 0.0 & 1.13 \times 10^{-2} \\ \hline 100.0 & 9.93 \times 10^{-3} \\ \hline 200.0 & 8.70 \times 10^{-3} \\ \hline 300.0 & 8.15 \times 10^{-3} \\ \hline \end{array}$$

Dinitrogen pentoxide \(\left(\mathrm{N}_{2} \mathrm{O}_{5}\right)\) decomposes as follows to nitrogen dioxide and nitrogen trioxide: $$ \mathrm{N}_{2} \mathrm{O}_{5}(g) \rightarrow \mathrm{NO}_{2}(g)+\mathrm{NO}_{3}(g) $$ Calculate the average rate of this reaction between consecutive measurement times in the following table. $$\begin{array}{cc} \text { Time (s) } & {\left[\mathrm{N}_{2} \mathrm{O}_{5}\right]} \\ & \text { (molecules/cm }^{3} \text { ) } \\ 0.00 & 1.500 \times 10^{12} \\ \hline 1.45 & 1.357 \times 10^{12} \\ \hline 2.90 & 1.228 \times 10^{12} \\ \hline 4.35 & 1.111 \times 10^{12} \\ \hline 5.80 & 1.005 \times 10^{12} \\ \hline \end{array}$$

Why are the units of the rate constants different for reactions of different order?

The rate of reaction does not increase with time for reactions with which order: zero, first, or second?

What effect does doubling the initial concentration of a reactant have on the half-life of a reaction that is second order in the reactant?

Suppose the decomposition reactions \(A \rightarrow B+C\) and \(X \rightarrow\) \(\mathrm{Y}+\mathrm{Z}\) are second order in \(\mathrm{A}\) and \(\mathrm{X}\), respectively, and both have the same rate constant. Under what conditions do the two reactions also have the same half-life?

For each of the following rate laws, determine the order with respect to each reactant and the overall reaction order. a. Rate \(=k[\mathrm{A}][\mathrm{B}]\) b. Rate \(=k[\mathrm{A}]^{2}[\mathrm{B}]\) c. Rate \(=k[\mathrm{A}][\mathrm{B}]^{3}\)

Determine the overall order of the following rate laws and the order with respect to each reactant. a. Rate \(=k[\mathrm{A}]^{2}[\mathrm{B}]^{1 / 2}\) b. Rate \(=k[\mathrm{A}]^{2}[\mathrm{B}][\mathrm{C}]\) c. Rate \(=k[\mathrm{A}][\mathrm{B}]^{3}[\mathrm{C}]^{1 / 2}\)

Write rate laws and determine the units of the rate constant (using the units \(M\) for concentration and s for time) for the following reactions: a. The reaction of oxygen atoms with \(\mathrm{NO}_{2}\) is first order in both reactants. b. The reaction between \(\mathrm{NO}\) and \(\mathrm{Cl}_{2}\) is second order in NO and first order in \(\mathrm{Cl}_{2}\). c. The reaction between \(\mathrm{Cl}_{2}\) and chloroform \(\left(\mathrm{CHCl}_{3}\right)\) is first order in \(\mathrm{CHCl}_{3}\) and one-half order in \(\mathrm{Cl}_{2}\) "d. The decomposition of ozone \(\left(\mathrm{O}_{3}\right)\) to \(\mathrm{O}_{2}\) is second order in \(\mathrm{O}_{3}\) and an order of -1 in \(\mathrm{O}\) atoms.

Compounds \(A\) and \(B\) react to give a single product, \(C .\) Write the rate law for each of the following cases and determine the units of the rate constant by using the units \(M\) for concentration and s for time: a. The reaction is first order in \(A\) and second order in \(B\). b. The reaction is first order in \(A\) and second order overall. c. The reaction is independent of the concentration of \(\mathrm{A}\) and second order overall. d. The reaction is second order in both \(\mathrm{A}\) and \(\mathrm{B}\).

Predict the rate law for the reaction \(2 \mathrm{Br} \mathrm{O}(g) \rightarrow \mathrm{Br}_{2}(g)+\) \(\mathrm{O}_{2}(g)\) if: a. The rate doubles when \([\mathrm{Br} \mathrm{O}]\) doubles. b. The rate quadruples when \([\mathrm{BrO}]\) doubles. c. The rate is halved when \([\mathrm{BrO}]\) is halved. d. The rate is unchanged when \([\mathrm{BrO}]\) is doubled.

Predict the rate law for the reaction \(\mathrm{NO}(g)+\mathrm{Br}_{2}(g) \rightarrow\) \(\mathrm{NOBr}_{2}(g)\) if: a. The rate doubles when \([\mathrm{NO}]\) is doubled and \(\left[\mathrm{Br}_{2}\right]\) remains constant. b. The rate doubles when \(\left[\mathrm{Br}_{2}\right]\) is doubled and \([\mathrm{NO}]\) remains constant. c. The rate increases by 1.56 times when \([\mathrm{NO}]\) is increased 1.25 times and \(\left[\mathrm{Br}_{2}\right]\) remains constant. d. The rate is halved when \([\mathrm{NO}]\) is doubled and \(\left[\mathrm{Br}_{2}\right]\) remains constant.

The rate of the reaction: $$ \mathrm{NO}(g)+\mathrm{O}_{3}(g) \rightarrow \mathrm{NO}_{2}(g)+\mathrm{O}_{2}(g) $$ quadruples when the concentrations of \(\mathrm{NO}\) and \(\mathrm{O}_{3}\) are doubled. Does this prove that the reaction is first order in both reactants? Why or why not?

The reaction between chlorine monoxide and nitrogen dioxide is second order overall and $$ \mathrm{ClO}(g)+\mathrm{NO}_{2}(g)+\mathrm{M}(g) \rightarrow \mathrm{C} 10 \mathrm{NO}_{2}(g)+\mathrm{M}(g) $$ produces chlorine nitrate (CIONO,). A third molecule (M) takes part in the reaction but is unchanged by it. The reaction is first order in \(\mathrm{NO}_{2}\) and in ClO. a. Write the rate law for this reaction. b. What is the reaction order with respect to M?

Rate Laws for Destruction of Tropospheric Ozone The reaction of \(\mathrm{NO}_{2}\) with ozone produces \(\mathrm{NO}_{3}\) in a second-order reaction overall: $$ \mathrm{NO}_{2}(g)+\mathrm{O}_{3}(g) \rightarrow \mathrm{NO}_{3}(g)+\mathrm{O}_{2}(g) $$ a. Write the rate law for the reaction if the reaction is first order in each reactant. b. The rate constant for the reaction is \(1.93 \times 10^{4} M^{-1} \mathrm{s}^{-1}\) at \(298 \mathrm{K}\). What is the rate of the reaction when $$ \left[\mathrm{NO}_{2}\right]=1.8 \times 10^{-8} \mathrm{Mand}\left[\mathrm{O}_{3}\right]=1.4 \times 10^{-7} \mathrm{MP} $$ c. What is the rate of formation of \(\mathrm{NO}_{3}\) under these conditions? d. What happens to the rate of the reaction if the concentration of \(\mathrm{O}_{3}(g)\) is doubled?

Sources of Nitric Acid in the Atmosphere The reaction between \(\mathrm{N}_{2} \mathrm{O}_{5}\) and water, $$ \mathrm{N}_{2} \mathrm{O}_{5}(g)+\mathrm{H}_{2} \mathrm{O}(g) \rightarrow 2 \mathrm{HNO}_{3}(g) $$ is a source of nitric acid in the atmosphere. a. The reaction is first order in each reactant. Write the rate law for the reaction. b. When \(\left[\mathrm{N}_{2} \mathrm{O}_{5}\right]\) is \(0.132 \mathrm{m} \mathrm{M}\) and \(\left[\mathrm{H}_{2} \mathrm{O}\right]\) is \(230 \mathrm{m} M,\) the rate of the reaction is \(4.55 \times 10^{-4} \mathrm{m} M^{-1} \mathrm{min}^{-1} .\) What is the rate constant for the reaction?

Each of the following reactions is first order in the reactants and second order overall. Which reaction is fastest if the initial concentrations of the reactants are the same? All reactions are at \(298 \mathrm{K}\) a. \(\mathrm{ClO}_{2}(g)+\mathrm{O}_{3}(g) \rightarrow \mathrm{ClO}_{3}(g)+\mathrm{O}_{2}(g)\) \(k=3.0 \times 10^{-19} \mathrm{cm}^{3} /(\text { molecule } \cdot \mathrm{s})\) b. \(\mathrm{ClO}_{2}(g)+\mathrm{NO}(g) \rightarrow \mathrm{NO}_{2}(g)+\mathrm{ClO}(g)\) \(k=3.4 \times 10^{-13} \mathrm{cm}^{3} /(\mathrm{molecule} \cdot \mathrm{s})\) c. \(\mathrm{ClO}(g)+\mathrm{NO}(g) \rightarrow \mathrm{Cl}(g)+\mathrm{NO}_{2}(g)\) \(k=1.7 \times 10^{-11} \mathrm{cm}^{3} /(\text { molecule } \cdot \mathrm{s})\) d. \(\mathrm{ClO}(g)+\mathrm{O}_{3}(g) \rightarrow \mathrm{ClO}_{2}(g)+\mathrm{O}_{2}(g)\) \(k=1.5 \times 10^{-17} \mathrm{cm}^{3} /(\text { molecule } \cdot \mathrm{s})\)

Two reactions in which there is a single reactant have nearly the same magnitude rate constant. One is first order; the other is second order. a. If the initial concentrations of the reactants are both \(1.0 \mathrm{mM},\) which reaction will proceed at the higher rate? b. If the initial concentrations of the reactants are both 2.0 \(M,\) which reaction will proceed at the higher rate?

The rate constant for the decomposition of \(\mathrm{N}_{2} \mathrm{O}_{5}\) to \(\mathrm{NO}_{2}\) and \(\mathrm{O}_{2}\) $$ 2 \mathrm{N}_{2} \mathrm{O}_{5}(g) \rightarrow 4 \mathrm{NO}_{2}(g)+\mathrm{O}_{2}(g) $$ is \(3.4 \times 10^{-5} \mathrm{s}^{-1}\) at \(298 \mathrm{K}\). What is the rate law expression for the reaction at \(298 \mathrm{K} ?\)

Hydroperoxyl Radicals in the Atmosphere During a smog event, trace amounts of many highly reactive substances are present in the atmosphere. One of these is the hydroperoxyl radical, \(\mathrm{HO}_{2},\) which reacts with sulfur trioxide, \(\mathrm{SO}_{3} .\) The rate constant for the reaction $$ 2 \mathrm{HO}_{2}(g)+\mathrm{SO}_{3}(g) \rightarrow \mathrm{H}_{2} \mathrm{SO}_{3}(g)+2 \mathrm{O}_{2}(g) $$ at \(298 \mathrm{K}\) is \(2.6 \times 10^{11} M^{-1} \mathrm{s}^{-1} .\) The initial rate of the reaction doubles when the concentration of \(\mathrm{SO}_{3}\) or \(\mathrm{HO}_{2}\) is doubled. What is the rate law for the reaction?

The following kinetic data were obtained at \(298 \mathrm{K}\) for the reaction: $$\begin{array}{cccc} \hline \text { Experiment } & \left[\mathrm{ClO}_{2}\right]_{0}(\mathrm{M}) & \left[\mathrm{OH}^{-}\right]_{0}(\mathrm{M}) & \begin{array}{c} \text { Initial Rate } \\ (\mathrm{M} / \mathrm{s}) \end{array} \\ \hline 1 & 0.060 & 0.030 & 0.0248 \\ \hline 2 & 0.020 & 0.030 & 0.00827 \\ \hline 3 & 0.020 & 0.090 & 0.0247 \\ \hline \end{array}$$ Determine the rate law and the rate constant for this reaction at \(298 \mathrm{K}.\) The following kinetic data were collected at \(298 \mathrm{K}\) for the reaction of ozone with nitrite ion, producing nitrate and oxygen: $$ \mathrm{NO}_{2}^{-}(a q)+\mathrm{O}_{3}(g) \rightarrow \mathrm{NO}_{3}^{-}(a q)+\mathrm{O}_{2}(g) $$ $$\begin{array}{cccc} \hline \text { Experiment } & \left[\mathrm{NO}_{2}\right]_{0}(\mathrm{M}) & \left[\mathrm{O}_{3}\right]_{0}(\mathrm{M}) & \begin{array}{c} \text { Initial Rate } \\ (M / \mathrm{s}) \end{array} \\ \hline 1 & 0.0100 & 0.0050 & 25 \\ \hline 2 & 0.0150 & 0.0050 & 37.5 \\ \hline 3 & 0.0200 & 0.0050 & 50.0 \\ \hline 4 & 0.0200 & 0.0200 & 200.0 \\ \hline \end{array}$$ Determine the rate law for the reaction and the value of the rate constant.

Hydrogen gas reduces \(\mathrm{NO}\) to \(\mathrm{N}_{2}\) in the following reaction: $$ 2 \mathrm{H}_{2}(g)+2 \mathrm{NO}(g) \rightarrow 2 \mathrm{H}_{2} \mathrm{O}(g)+\mathrm{N}_{2}(g) $$ The initial reaction rates of four mixtures of \(\mathrm{H}_{2}\) and \(\mathrm{NO}\) were measured at \(900^{\circ} \mathrm{C}\) with the following results: $$\begin{array}{cccc} \hline \text { Experiment } & \left[\mathrm{H}_{2}\right]_{0}(\mathrm{M}) & [\mathrm{NO}]_{0}(\mathrm{M}) & \begin{array}{c} \text { Initial Rate } \\ (M / \mathrm{s}) \end{array} \\ \hline 1 & 0.212 & 0.136 & 0.0248 \\ \hline 2 & 0.212 & 0.272 & 0.0991 \\ \hline 3 & 0.424 & 0.544 & 0.793 \\ \hline 4 & 0.848 & 0.544 & 1.59 \\ \hline \end{array}$$ Determine the rate law and the rate constant for the reaction at \(900^{\circ} \mathrm{C}.\)

The rate of the reaction $$ \mathrm{NO}_{2}(g)+\mathrm{CO}(g) \rightarrow \mathrm{NO}(g)+\mathrm{CO}_{2}(g) $$ was determined in three experiments at \(225^{\circ} \mathrm{C} .\) The results are given in the following table. $$\begin{array}{cccc} \hline & & & \begin{array}{c} \text { Initial Rate, } \\ -\mathbf{\Delta}\left[\mathrm{NO}_{2}\right] / \Delta t \end{array} \\ \text { Experiment } & \left[\mathrm{NO}_{2}\right]_{0}(\mathrm{M}) & [\mathrm{CO}]_{0}(M) & (M / \mathrm{s}) \\ \hline 1 & 0.263 & 0.826 & 1.44 \times 10^{-5} \\ \hline 2 & 0.263 & 0.413 & 1.44 \times 10^{-5} \\ \hline 3 & 0.526 & 0.413 & 5.76 \times 10^{-5} \\ \hline \hline \end{array}$$ a. Determine the rate law for the reaction. b. Calculate the value of the rate constant at \(225^{\circ} \mathrm{C}\) c. Calculate the rate of formation of \(\mathrm{CO}_{2}\) when \(\left[\mathrm{NO}_{2}\right]=\) \([\mathrm{CO}]=0.500 \mathrm{M}.\)

The reaction between propionaldehyde (CH \(_{3} \mathrm{CH}_{2} \mathrm{CHO}\) ) and hydrocyanic acid (HCN) has been studied in aqueous solution at \(25^{\circ} \mathrm{C}\). Concentrations of reactants as a function of time are shown in the following table. a. What is the average rate of consumption of HCN from \(11.12 \mathrm{min}\) to \(40.35 \mathrm{min} ?\) b. What is the average rate of consumption of propionaldehyde over that same period? $$\begin{array}{ccc} \text { Time (min) } & {\left[\mathrm{CH}_{3} \mathrm{CH}_{2} \mathrm{CHO}\right](\mathrm{M})} & {[\mathrm{HCN}](\mathrm{M})} \\ 3.28 & 0.0384 & 0.0657 \\ \hline 11.12 & 0.0346 & 0.0619 \\ \hline 24.43 & 0.0296 & 0.0569 \\ \hline 40.35 & 0.0242 & 0.0515 \\ \hline 67.22 & 0.0190 & 0.0463 \\ \hline \end{array}$$

Hydrogen peroxide decomposes spontaneously into water and oxygen gas via a first-order reaction: $$ 2 \mathrm{H}_{2} \mathrm{O}_{2}(a q) \rightarrow 2 \mathrm{H}_{2} \mathrm{O}(g)+\mathrm{O}_{2}(g) $$ but in the absence of catalysts this reaction proceeds very slowly. If a small amount of a salt containing the \(\mathrm{Fe}^{3+}\) ion is added to a \(0.437 M\) solution of \(\mathrm{H}_{2} \mathrm{O}_{2}\) in water, the reaction proceeds with a half-life of 17.3 min. What is the concentration of the solution after 10.0 min under these conditions?

Yogurt Expiration Date Labels of many food products have expiration dates, at which point they are typically removed from supermarket shelves. A particular natural yogurt degrades with a half-life of 45 days. The manufacturer of the yogurt wants unsold product pulled from the shelves when it degrades to no more than \(80 \%\) of its original quality. Assume the degradation process is first order. What should be the "best if used before" date on the container with respect to the date the yogurt was packaged?

Acetoacetic acid, \(\mathrm{CH}_{3} \mathrm{COCH}_{2} \mathrm{COOH},\) decomposes in aqueous acidic solution to form acetone and carbon dioxide: $$ =\mathrm{CH}_{3} \mathrm{COCH}_{2} \mathrm{COOH}(a q) \rightarrow \mathrm{CH}_{3} \mathrm{COCH}_{3}(a q)+\mathrm{CO}_{2}(g) $$ The reaction is first order. At room temperature, the half-life of the reactant is 139 min. a. What is the rate constant of the decomposition reaction? b. If the initial concentration of acetoacetic acid is \(2.75 M\) what is its concentration after 5 hours?

Laughing Gas Nitrous oxide ( \(\mathrm{N}_{2} \mathrm{O}\) ) is used as an anesthetic (laughing gas) and in acrosol cans to produce whipped cream. It is a potent greenhouse gas and decomposes slowly to \(\mathrm{N}_{2}\) and \(\mathrm{O}_{2}\) : $$ 2 \mathrm{N}_{2} \mathrm{O}(g) \rightarrow 2 \mathrm{N}_{2}(g)+\mathrm{O}_{2}(g) $$ a. If the plot of \(\ln \left[\mathrm{N}_{2} \mathrm{O}\right]\) as a function of time is linear, what is the rate law for the reaction? b. How many half-lives will it take for the concentration of the \(\mathrm{N}_{2} \mathrm{O}\) to reach \(6.25 \%\) of its original concentration? [Hint: The amount of reactant remaining after time \(t\left(A_{i}\right)\) is related to the amount initially present \(\left(A_{0}\right)\) by the equation \(A / A_{0}=(0.5)^{n},\) where \(n\) is the number of half-lives in time \(t .]\)

Tracing Phosphorus in Organisms Radioactive isotopes such as \(^{32} \mathrm{P}\) are used to follow biological processes. The following radioactivity data (in relative radioactivity values) were collected for a sample containing \(^{32} \mathrm{P}\) : $$\begin{array}{c} \text { Radioactivity (relative } \\ \text { Time (days) } \text { radioactivity values) } \\ 0 & 10.0 \\ \hline 1 & 9.53 \\ \hline 2 & 9.08 \\ \hline 5 & 7.85 \\ \hline 10 & 6.16 \\ \hline 20 & 3.79 \\ \hline \end{array}$$ a. Write the rate law for the decay of \(^{32} \mathrm{P}\). b. Determine the value of the first-order rate constant. c. Determine the half-life of \(^{32} \mathrm{P}\).