Solution: Write the reaction rate expressions for the

What is meant by the rate of a chemical reaction? What are the units of the rate of a reaction?

Write the rate expressions for the following reaction : \(\mathrm{CH}_{4}(\mathrm{~g})+2 \mathrm{O}_{2}(\mathrm{~g}) \rightarrow \mathrm{CO}_{2}(g)+2 \mathrm{H}_{2} \mathrm{O}(\mathrm{g})\)

Write a balanced equation for a gas-phase reaction whose rate is given by \(\text { rate }=-\frac{1}{2} \frac{\Delta[\mathrm{NOCl}]}{\Delta t}=\frac{1}{2} \frac{\Delta[\mathrm{NO}]}{\Delta t}=\frac{\Delta\left[\mathrm{Cl}_{2}\right]}{\Delta t}\)

Distinguish between average rate and instantaneous rate. Which of the two rates gives us an unambiguous measurement of reaction rate? Why?

Consider the reaction \(4 \mathrm{PH}_{3}(g) \rightarrow \mathrm{P}_{4}(g)+6 \mathrm{H}_{2}(g)\) Suppose that, at a particular moment during the reaction, molecular hydrogen is being formed at the rate of 0.078 M/s. (a) At what rate is \(P_{4}\) being formed? (b) At what rate is \(P H_{3}\) reacting?

The relative rates of the reaction 2A + B ? products shown in the diagrams (a)-(c) are 1:2:4. The red spheres represent A molecules and the green spheres represent B molecules. Write a rate law for this reaction.

What are the advantages of measuring the initial rate of a reaction?

The reaction of peroxydisulfate ion (\(\mathrm{S}_{2} \mathrm{O}_{8}^{2-}\)) with iodide ion (\(I^{-}\)) is \(S_{2} O_{8}^{2-}(a q)+3 I^{-}(a q) \rightarrow 2 S O_{4}^{2-}(a q)+I_{3}^{-}(a q)\) From the following data collected at a certain temperature, determine the rate law and calculate the rate constant.

Consider the first-order reaction A ? B in which A molecules (blue spheres) are converted to B molecules (orange spheres). (a) What are the half-life and rate constant for the reaction? (b) How many molecules of A and B are present at t = 20 s and t = 30 s?

Can you suggest two reactions that are very slow (take days or longer to complete) and two reactions that are very fast (reactions that are over in minutes or seconds)?

The reaction 2A ? B is first order in A with a rate constant of \(2.8 \times 10^{-2} s^{-1}\) at 80°C. How long (in seconds) will it take for A to decrease from 0.88M to 0.14M?

Consider the reaction A ? products. The half-life of the reaction depends on the initial concentration of A. Which of the following statements is inconsistent with the given information? (a) The half-life of the reaction decreases as the initial concentration increases. (b) A plot of In \([A]_{t}\) versus t yields a straight line. (c) Doubling the concentration of A quadruples the rate.

Write the reaction rate expressions for the following reactions in terms of the disappearance of the reactants and the appearance of products: (a) \(\mathrm{H}_{2}(g)+\mathrm{I}_{2}(g) \rightarrow 2 \mathrm{HI}(g)\) (b) \(5 \mathrm{Br}^{-}(a q)+\mathrm{BrO}_{3}^{-}(a q)+6 \mathrm{H}^{+}(a q) \rightarrow 3 \mathrm{Br}_{2}(a q)+3 \mathrm{H}_{2} \mathrm{O}(\mathrm{l})\)

Ethyl iodide (\(\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{I}\)) decomposes at a certain temperature in the gas phase as follows: \(\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{I}(\mathrm{g}) \rightarrow \mathrm{C}_{2} \mathrm{H}_{4}(\mathrm{~g})+\mathrm{HI}(\mathrm{g})\) From the following data determine the order.of the reaction and the rate constant.

(a) What can you deduce about the magnitude of the activation energy of a reaction if its rate constant changes appreciably with a small change in temperature? (b) If a reaction occurs every time two reacting molecules collide, what can you say about the orientation factor and the activation energy of the reaction?

Write the reaction rate expressions for the following reactions in terms of the disappearance of the reactants and the appearance of products: (a) \(2 \mathrm{H}_{2}(g)+\mathrm{O}_{2}(g) \rightarrow 2 \mathrm{H}_{2} \mathrm{O}(g)\) (b) \(4 \mathrm{NH}_{3}(g)+5 \mathrm{O}_{2}(g) \rightarrow 4 \mathrm{NO}(g)+6 \mathrm{H}_{2} \mathrm{O}(g)\)

The rate law for the reaction \(\mathrm{H}_{2}+2 \mathrm{IBr} \rightarrow \mathrm{I}_{2}+2 \mathrm{HBr}\) is \(\text { rate }=k\left[\mathrm{H}_{2}\right][\mathrm{IBr}]\).Given that HI is an intermediate, write a two-step mechanism for the reaction.

Consider the reaction \(2 \mathrm{NO}(g)+\mathrm{O}_{2}(\mathrm{~g}) \rightarrow 2 \mathrm{NO}_{2}(g)\) Suppose that at a particular moment during the reaction nitric oxide (NO) is reacting at the rate of 0.066 M/s. (a) At what rate is \(\mathrm{NO}_{2}\) being formed? (b) At what rate is molecular oxygen reacting?

The reaction 2A ? B is second order with a rate constant of 51/M ? min at 24°C. (a) Starting with \([A]_{0}\) = 0.0092 M, how long will it take for \([A]_{t}=3.7 \times 10^{-3}\) M? (b) Calculate the half-life of the reaction.

Which of the following is false regarding catalysis: (a) \(E_{a}\) is lower for a catalyzed reaction . (b) \(\Delta H_{r x n}^{\circ}\) is lower for a catalyzed reaction. (c) A catalyzed reaction has a different mechanism.

Consider the reaction \(N_{2}(g)+3 H_{2}(g) \rightarrow 2 N H_{3}(g)\) Suppose that at a particular moment during the reaction molecular hydrogen is reacting at the rate of 0.074 M/s. (a) At what rate is ammonia being formed? (b) At what rate is molecular nitrogen reacting?

The second-order rate constant for the decomposition of nitrous oxide (\(\mathrm{N}_{2} \mathrm{O}\)) into nitrogen molecule and oxygen atom has been measured at different temperatures: Determine graphically the activation energy for the reaction.

Explain what is meant by the rate law of a reaction.

The first-order rate constant for the reaction of methyl chloride (\(\mathrm{CH}_{3} \mathrm{Cl}\)) with water to produce methanol (\(\mathrm{CH}_{3} \mathrm{OH}\)) and hydrochloric acid (HCl) is \(3.32 \times 10^{-10} \mathrm{s}^{-1}\) at 25°C. Calculate the rate constant at 40°C if the activation energy is 116 kJ/mol.

What are the units for the rate constants of zero-order, first-order, and second-order reactions?

The reaction between \(\mathrm{NO}_{2}\) and CO to produce NO and \(\mathrm{CO}_{2}\) is believed to occur via two steps: Step 1: \(\mathrm{NO}_{2}+\mathrm{NO}_{2} \rightarrow \mathrm{NO}+\mathrm{NO}_{3}\) Step 2: \(\mathrm{NO}_{3}+\mathrm{CO} \rightarrow \mathrm{NO}_{2}+\mathrm{CO}_{2}\) The experimental rate law is \(\text { rate }=k\left[\mathrm{NO}_{2}\right]^{2}\). (a) Write the equation for the overall reaction. (b) Identify the intermediate. (c) What can you say about the relative rates of steps 1 and 2?

Consider the zero-order reaction: A ? product. (a) Write the rate law for the reaction. (b) What are the units for the rate constant? (c) Plot the rate of the reaction versus [A].

On which of the following properties does the rate constant of a reaction depend? (a) reactant concentrations, (b) nature of reactants, (c) temperature.

The rate for law for the reaction \(\mathrm{NH}_{4}^{+}(a q)+\mathrm{NO}_{2}^{-}(a q) \rightarrow \mathrm{N}_{2}(g)+2 \mathrm{H}_{2} \mathrm{O}(l)\) is given by \(\text { rate }=k\left[\mathrm{NH}_{4}^{+}\right]\left[\mathrm{NO}_{2}^{-}\right]\). At 25°C, the rate constant is \(3.0 \times 10^{-4} / \mathrm{M} \cdot \mathrm{s}\). Calculate the rate of the reaction at this temperature if \(\left[N H_{4}^{+}\right]\) = 0.26 M and \(\left[\mathrm{NO}_{2}^{-}\right]\) = 0.080 M.

Use the data in Table 13.2 to calculate the rate of the reaction at the time when [\(F_{2}\)] = 0.010 M and [\(\mathrm{ClO}_{2}\)] = 0.020 M.

Consider the reaction A + B ? products From the following data obtained at a certain temperature, determine the order of the reaction and calculate the rate constant:

Consider the reaction X + Y ? Z From the following data, obtained at 360 K, (a) determine the order of the reaction, and (b) determine the initial rate of disappearance of X when the concentration of X is 0.30 M and that of Y is 0.40 M.

Determine the overall orders of the reactions to which the following rate laws apply: (a) \(\text { rate } =k\left[\mathrm{NO}_{2}\right]^{2}\) (b) rate = k (c) \(\text { rate } =k\left[\mathrm{H}_{2}\right]\left[\mathrm{Br}_{2}\right]^{\frac{1}{2}}\) (d) \(\text { rate }=k[N O]^{2}\left[O_{2}\right]\)

Consider the reaction A ? B The rate of the reaction is \(1.6 \times 10^{-2}\) M/s when the concentration of A is 0.35 M. Calculate the rate constant if the reaction is (a) first order in A and (b) second order in A.

Cyclobutane decomposes to ethylene according to the equation \(C_{4} H_{8}(g) \rightarrow 2 C_{2} H_{4}(g)\) Determine the order of the reaction and the rate constant based on the following pressures, which were recorded when the reaction was carried out at 430°C in a constant-volume vessel.

The following gas-phase reaction was studied at 290°C by observing the change in pressure as a function of time in a constant-volume vessel: \(\mathrm{ClCO}_{2} \mathrm{CCl}_{3}(g) \rightarrow 2 \mathrm{COCl}_{2}(g)\) Determine the order of the reaction and the rate constant based on the following data:

Write an equation relating the concentration of a reactant A at t = 0 to that at t = t for a first-order reaction. Define all the terms and give their units. Do the same for a second-order reaction.

Define half-life. Write the equation relating the half-life of a first-order reaction to the rate constant.

Write the equations relating the half-life of a second- order reaction to the rate constant. How does it differ from the equation for a first-order reaction?

For a first-order reaction, how long will it take for the concentration of reactant to fall to one-eighth its original value? Express your answer in terms of the half-life \(\left(t_{\frac{1}{2}}\right)\) and in terms of the rate constant k.

What is the half-life of a compound if 75 percent of a given sample of the compound decomposes in 60 min? Assume first-order kinetics.

The thermal decomposition of phosphine (\(\mathrm{PH}_{3}\)) into phosphorus and molecular hydrogen is a first-order reaction: \(4 \mathrm{PH}_{3}(g) \rightarrow \mathrm{P}_{4}(g)+6 \mathrm{H}_{2}(g)\) The half-life of the reaction is 35.0 s at 680°C. Calculate (a) the first-order rate constant for the reaction and (b) the time required for 95 percent of the phosphine to decompose.

The rate constant for the second-order reaction \(2 N O B r(g) \rightarrow 2 N O(g)+B r_{2}(g)\) is \(0.80 / M \cdot s\) at 10°C. (a) Starting with a concentration of 0.086 M, calculate the concentration of NOBr after 22 s. (b) Calculate the half-lives when \({[N O B r]_{0}}\) = 0.072 M and \({[N O B r]_{0}}\) = 0.054 M.

The rate constant for the second-order reaction \(2 \mathrm{NO}_{2}(g) \rightarrow 2 \mathrm{NO}(g)+\mathrm{O}_{2}(g)\) is \(0.54 / M \cdot s\) at 300°C. How long (in seconds) would it take for the concentration of \(\mathrm{NO}_{2}\) to decrease from 0.62 M to 0.28 M?

Consider the first-order reaction A ? B shown here. (a) What is the rate constant of the reaction? (b) How many A (yellow) and B (blue) molecules are present at t = 20 s and 30 s?

The reaction X ? Y shown here follows first-order kinetics. Initially different amounts of X molecules are placed in three equal-volume containers at the same temperature. (a) What are the relative rates of the reaction in these three containers? (b) How would the relative rates be affected if the volume of each container were doubled? (c) What arc the relative half-lives of the reactions in (i) to (iii)?

Write the Arrhenius equation and define all terms.

Define activation energy. What role does activation energy play in chemical kinetics?

Use the Arrhenius equation to show why the rate constant of a reaction (a) decreases with increasing activation energy and (b) increases with increasing temperature.

The burning of methane in oxygen is a highly exothermic reaction. Yet a mixture of methane and oxygen gas can be kept indefinitely without any apparent change. Explain.

Sketch a potential energy versus reaction progress plot for the following reactions: (a) \(S(s)+O_{2}(g) \rightarrow S \mathrm{O}_{2}(g) \Delta H^{\circ}=-296 \mathrm{~kJ} / \mathrm{mol}\) (b) \(\mathrm{Cl}_{2}(g) \rightarrow \mathrm{Cl}(\mathrm{g})+\mathrm{Cl}(\mathrm{g}) \Delta H^{\circ}=243 \mathrm{~kJ} / \mathrm{mol}\)

The reaction \(H+H_{2} \rightarrow H_{2}+H\) has been studied for many years. Sketch a potential energy versus reaction progress diagram for this reaction.

(1) The diagram in (a) shows the plots of In k versus 1/T for two first-order reactions, where k is the rate constant and T is the absolute temperature. Which reaction has a greater activation energy? (2) The diagram in (b) shows the plots for a first-order reaction at two different temperatures. Which plot corresponds to a higher temperature?

Given the same reactant concentrations, the reaction \(\mathrm{CO}(\mathrm{g})+\mathrm{Cl}_2(\mathrm{~g}) \longrightarrow \mathrm{COCl}_2(\mathrm{~g})\) at \(250^{\circ} \mathrm{C}\) is \(1.50 \times 10^3\) times as fast as the same reaction at \(150^{\circ} \mathrm{C}\). Calculate the activation energy for this reaction. Assume that the frequency factor is constant.

Some reactions are described as parallel in that the reactant simultaneously forms different products with different rate constants. An example is and \(A \stackrel{k_1}{\longrightarrow} \text { B }\) \(A \stackrel{k_2}{\longrightarrow} C\) The activation energies are 45.3 kJ/mol for \(k_1\) and 69.8 kJ/mol for \(k_2\). If the rate constants are equal at 320 K, at what temperature will \(k_1 / k_2=2.00\) ?

Variation of the rate constant with temperature for the first-order reaction \(2 \mathrm{~N}_{2} \mathrm{O}_{5}(g) \rightarrow 2 \mathrm{~N}_{2} \mathrm{O}_{4}(g)+\mathrm{O}_{2}(g)\) is given in the following table. Determine graphically the activation energy for the reaction.

For the reaction \(\mathrm{NO}(\mathrm{g})+\mathrm{O}_3(\mathrm{~g}) \longrightarrow \mathrm{NO}_2(\mathrm{~g})+\mathrm{O}_2(\mathrm{~g})\) the frequency factor A is \(8.7 \times 10^{12} \mathrm{~s}^{-1}\) and the activation energy is \(63 \mathrm{~kJ} / \mathrm{mol}\). What is the rate constant for the reaction at \(75^{\circ} \mathrm{C}\)?

The rate constant of a first-order reaction is \(4.60 \times 10^{-4} s^{-1}\) at 350°C. If the activation energy is 104 kJ/mol, calculate the temperature at which its rate constant is \(8.80 \times 10^{-4} s^{-1}\).

The rate constants of some reactions double with every 10-degree rise in temperature. Assume that a reaction takes place at 295 K and 305 K. What must the activation energy be for the rate constant to double as described?

Consider the first-order reaction \(\mathrm{CH}_{3} \mathrm{NC}(\mathrm{g}) \rightarrow \mathrm{CH}_{3} \mathrm{CN}(\mathrm{g})\) Given that the frequency factor and activation energy for the reaction are \(3.98 \times 10^{13} \mathrm{s}^{-1}\) and 161 kJ/mol, respectively, calculate the rate constant at 600°C.

Consider the second-order reaction \(\mathrm{NO}(g)+\mathrm{Cl}_{2}(g) \rightarrow \mathrm{NOCl}(g)+\mathrm{Cl}(g)\) Given that the frequency factor and activation energy for the reaction are \(4.0 \times 10^{9} / \mathrm{M} \cdot \mathrm{s}\) and 85 kJ/mol, respectively, calculate the rate constant at 500°C.

The rate at which tree crickets chirp is \(2.0 \times 10^{2}\) per minute at 27°C but only 39.6 per minute at 5°C. From these data, calculate the "activation energy" for the chirping process. (Hint: The ratio of rates is equal to the ratio of rate constants.)

The diagram here describes the initial state of the reaction \(A_{2}+B_{2} \rightarrow 2 A B\). Suppose the reaction is carried out at two temperatures as shown below. Which picture represents the result at the higher temperature? (The reaction proceeds for the same amount of time at both temperatures.)

What do we mean by the mechanism of a reaction? What is an elementary step? What is the molecularity of a reaction?

Classify each of the following elementary steps as unimolecular, bimolecular, or termolecular.

Reactions can be classified as unimolecular, bimolecular, and so on. Why are there no zero-molecular reactions? Explain why termolecular reactions are rare.

Determine the molecularity and write the rate law for each of the following elementary steps: (a) X ? products (b) X + Y ? products (c) X + Y + Z ? products (d) X + X ? products (e) X + 2Y ? products

What is the rate-determining step of a reaction? Give an everyday analogy to illustrate the meaning of "rate determining."

Specify which of the following species cannot be isolated in a reaction: activated complex, product, intermediate.

The equation for the combustion of ethane (\(C_{2} H_{6}\)) is \(2 \mathrm{C}_{2} \mathrm{H}_{6}(g)+7 \mathrm{O}_{2}(g) \rightarrow 4 \mathrm{CO}_{2}(g)+6 \mathrm{H}_{2} \mathrm{O}(\mathrm{l})\) Explain why it is unlikely that this equation also represents the elementary step for the reaction.

The rate law for the reaction \(2 \mathrm{NO}(\mathrm{g})+\mathrm{Cl}_{2}(\mathrm{~g}) \rightarrow 2 \mathrm{NOCl}(\mathrm{g})\) is given by rate = \(k[\mathrm{NO}]\left[\mathrm{Cl}_{2}\right]\). (a) What is the order of the reaction? (b) A mechanism involving the following steps has been proposed for the reaction: \(\mathrm{NO}(\mathrm{g})+\mathrm{Cl}_{2}(g) \rightarrow \mathrm{NOCl}_{2}(g)\) \(\mathrm{NOCl}_{2}(g)+\mathrm{NO}(\mathrm{g}) \rightarrow 2 \mathrm{NOCl}(g)\) If this mechanism is correct, what does it imply about the relative rates of these two steps?

For the reaction \(X_{2}\) + Y + Z ? XY + XZ it is found that doubling the concentration of \(X_{2}\) doubles the reaction rate, tripling the concentration of Y triples the rate, and doubling the concentration of Z has no effect. (a) What is the rate law for this reaction? (b) Why is it that the change in the concentration of Z has no effect on the rate? (c) Suggest a mechanism for the reaction that is consistent with the rate law.

The rate law for the decomposition of ozone to molecular oxygen \(2 \mathrm{O}_{3}(g) \rightarrow 3 \mathrm{O}_{2}(g)\) is \(\text{ rate }=k\frac{\left[\mathrm{O}_3\right]^2}{\left[\mathrm{O}_2\right]}\) The mechanism proposed for this process is Derive the rate law from these elementary steps. Clearly state the assumptions you use in the derivation. Explain why the rate decreases with increasing \(O_{2}\) concentration.

The rate law for the reaction \(2 \mathrm{H}_{2}(g)+2 N O(g) \rightarrow \mathrm{N}_{2}(g)+2 \mathrm{H}_{2} \mathrm{O}(g)\) is rate = \(k\left[\mathrm{H}_{2}\right][\mathrm{NO}]^{2}\). Which of the following mechanisms can be ruled out on the basis of the observed rate expression?

The rate law for the reaction \(2 \mathrm{H}_{2}(g)+2 N O(g) \rightarrow \mathrm{N}_{2}(g)+2 \mathrm{H}_{2} \mathrm{O}(g)\) is rate = \(k\left[\mathrm{H}_{2}\right][\mathrm{NO}]^{2}\). Which of the following mechanisms can be ruled out on the basis of the observed rate expression?

What are the characteristics of a catalyst?

A certain reaction is known to proceed slowly at room temperature. Is it possible to make the reaction proceed at a faster rate without changing the temperature?

Distinguish between homogeneous catalysis and heterogeneous catalysis. Describe three important industrial processes that utilize heterogeneous catalysis.

Are enzyme-catalyzed reactions examples of homogeneous or heterogeneous catalysis? Explain.

The diagram shown here represents a two-step mechanism. (a) Write the equation for each step and the overall reaction. (b) Identify the intermediate and catalyst. The color codes are A = green and B = red.

Consider the following mechanism for the enzyme-catalyzed reaction: Derive an expression for the rate law of the reaction in terms of the concentrations of E and S.(Hint: To solve for [ES], make use of the fact that, at equilibrium, the rate of forward reaction is equal to the rate of the reverse reaction.)

The following diagrams represent the progress of the reaction A ? B, where the red spheres represent A molecules and the green spheres represent B molecules. Calculate the rate constant of the reaction.

The following diagrams show the progress of the reaction 2A ? \(A_{2}\). Determine whether the reaction is first order or second order and calculate the rate constant.

Suggest experimental means by which the rates of the following reactions could be followed: (a) \(\mathrm{CaCO}_{3}(s) \rightarrow \mathrm{CaO}(\mathrm{s})+\mathrm{CO}_{2}(g)\) (b) \(\mathrm{Cl}_{2}(g)+2 \mathrm{Br}^{-}(a q) \rightarrow \mathrm{Br}_{2}(a q)+2 \mathrm{Cl}^{-}(a q)\) (c) \(\mathrm{C}_{2} \mathrm{H}_{6}(g) \rightarrow \mathrm{C}_{2} \mathrm{H}_{4}(g)+\mathrm{H}_{2}(g)\) (d) \(\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{I}(\mathrm{g})+\mathrm{H}_{2} \mathrm{O}(\mathrm{l}) \rightarrow \mathrm{C}_{2} \mathrm{H}_{5} \mathrm{OH}(\mathrm{aq})+\mathrm{H}^{+}(a q)+\mathrm{I}^{-}(a q)\)

List four factors that influence the rate of a reaction.

"The rate constant for the reaction \(\mathrm{NO}_{2}(g)+\mathrm{CO}(g) \rightarrow \mathrm{NO}(g)+\mathrm{CO}_{2}(g)\) is \(1.64 \times 10^{-6} / \mathrm{M} \cdot \mathrm{s}\)." What is incomplete about this statement?

In a certain industrial process involving a heterogeneous catalyst, the volume of the catalyst (in the shape of a sphere) is 10.0 \(\mathrm{cm}^{3}\). Calculate the surface area of the catalyst. If the sphere is broken down into eight spheres, each having a volume of 1.25 \(\mathrm{cm}^{3}\), what is the total surface area of the spheres? Which of the two geometric configurations of the catalyst is more effective? (The surface area of a sphere is 4 \(\pi r^{2}\), where r is the radius of the sphere.) Based on your analysis here, explain why it is sometimes dangerous to work in grain elevators.

Use the data in Example 13.5 to determine graphically the half-life of the reaction.

The following data were collected for the reaction between hydrogen and nitric oxide at 700°C: \(2 \mathrm{H}_{2}(\mathrm{~g})+2 \mathrm{NO}(\mathrm{g}) \rightarrow 2 \mathrm{H}_{2} \mathrm{O}(\mathrm{g})+\mathrm{N}_{2}(g)\) (a) Determine the order of the reaction, (b) Calculate the rate constant, (c) Suggest a plausible mechanism that is consistent with the rate law. (Hint: Assume that the oxygen atom is the intermediate.)

When methyl phosphate is heated in acid solution, it reacts with water: \(\mathrm{CH}_{3} \mathrm{OPO}_{3} \mathrm{H}_{2}+\mathrm{H}_{2} \mathrm{O} \rightarrow \mathrm{CH}_{3} \mathrm{OH}+\mathrm{H}_{3} \mathrm{PO}_{4}\) If the reaction is carried out in water enriched with \({ }^{18} \mathrm{O}\), the oxygen-18 isotope is found in the phosphoric acid product but not in the methanol. What does this tell us about the mechanism of the reaction?

The reaction 2A + 3B ? C is first order with respect to A and B. When the initial concentrations are [A] = \(1.6 \times 10^{-2}\) M and [B] = \(2.4 \times 10^{-3}\) M, the rate is \(4.1 \times 10^{-4}\) M/s. Calculate the rate constant of the reaction.

Which of the following equations best describes the diagram shown above: (a) A ? B, (b) A ? 3B, (c) 3A ? B?

The bromination of acetone is acid-catalyzed: The rate of disappearance of bromine was measured for several different concentrations of acetone, bromine, and \(H^{+}\) ions at a certain temperature: (a) What is the rate law for the reaction? (b) Determine the rate constant, (c) The following mechanism has been proposed for the reaction: Show that the rate law deduced from the mechanism is consistent with that shown in (a).

The decomposition of \(\mathrm{N}_{2} \mathrm{O}\) to \(\mathrm{N}_{2}\) and \(\mathrm{O}_{2}\) is a first-order reaction. At 730°C the half-life of the reaction is \(3.58 \times 10^{3}\) min. If the initial pressure of \(\mathrm{N}_{2} \mathrm{O}\) is 2.10 atm at 730°C, calculate the total gas pressure after one half-life. Assume that the volume remains constant.

The reaction \(\mathrm{S}_{2} \mathrm{O}_{8}^{2-}+2 \mathrm{I}^{-} \rightarrow 2 \mathrm{SO}_{4}^{2-}+\mathrm{I}_{2}\) proceeds slowly in aqueous solution, but it can be catalyzed by the \(\mathrm{Fe}^{3+}\) ion. Given that \(\mathrm{Fe}^{3+}\) can oxidize \(\mathrm{I}^{-}\) and \(\mathrm{Fe}^{2+}\) can reduce \(\mathrm{S}_{2} \mathrm{O}_{8}^{2-}\), write a plausible two-step mechanism for this reaction. Explain why the uncatalyzed reaction is slow.

What are the units of the rate constant for a third-order reaction?

The integrated rate law for the zero-order reaction \(A \rightarrow B \text { is }[A]_{t}=[A]_{0}-k t\). (a) Sketch the following plots: (i) rate versus \([A]_{t}\) and (ii) \([A]_{t}\) versus t. (b) Derive an expression for the half-life of the reaction. (c) Calculate the time in half-lives when the integrated rate law is no longer valid, that is, when \([A]_{t}=0\).

A flask contains a mixture of compounds A and B. Both compounds decompose by first-order kinetics. The half-lives are 50,0 min for A and 18.0 min for B. If the concentrations of A and B are equal initially, how long will it take for the concentration of A to be four times that of B?

Shown here are plots of concentration of reactant versus time for two first-order reactions at the same temperature. In each case, determine which reaction has a greater rate constant.

The diagrams here represent the reaction \(A+B \rightarrow C\) carried out under different initial concentrations of A and B. Determine the rate law of the reaction. The color codes are A = red, B = green, C = blue.)

Referring to Example 13.5, explain how you would measure the partial pressure of azomethane experimentally as a function of time.

The rate law for the reaction \(2 \mathrm{NO}_{2}(g) \rightarrow \mathrm{N}_{2} \mathrm{O}_{4}(g)\) is rate = \(k\left[N O_{2}\right]^{2}\). Which of the following changes will change the value of k? (a) The pressure of \(\mathrm{NO}_{2}\) is doubled. (b) The reaction is run in an organic solvent. (c) The volume of the container is doubled. (d) The temperature has decreased. (e) A catalyst is added to the container.

The reaction of \(G_{2}\) with \(E_{2}\) to form 2EG is exothermic, and the reaction of \(G_{2}\) with \(X_{2}\) to form 2XG is endothermic. The activation energy of the exothermic reaction is greater than that of the endothermic reaction. Sketch the potential energy profile diagrams for these two reactions on the same graph.

In the nuclear industry, workers use a rule of thumb that the radioactivity from any sample will be relatively harmless after 10 half-lives. Calculate the fraction of a radioactive sample that remains after this time period. (Hint: Radioactive decays obey first-order kinetics.)

Briefly comment on the effect of a catalyst on each of the following: (a) activation energy, (b) reaction mechanism, (c) enthalpy of reaction, (d) rate of forward step, (e) rate of reverse step.

When 6 g of granulated Zn is added to a solution of 2 M HCl in a beaker at room temperature, hydrogen gas is generated. For each of the following changes (at constant volume of the acid) state whether the rate of hydrogen gas evolution will be increased, decreased, or unchanged: (a) 6 g of powdered Zn is used; (b) 4 g of granulated Zn is used; (C) 2 M acetic acid is used instead of 2 M HCl; (d) temperature is raised to \(40^{\circ} \mathrm{C}\).

Strictly speaking, the rate law derived for the reaction in Problem 13.74 applies only to certain concentrations of \(H_{2}\). The general rate law for the reaction takes the form \(\text { rate }=\frac{k_{1}[N O]^{2}\left[H_{2}\right]}{1+k_{2}\left[H_{2}\right]}\) where \(k_{1}\) and \(k_{2}\) are constants. Derive rate law expressions under the conditions of very high and very low hydrogen concentrations. Does the result from Problem 13.74 agree with one of the rate expressions here?

A certain first-order reaction is 35.5 percent complete in 4.90 min at \(25^{\circ} \mathrm{C}\). What is its rate constant?

The decomposition of dinitrogen pentoxide has been studied in carbon tetrachloride solvent (CCI) at a certain temperature: \(2 \mathrm{~N}_{2} \mathrm{O}_{5} \rightarrow 4 \mathrm{NO}_{2}+\mathrm{O}_{2}\) Determine graphically the rate law for the reaction and calculate the rate constant.

The thermal decomposition of \(\mathrm{N}_{2} \mathrm{O}_{5}\) obeys first-order kinetics. At \(45^{\circ} \mathrm{C}\), a plot of In \(\left[\mathrm{N}_{2} \mathrm{O}_{5}\right]\) versus t gives a slope of \(6.18 \times 10^{-4}\ \mathrm{min}^{-1}\). What is the half-life of the reaction?

When a mixture of methane and bromine is exposed to visible light, the following reaction occurs slowly: \(\mathrm{CH}_{4}(g)+\mathrm{Br}_{2}(g) \rightarrow \mathrm{CH}_{3} \mathrm{Br}(g)+\mathrm{HBr}(g)\) Suggest a reasonable mechanism for this reaction. (Hint: Bromine vapor is deep red; methane is colorless.)

The rate of the reaction between \(\mathrm{H}_{2}\) and \(I_{2}\) to form HI (discussed on p. 598) increases with the intensity of visible light. (a) Explain why this fact supports the two-step mechanism given. (The color of \(I_{2}\) vapor is shown on p. 504.) (b) Explain why the visible light has no effect on the formation of H atoms.

The carbon-14 decay rate of a sample obtained from a young tree is 0.260 disintegration per second per gram of the sample. Another wood sample prepared from an object recovered at an archaeological excavation gives a decay rate of 0.186 disintegration per second per gram of the sample. What is the age of the object? (Hint: See Chemistry in Action essay on p. 588.)

Consider the following elementary step: \(X+2 Y \rightarrow X Y_{2}\) (a) Write a rate law for this reaction. (b) If the initial rate of formation of \(X Y_{2}\) is \(3.8 \times 10^{-3}\ \mathrm{M} / \mathrm{s}\) and the initial concentrations of X and Y are 0.26 M and 0.88 M, what is the rate constant of the reaction?

In recent years ozone in the stratosphere has been depleted at an alarmingly fast rate by chlorofluorocarbons (CFCs). A CFC molecule such as \(\mathrm{CFCl}_{3}\) is first decomposed by UV radiation: \(\mathrm{CFCl}_{3} \rightarrow \mathrm{CFCl}_{2}+\mathrm{Cl}\) The chlorine radical then reacts with ozone as follows: \(\mathrm{Cl}+\mathrm{O}_{3} \rightarrow \mathrm{ClO}+\mathrm{O}_{2}\) \(\mathrm{ClO}+\mathrm{O} \rightarrow \mathrm{Cl}+\mathrm{O}_{2}\) The O atom is from the photochemical decomposition of \(\mathrm{O}_{2}\) molecules. (a) Write the overall reaction for the last two steps. (b) What are the roles of Cl and CIO? (c) Why is the fluorine radical not important in this mechanism? (d) One suggestion to reduce the concentration of chlorine radicals is to add hydrocarbons such as ethane \(\left(C_{2} H_{6}\right)\) to the stratosphere. How will this work? (e) Draw potential energy versus reaction progress diagrams for the uncatalyzed and catalyzed (by Cl) destruction of ozone: \(\mathrm{O}_{3}+\mathrm{O} \rightarrow 2 \mathrm{O}_{2}\). Use the thermodynamic data in Appendix 3 to determine whether the reaction is exothermic or endothermic.

Chlorine oxide (CIO), which plays an important role in the depletion of ozone (see Problem 13.101), decays rapidly at room temperature according to the equation \(2 \mathrm{ClO}(g) \rightarrow \mathrm{Cl}_{2}(g)+\mathrm{O}_{2}(g)\) From the following data, determine the reaction order and calculate the rate constant of the reaction

A compound X undergoes two simultaneous first order reactions as follows: \(X \rightarrow Y\) with rate constant \(k_{1}\) and \(X \rightarrow Z\) with rate constant \(k_{2}\). The ratio of \(k_{1} / k_{2}\) at \(40^{\circ} \mathrm{C}\) is 8.0. What is the ratio at \(300^{\circ} \mathrm{C}\)? Assume that the frequency factors of the two reactions are the same.

Consider a car fitted with a catalytic converter. The first 5 minutes or so after it is started are the most polluting. Why?

The following scheme in which A is converted to B, which is then converted to C is known as a consecutive reaction. \(A \rightarrow B \rightarrow C\) Assuming that both steps are first order, sketch on the same graph the variations of [A], [B], and [C] with time.

Hydrogen and iodine monochloride react as follows: \(\mathrm{H}_{2}(g)+2 \mathrm{ICl}(g) \rightarrow 2 \mathrm{HCl}(g)+I_{2}(g)\) The rate law for the reaction is rate = \(k\left[\mathrm{H}_{2}\right][\mathrm{ICl}]\). Suggest a possible mechanism for the reaction.

The rate law for the following reaction \(\mathrm{CO}(g)+\mathrm{NO}_{2}(g) \rightarrow \mathrm{CO}_{2}(g)+N O(g)\) is rate = \(k\left[\mathrm{NO}_{2}\right]^{2}\). Suggest a plausible mechanism for the reaction, given that the unstable species \(\mathrm{NO}_{3}\) is an intermediate.

Radioactive plutonium-239 \((\left.t_{\frac{1}{2}}=2.44 \times 10^{5}\ y r\right)\) is used in nuclear reactors and atomic bombs. If there are \(5.0 \times 10^{2}\ \mathrm{g}\) of the isotope in a small atomic bomb, how long will it take for the substance to decay to \(1.0 \times 10^{2}\ \mathrm{g}\), too small an amount for an effective bomb?

Many reactions involving heterogeneous catalysts are zero order; that is, rate = k. An example is the decomposition of phosphine \(\left(\mathrm{PH}_{3}\right)\) over tungsten (W): \(4 \mathrm{PH}_{3}(g) \rightarrow \mathrm{P}_{4}(g)+6 \mathrm{H}_{2}(g)\) It is found that the reaction is independent of \(\left[\mathrm{PH}_{3}\right]\) as long as phosphine's pressure is sufficiently high \((\geq 1\ \mathrm{atm})\). Explain.

Thallium(I) is oxidized by cerium(IV) as follows: \(T I^{+}+2 C e^{4+} \rightarrow T I^{3+}+2 C e^{3+}\) The elementary steps, in the presence of Mn(II), are as follows: \(C e^{4+}+M n^{2+} \rightarrow C e^{3+}+M n^{3+}\) \(\mathrm{Ce}^{4+}+\mathrm{Mn}^{3+} \rightarrow \mathrm{Ce}^{3+}+\mathrm{Mn}^{4+}\) \(T I^{+}+M n^{4+} \rightarrow T I^{3+}+M n^{2+}\) (a) Identify the catalyst, intermediates, and the rate-determining step if the rate law is rate \(=k\left[\mathrm{Ce}^{4+}\right]\left[\mathrm{Mn}^{2+}\right]\). (b) Explain why the reaction is slow without the catalyst. (c) Classify the type of catalysis (homogeneous or heterogeneous).

Sucrose \(\left(C_{12} H_{22} O_{11}\right)\), commonly called table sugar, undergoes hydrolysis (reaction with water) to produce fructose \(\left(\mathrm{C}_{6} \mathrm{H}_{12} \mathrm{O}_{6}\right)\) and glucose \(\left(\mathrm{C}_{6} \mathrm{H}_{12} \mathrm{O}_{6}\right)\): \(\mathrm{C}_{12} \mathrm{H}_{22} \mathrm{O}_{11}+\mathrm{H}_{2} \mathrm{O} \rightarrow \mathrm{C}_{6} \mathrm{H}_{12} \mathrm{O}_{6}+\mathrm{C}_{6} \mathrm{H}_{12} \mathrm{O}_{6}\) fructose glucose This reaction is of considerable importance in the candy industry. First, fructose is sweeter than sucrose. Second, a mixture of fructose and glucose, called invert sugar, does not crystallize, so the candy containing this sugar would be chewy rather than brittle as candy containing sucrose crystals would be. (a) From the following data determine the order of the reaction. (b) How long does it take to hydrolyze 95 percent of sucrose? (c) Explain why the rate law does not include \(\left[\mathrm{H}_{2} \mathrm{O}\right]\) even though water is a reactant.

The first-order rate constant for the decomposition of dimethyl ether \(\left(\mathrm{CH}_{3}\right)_{2} \mathrm{O}(\mathrm{g}) \rightarrow \mathrm{CH}_{4}(g)+\mathrm{H}_{2}(g)+\mathrm{CO}(g)\) is \(3.2 \times 10^{-4}\ \mathrm{s}^{-1}\) at \(450^{\circ} \mathrm{C}\). The reaction is carried out in a constant-volume flask. Initially only dimethyl ether is present and the pressure is 0.350 atm. What is the pressure of the system after 8.0 min? Assume ideal behavior.

At \(25^{\circ} \mathrm{C}\), the rate constant for the ozone-depleting reaction \(O(g)+O_{3}(g) \rightarrow 2 O_{2}(g)\) is \(7.9 \times 10^{-15}\ \mathrm{cm}^{3} / \text { molecule } \cdot \mathrm{s}\). Express the rate constant in units of \(1 / M \cdot S\).

Consider the following elementary steps for a consecutive reaction: \(A\xrightarrow{{k_{1}}}B\xrightarrow{{k_{2}}}C\) (a) Write an expression for the rate of change of B. (b) Derive an expression for the concentration of B under steady-state conditions; that is, when B is decomposing to C at the same rate as it is formed from A.

Ethanol is a toxic substance that, when consumed in excess, can impair respiratory and cardiac functions by interference with the neurotransmitters of the nervous system. In the human body, ethanol is metabolized by the enzyme alcohol dehydrogenase to acetaldehyde, which causes "hangovers." (a) Based on your knowledge of enzyme kinetics, explain why binge drinking (that is, consuming too much alcohol too fast) can prove fatal. (b) Methanol is even more toxic than ethanol. It is also metabolized by alcohol dehydrogenase, and the product, formaldehyde, can cause blindness or death. An antidote to methanol poisoning is ethanol. Explain how this procedure works.

trontium-90, a radioactive isotope, is a major product of an atomic bomb explosion. It has a half-life of 28.1 yr. (a) Calculate the first-order rate constant for nuclear decay. (b) Calculate the fraction of \({ }^{90} \mathrm{Sr}\) that remains after 10 half-lives. (c) Calculate the number of years required for 99.0 percent of \({ }^{90} \mathrm{Sr}\) to disappear.

Consider the potential energy profiles for the following three reactions (from left to right). (1) Rank the rates (slowest to fastest) of the reactions. (2) Calculate \(\Delta H\) for each reaction and determine which reaction(s) are exothermic and which reaction(s) are endothermic. Assume the reactions have roughly the same frequency factors.

Consider the following potential energy profile for the \(A \rightarrow D\) reaction. (a) How many elementary steps are there? (b) How many intermediates are formed? (c) Which step is rate determining? (d) Is the overall reaction exothermic or endothermic?

A factory that specializes in the refinement of transition metals such as titanium was on fire. The firefighters were advised not to douse the fire with water. Why?

The activation energy for the decomposition of hydrogen peroxide \(2 \mathrm{H}_{2} \mathrm{O}_{2}(a q) \rightarrow 2 \mathrm{H}_{2} \mathrm{O}_{2}(l)+\mathrm{O}_{2}(g)\) is 42 kJ/mol, whereas when the reaction is catalyzed by the enzyme catalase, it is 7.0 kJ/mol. Calculate the temperature that would cause the uncatalyzed reaction to proceed as rapidly as the enzyme catalyzed decomposition at \(20^{\circ} \mathrm{C}\). Assume the frequency factor A to be the same in both cases.

The activity of a radioactive sample is the number of nuclear disintegrations per second, which is equal to the first-order rate constant times the number of radioactive nuclei present. The fundamental unit of radioactivity is the curie (Ci), where 1 Ci corresponds to exactly \(3.7 \times 10^{10}\) disintegrations per second. This decay rate is equivalent to that of 1 g of radium-226. Calculate the rate constant and half-life for the radium decay. Starting with 1.0 g of the radium sample, what is the activity after 500 yr? The molar mass of Ra-226 is 226.03 g/mol.

To carry out metabolism, oxygen is taken up by hemoglobin (Hb) to form oxyhemoglobin \(\left(\mathrm{HbO}_{2}\right)\) according to the simplified equation \(Hb(aq)+O_{2}(aq)\xrightarrow{{k}}HbO_{2}(aq)\) where the second-order rate constant is \(2.1 \times 10^{5} / M \cdot s\) at \(37^{\circ} \mathrm{C}\). (The reaction is first order in Hb and \(\mathrm{O}_{2}\).) For an average adult, the concentrations of Hb and \(\mathrm{O}_{2}\) in the blood at the lungs are \(8.0 \times 10^{-6}\ M\) and \(1.5 \times 10^{-6}\ \mathrm{M}\), respectively. (a) Calculate the rate of formation of \(\mathrm{HbO}_{2}\). (b) Calculate the rate of consumption of \(\mathrm{O}_{2}\). (c) The rate of formation of \(\mathrm{HbO}_{2}\) increases to \(1.4 \times 10^{-4}\ \mathrm{M} / \mathrm{s}\) during exercise to meet the demand of increased metabolism rate. Assuming the Hb concentration to remain the same, what must be the oxygen concentration to sustain this rate of \(\mathrm{HbO}_{2}\) formation?

At a certain elevated temperature, ammonia decomposes on the surface of tungsten metal as follows: \(2 \mathrm{NH}_{3} \rightarrow \mathrm{N}_{2}+3 \mathrm{H}_{2}\) From the following plot of the rate of the reaction versus the pressure of \(\mathrm{NH}_{3}\), describe the mechanism of the reaction.

The following expression shows the dependence of the half-life of a reaction \(\left(t_{\frac{1}{2}}\right)\) on the initial reactant concentration \([A]_{0}\). \(t_{\frac{1}{2}} \infty \frac{1}{[A]_{0}^{n-1}}\) where n is the order of the reaction. Verify this dependence of zero-, first-, second-order reactions.

Polyethylene is used in many items, including water pipes, bottles, electrical insulation, toys, and mailer envelopes. It is a polymer, a molecule with a very high molar mass made by joining many ethylene molecules together. (Ethylene is the basic unit, or monomer for polyethylene.) The initiation step is \(\mathrm{R}_2 \stackrel{k_1}{\longrightarrow} 2 \mathrm{R} \cdot \quad \text { initiation }\) The \(\mathrm{R} \cdot\) species (called a radical) reacts with an ethylene molecule M} to generate another radical \(\mathrm{R} \cdot \mathrm{M} \longrightarrow \mathrm{M}_1 \text {. }\) Reaction of \(\mathrm{M}_1 \cdot\) with another monomer leads to the growth or propagation of the polymer chain: \(\mathrm{M}_1 \cdot+\mathrm{M} \stackrel{k_{\mathrm{p}}}{\longrightarrow} \mathrm{M}_2 \text {. propagation }\) This step can be repeated with hundreds of monomer units. The propagation terminates when two radicals combine \(\mathrm{M}^{\prime} \cdot+\mathrm{M}^{\prime \prime} \cdot \stackrel{k_1}{\longrightarrow} \mathrm{M}^{\prime}-\mathrm{M}^{\prime \prime}\) termination The initiator frequently used in the polymerization of ethylene is benzoyl peroxide \(\left[\left(\mathrm{C}_6 \mathrm{H}_5 \mathrm{COO}\right)_2\right]\): \(\left[\left(\mathrm{C}_6 \mathrm{H}_5 \mathrm{COO}\right)_2\right] \longrightarrow 2 \mathrm{C}_6 \mathrm{H}_5 \mathrm{COO} \text {. }\) This is a first-order reaction. The half-life of benzoyl peroxide at \(100^{\circ} \mathrm{C}\) is 19.8 min. (a) Calculate the rate constant (in \(\mathrm{min}^{-1}\) ) of the reaction. (b) If the halflife of benzoyl peroxide is 7.30 h, or 438 min, at \(70^{\circ} \mathrm{C}\), what is the activation energy (in kJ/mol) ) for the decomposition of benzoyl peroxide? (c) Write the rate laws for the elementary steps in the above polymerization process, and identify the reactant, product, and intermediates. (d) What condition would favor the growth of long, high-molar-mass polyethylenes?

The rate constant for the gaseous reaction \(\mathrm{H}_{2}(g)+I_{2}(g) \rightarrow 2 H I(g)\) is \(2.42 \times 10^{-2} / M \cdot s\) at \(400^{\circ} \mathrm{C}\). Initially an equimolar sample of \(H_{2}\) and \(I_{2}\) is placed in a vessel at \(400^{\circ} \mathrm{C}\) and the total pressure is 1658 mmHg. (a) What is the initial rate (M/min) of formation of HI? (b) What are the rate of formation of HI and the concentration of HI (in molarity) after 10.0 min?

When the concentration of A in the reaction \(A \rightarrow B\) was changed from 1.20 M to 0.60 M, the half-life increased from 2.0 min to 4.0 min at \(25^{\circ} \mathrm{C}\). Calculate the order of the reaction and the rate constant. (Hint: Use the equation in Problem 13.124.)

At a certain elevated temperature, ammonia decomposes on the surface of tungsten metal as follows: \(\mathrm{NH}_{3} \rightarrow \frac{1}{2} N_{2}+\frac{3}{2} H_{2}\) The kinetic data are expressed as the variation of the half-life with the initial pressure of NH3: P (mmHg) 264 130 59 16 11 \(t_{\frac{1}{2}}(s)\) 456 228 102 60 (a) Determine the order of the reaction. (b) How does the order depend on the initial pressure? (c) How does the mechanism of the reaction vary with pressure? (Hint: You need to use the equation in Problem 13.124 and plot log t. versus log P.)

The activation energy for the reaction \(N_{2} O(g) \rightarrow N_{2}(g)+O(g)\) is \(2.4 \times 10^{2}\ \mathrm{kJ} / \mathrm{mol}\) at 600 K. Calculate the percentage of the increase in rate from 600 K to 606 K. Comment on your results.

The rate of a reaction was followed by the absorption of light by the reactants and products as a function of wavelengths \(\left(\lambda_{1}, \lambda_{2}, \lambda_{3}\right)\) as time progresses. Which of the following mechanisms is consistent with the experimental data? (a) \(A \rightarrow B,\ A \rightarrow C\) (b) \(A \rightarrow B+C\) (c) \(A \rightarrow B,\ B \rightarrow C+D\) (d) \(A \rightarrow B,\ B \rightarrow C\)

A gas mixture containing \(\mathrm{CH}_{3}\) fragments, \(C_{2} H_{6}\) molecules, and an inert gas (He) was prepared at 600 K with a total pressure of 5.42 atm. The elementary reaction \(\mathrm{CH}_{3}+\mathrm{C}_{2} \mathrm{H}_{6} \rightarrow \mathrm{CH}_{4}+\mathrm{C}_{2} \mathrm{H}_{5}\) has a second-order rate constant of \(3.0 \times 10^{4}\ / \mathrm{M} \cdot \mathrm{s}\). Given that the mole fractions of \(\mathrm{CH}_{3}\) and \(C_{2} H_{6}\) are 0.00093 and 0.00077, respectively, calculate the initial rate of the reaction at this temperature.

To prevent brain damage, a drastic medical procedure is to lower the body temperature of someone who has suffered cardiac arrest. What is the physiochemical basis for this treatment?

The activation energy (E) for the reaction \(2 \mathrm{~N}_{2} \mathrm{O}(\mathrm{g}) \rightarrow 2 \mathrm{N}_{2}(\mathrm{g})+\mathrm{O}_{2}(\mathrm{g})\ \Delta \mathrm{H}^{\circ}=-164\ \mathrm{kJ} / \mathrm{mol}\) is 240 kJ/mol. What is \(E_{a}\) for the reverse reaction?

The rate constants for the first-order decomposition of an organic compound in solution are measured at several temperatures: \(k\left(s^{-1}\right)\) 0.00492 0.0216 0.0950 0.326 1.15 T (K) 278 288 298 308 318 Determine graphically the activation energy and frequency factor for the reaction.

Assume that the formation of nitrogen dioxide: \(2 \mathrm{NO}(g)+\mathrm{O}_{2}(g) \rightarrow 2 \mathrm{NO}_{2}(g)\) is an elementary reaction. (a) Write the rate law for this reaction. (b) A sample of air at a certain temperature is contaminated with 2.0 ppm of NO by volume. Under these conditions, can the rate law be simplified? If so, write the simplified rate law. (c) Under the condition described in (b), the half-life of the reaction has been estimated to be \(6.4 \times 10^{3}\ \mathrm{min}\). What would be the half-life if the initial concentration of NO were 10 ppm?

An instructor performed a lecture demonstration of the thermite reaction (see p. 259). He mixed aluminum with iron(III) oxide in a metal bucket placed on a block of ice. After the extremely exothermic reaction started, there was an enormous bang, which was not characteristic of thermite reactions. Give a plausible chemical explanation for the unexpected sound effect. The bucket was open to air.

Account for the variation of the rate of an enzyme catalyzed reaction versus temperature shown here. What is the approximate temperature that corresponds to the maximum rate in the human body?

Is the rate constant (k) of a reaction more sensitive to changes in temperature if \(E_{a}\) is small or large?

Shown here is a plot of \([A]_{t}\) versus t for the reaction \(A \rightarrow\) product. (a) Determine the order and the rate constant of the reaction. (b) Estimate the initial rate and the rate at 30 s.

What are the shortest and longest times (in years) that can be estimated by carbon-14 dating?

In addition to chemical and biological systems, kinetic treatments can sometimes be applied to behavioral and social processes such as the evolution of technology. For example, in 1965. Gordon Moore, a co-founder of Intel, described a trend that the number of transistors on an integrated circuit (N) roughly doubles every 1.5 yr. Now referred to as Moore's law, this trend has persisted for the past several decades. A plot of In N versus year is shown here. (a) Determine the rate constant for the growth in the number of transistors on an integrated circuit. (b) Based on the rate constant, how long does it take for N to double? (c) If Moore's law continues until the end of the century, what will be the number of transistors on an integrated circuit in the year 2100? Comment on your result.

Calculate the half-life of the decomposition of \(\mathrm{N}_{2} \mathrm{O}_{5}\), discussed on p. 579.

How does a catalyst increase the rate of a reaction?

The concentrations of enzymes in cells are usually quite small. What is the biological significance of this fact?

The rate of the reaction \(\mathrm{CH}_{3} \mathrm{COOC}_{2} \mathrm{H}_{5}(a q)+\mathrm{H}_{2} \mathrm{O}(\mathrm{l}) \rightarrow \mathrm{CH}_{3} \mathrm{COOH}(a q)+\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{OH}(a q)\) shows first-order characteristics - that is, rate = \(k\left[\mathrm{CH}_{3} \mathrm{COOC}_{2} \mathrm{H}_{5}\right]\) - even though this is a second-order reaction (first order in \(\mathrm{CH}_{3} \mathrm{COOC}_{2} \mathrm{H}_{5}\) and first order in \(\mathrm{H}_{2} \mathrm{O}\)). Explain.

A protein molecule, P, of molar mass M dimerizes when it is allowed to stand in solution at room temperature. A plausible mechanism is that the protein molecule is first denatured (that is, loses its activity due to a change in overall structure) before it dimerizes: \(P\xrightarrow{{k}}P^{*}\) (denatured) slow \(2P\rightarrow P_{2}\) fast where the asterisk denotes a denatured protein molecule. Derive an expression for the average molar mass (of P and \(P_{2}\)), \(\overline{\mathscr{M}}\), in terms of the initial protein concentration \([P]_{0}\) and the concentration at time t, \([P]_{t}\), and \({\mathscr{M}}\). Describe how you would determine k from molar mass measurements.