578 CHAPTER 14 Chemical Kinetics Applying What You've Learned It takes as little as 5 mL (l tsp) of methanol to cause permanent blindness or death; and unlike ethanol, methanol can be absorbed in toxic amounts by ingestion, inhalation of vapor, or absorption through the skin. Nevertheless, methanol is present in a number of common household products including antifreeze, windshield-washing fluid, and paint remov er. One method that ha s been used to synthesize methanol is the combination of car- bon monoxide and hydrogen gases at 100 o e: CO(g) + 2H2(g) +. CH 3 0H(g) This reaction is catalyzed by a nickel compound. Problems: a) Write the expression for the rate of this reaction in terms of [CO]. [I ~~ Sample Problem 14.1] b) Write the expression for the rate of this reaction in terms of [H 2 ] and in terms of [CH 3 0H ]. [ ~~ Sample Problem 14.2] c) Given the following table of experimental data at 100°C, determine the rate law and the rate constant for the reaction. Then, determine the initial rate of the reaction when the starting concentration of CO is 16.5 M. [ ~. Sample Problem 14.3] Experiment 1 2 3 [CO] (M) 5.60 5.60 11.2 [H 2 ] (M) 11.2 22.4 11.2 Initial Rate (Mis) 0.952 0.952 1.90 d) Calculate the time required for the concentration of CO to be reduced from 16.5 M to 1.91 M. [ ~~ Sample Problem 14.4] e) Calculate tll2 of the reaction. [ ~. Sample Problem 14.6] f) Given that k is 3.0 S -1 at 200°C, calculate Ea of the reaction. [ ~. Sample Problem 14.9] g) Use the calculated value of Ea to determine the value of k at 180°C. [I •• Sample Problem 14 .10] 1 I ) IW I~ . t; " 1 CHAPTER SUMMARY Section 14.1 • The rate of a chemical reaction is the change in concentration of reactants or products over time. Rates may be expressed as an average rate over a given time interval or as an instantaneous rate. • The rate constant (k) is a proportionality constant that relates the rate of reaction with the concentration(s) of reactant(s). The rate constant k for a given reaction changes only with temperature. Section 14.2 • The rate law is an equation that expresses the relationship between rate and reactant concentration(s). In general, the rate law for the reaction of A and B is rate = k[Ay[BY • The reaction order is the power to which the concentration of a given reactant is raised in the rate law equation. The overall reaction order is the sum of the powers to which reactant concentrations are raised in the rate law. • The initial rate is the instantaneous rate of reaction when the reactant concentrations are starting concentrations. • The rate law and reaction order must be determined by comparing changes in the initial rate with changes in starting reactant concentrations. In general, the rate law cannot be determined solely from the balanced equation. Section 14.3 • The integrated rate law can be used to determine reactant concentrations after a specified period of time. It can also be used to determine how long it will take to reach a specified reactant concentration. • The rate of afirst-order reaction is proportional to the concentration of a single reactant. The rate of a second-order reaction is proportional to the product of two reactant concentrations ([A][B]), or on the concentration of a single reactant squared ([A]2 or [B]z ). The rate of a zeroth-order reaction does not depend on reactant concentration. • The half-life (t1l2) of a reaction is the time it takes for half of a reactant to be consumed. The half-life is constant for first-order reactions, and it can be used to determine the rate constant of the reaction. Section 14.4 • Collision theory explains why the rate constant, and therefore the reaction rate, increases with increasing temperature. The relationship KEyWORDS Activated complex, 564 Activation energy (Ea)' 563 Arrhenius equation, 564 Bimolecular, 569 Catalyst, 573 Collision theory, 562 Effective collision, 562 Elementary reaction, 568 Enzymes, 575 First-order reaction, 555 Half-life (t IlZ ), 558 Heterogeneous catalysis, 574 Homogeneous catalysis, 575 Initial rate, 551 Instantaneous rate, 546 Integrated rate law, 556 • • KEY WORDS 579 between temperature and the rate constant is expressed by the Arrhenius equation. Reactions occur when molecules of sufficient energy (and appropriate orientation) collide. Effective collisions are those that result in the formation of an activated complex, also called a transition state. Only effective collisions can result in product formation. The activation energy (Ea) is the minimum energy that colliding molecules must possess in order for the collision to be effective. Section 14.5 • • A reaction mechanism may consist of a series of steps, called elementary reactions. Unlike rate laws in general, the rate law for an elementary reaction can be written from the balanced equation, using the stoichiometric coefficient for each reactant species as its exponent in the rate law. A species that is produced in one step of a reaction mechanism and subsequently consumed in another step is called an intermediate. A species that is first consumed and later regenerated is called a catalyst. Neither intermediates nor catalysts appear in the overall balanced equation. • The rate law of each step in a reaction mechanism indicates the molecularity or overall order of the step. A unimolecular step is first order, involving just one molecule; a bimolecular step is second order, involving the collision of two molecules; and a termolecular step is third order, involving the collision of three molecules. Termolecular processes are relatively rare. • If one step in a reaction is much slower than all the other steps, it is the rate-determining step. The rate-determining step has a rate law identical to the experimental rate law. Section 14.6 • A catalyst speeds up a reaction, usually by lowering the value of the activation energy. Catalysis refers to the process by which a catalyst increases the reaction rate. • Catalysis may be heterogeneous, in which the catalyst and reactants exist in different phases, or homogeneous, in which the catalyst and reactants exist in the same phase. • Enzymes are biological catalysts with high specificity for the reactions that they catalyze. Intermediate, 568 Molecularity, 568 Rate constant (k), 547 Rate-determining step, Rate law, 551 Rate of reaction, 548 569 Reaction mechanism, 568 Reaction order, 551 Second-order reaction, 560 Termolecular, 569 Transition state, 564 Unimolecular, 569 Zeroth-order reaction, 561 580 CHAPTER 14 Chemical Kinetics KEY EQUATIONS ~~ 14.1 14.2 14 .3 1 Ll[Al rate = - a !::: t rate = k[Ay[ BY I [All = -kt n [Alo 1 !::: [B 1 1 Ll[ C] 1 Ll[Dl b !::: t c!::: t d !::: t 14.4 14.5 In [All = -k t + In [Alo 0.693 t 1/2 = 14.6 14.7 14.8 14.9 14.10 14.11 k 1 = kt + 1 [All [Alo 1 t1/2 = k[Alo k = Ae -E.jRT Ea In k = InA =- RT QUESTIONS AND PROBLEM=S ==== =======- Section 14.1: Reaction Rates Review Questions 14.1 What is meant by the rate of a chemical reaction? What are the units of the rate of a reaction? 14.2 Distinguish between average rate and instantaneous rate. Which of the two rates gives us an unambiguous measurement of reaction rate? Why? 14.3 What are the advantages of measuring the initial rate of a reaction? 14.4 Identify 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). Problems 14.5 14.6 Write the reaction rate expressions for the following reactions in terms of the disappearance of the reactants and the appearance of products: (a) H 2 (g) + I2(g) • 2HI(g) (b) 5Br-(aq) + Br0 3(aq) + 6H +( aq) -_. 3Brzeaq) + 3H 2 0(l) Write the reaction rate expressions for the following reactions in terms of the disappearance of the reactants and the appearance of products: (a) 2H2(g) + 0 2(g) -_. 2H 2 0(g) (b) 4NH 3 (g) + 50 2 (g) • 4NO(g) + 6H z O(g) 14.7 Consider the reaction 2NO(g) + 0 2(g) +. 2N0 2 (g) Suppose that at a particular moment during the reaction nitric oxide (NO) is reacting at the rate of 0.066 MIs. (a) At what rate is N0 2 being formed? (b) At what rate is molecular oxygen reacting? 14.8 Consider the reaction Suppose that at a particular moment during the reaction molecular hydrogen is reacting at the rate of 0.082 MIs. (a) At what rate is ammonia being formed? (b) At what rate is molecular nitrogen reacting?, Section 14.2: Dependence of Reaction Rate on Reactant Concentration Review Questions 14.9 Explain what is meant by the rate law of a reaction. 14.10 Explain what is meant by the order of a reaction. 14.11 What are the units for the rate constants of first-order and second- order reactions? 14.12 Consider the zeroth-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 l. 14.13 The rate constant of a first-order reaction is 66 S- l What is the rate constant in units of minutes? 14.14 On which of the following properties does the rate constant of a reaction depend: (a) reactant concentrations, (b) nature of reactants, (c) temperature? Problems 14.15 The rate law for the reaction NH ~ (aq) + NO;-(aq) +. N 2 (g) + 2H 2 0(l) is given by rate = k[NH ~ ][NO;-]. At 25°C, the rate constant is 3.0 X 1O- 4 IM' s. Calculate the rate of the reaction at this temperature if [NH ~ l = 0.36 M and [N0 2 l = 0.075 M. 14.16 Use the data in Table 14.2 to calculate the rate of the reaction at the time when [F2l = 0.020 M and [CI0 2 l = 0.035 M. 14.17 Consider the reaction 14.18 A + B +. products From the following data obtained at a certain temperature, determine the order of the reaction and calculate the rate constant. [Al (M) [Bl (M) Rate (Mis) 1.50 1.50 3.20 X 10- 1 1.50 2.50 3.20 X 10- 1 3.00 1.50 6.40 X 10- 1 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 ofY is 0.40 M. Initial Rate of Disappearance of X (Mis) [Xl (M) [Yl (M) 0.053 0.10 0.50 0.127 0.20 0.30 1.02 0.40 0.60 0.254 0.20 0.60 0.509 0.40 0.30 14.19 Determine the overall orders of the reactions to which the following rate laws apply: (a) rate = k[N0 2 l 2 , (b) rate = k, (c) rate = k[H2lz[Brzlll2, (d) rate = k[NOlz[Ozl. 14.20 Consider the reaction 14.21 • A +'B The rate of the reaction is 1.6 X lO- z Mis when the concentration of A is 0.15 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 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. Time (s) P C 4 H S (mmHg) 0 400 2,000 316 4,000 248 6,000 196 8,000 155 10,000 122 QUESTIONS AND PROBLEMS 581 14.22 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: Determine the order of the reaction and the rate constant based on the following data. Time (s) o 181 513 1164 where P is the total pressure. P(mmHg) 15.76 18.88 22.79 27.08 Section 14.3: Dependence of Reactant Concentration on Time Review Questions 14.23 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. 14.24 14.25 14.26 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 telms of the half-life (tll2) and in terms of the rate constant k. Problems 14.27 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. 14.28 14.29 The thermal decomposition of phosphine ( PH 3 ) into phosphorus and molecular hydrogen is a first-order reaction: The half-life of the reaction is 35.0 s at 680°e. 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 2NOBr(g) +. 2NO(g) + BrzCg) is 0.801M . s at 10 0 e. (a) Starting with a concentration of 0.086 M, calculate the concentration of NOBr after 22 s. (b) Calculate the half-lives when [NOBrlo = 0.072 M and [NOBrlo = 0.054 M. 14.30 The rate constant for the second-order reaction is 0.541M . sat 300° e. How long (in seconds) would it take for the concentration of NO z to decrease from 0.65 M to 0.18 M? 582 CHAPTER 14 Chemical Kinetics 14.31 The second-order rate constant for the dimerization of a protein (P) p + p • P2 is 6.2 X 1O - 3 1M . s at 25° C. If the concentration of the protein is 2.7 X 10- 4 M, calculate the initial rate (Mis) of formation of P 2 . How long (in seconds) will it take to decrease the concentration of P to 2.7 X 10- 5 M ? 14.32 Consider the first-order reaction X • Y shown here. 14.33 (a) What is the half-life of the reaction? (b) Dra w pictures showing the number of X (red) and Y (blue) molecules at 20 s and at 30 s. • ••• • • •• •• • •••• t = 0 s • • • •• • • •• • •• •• • t = 10 s The reaction A B shown here follows first-order kinetics. Initially different amounts of A molecules are placed in three containers of equal volume 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 are the relative half-lives of the reactions in (i) to (iii)? (i) (ii) (iii) Section 14.4: Dependence of Reaction Rate on Temperature Review Questions 14.34 Define activation energy. What role does activation energy play in chemical kinetics? 14.35 Write the Arrhenius equation, and define all terms. 14.36 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. 14.37 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. 14.38 Sketch a potential energy versus reaction progress plot for the following reactions: (a) S(s) + 0 2(g) • S0 2(g) (b) CI 2 (g) • CI(g) + CI( g) I1Ho = -296 kJ/ mol I1W = 243 kJ/ mol 14.39 The reaction H + H2 • H2 + H has been studied for many years. Sketch a potential energy versus reaction progress diagram for this reaction. 14.40 Over the range of about +3 °C from normal body temperature, the metabolic rate, M T , is given by M T = M 37 ( 1.1 )';T, where M 37 is the normal rate (at 37°C) and I1T is the change in T. Discuss this equation in terms of a possible molecular interpretation. Problems 14.41 Variation of the rate constant with temperature for the first-order reaction 14.42 14.43 14.44 is gi ven in the following table. Determine graphically the activation energy for the reaction. T(K) k (8- 1 ) 298 1.74 X 10- 5 308 6.61 X 10- 5 318 2.51 X 10- 4 328 7.59 X 10- 4 338 2.40 X 10- 3 Given the same reactant concentrations, the reaction at 250°C is 1.50 X 10 3 times as fast as the same reaction at 150°C. Calculate the activation energy for this reaction. Assume that the frequency factor is constant. For the reaction the frequency factor A is 8.7 X 10 12 S- l and the activation energy is 63 kJ/mol. What is the rate constant for the reaction at 75°C? The rate constant of a first-order reaction is 4.60 X 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 X 10- 4 s - 1. 14.45 The rate constants of some reactions double with every 10° 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? 14.46 The rate at which tree crickets chirp is 2.0 X 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.) 14.47 The rate of bacterial hydrolysis of fish muscle is twice as great at 2.2°C as at -1.1 0 C. Estimate an Ea value for this reaction. Is there any relation to the problem of storing fish for food? 14.48 The activation energy for the denaturation of a protein is 39.6 kJ/mol. At what temperature will the rate of denaturation be 20 percent greater than its rate at 25°C? 14.49 Diagram A describes the initial state of reaction Diagram A Suppose the reaction is carried out at two different temperatures as shown in diagram B. Which picture represents the result at the higher temperature? (The reaction proceeds for the same amount of time at both temperatures .) (a) (b) Diagram B Section 14.5: Reaction Mechanisms Review Questions 14.50 What do we mean by the mechanism of a reaction? 14.51 What is an elementary step? What is the molecularity of a reaction? 14.52 Classify the following elementary reactions as unimolecular, bimolecular, or termolecular: 14.53 (a) 2NO + Br 2 -_. 2NOBr (b) CH 3 NC • CH 3 CN (c) SO + O 2 • SO z + 0 Reactions can be classified as unimolecular, bimolecular, and so on. Why are there no zero-molecular reactions? Explain why termolecular reactions are rare. 14.54 Determine the molecularity, and write the rate law for each of the following elementary steps: 14.55 (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 ev er yday analogy to illustrate the meaning of rate determining. 14.56 The equation for the combustion of ethane (C z H 6 ) is Explain why it is unlikely that this equation also represents the elementary step for the reaction. 14.57 Specify which of the following species cannot be isolated in a reaction: activated complex, product, intermediate. Problems 14.58 Classify each of the following elementary steps as unimolecular, bimolecular, or termolecular. + + • + ( a) • + (b) + • + (c) QUESTIONS AND PROBLEMS 583 14.59 The rate law for the reaction 2NO(g) + CI 2 (g) 2NOCI(g) is given by rate = k[NO][CI 2 ]. (a) What is the order of the reaction? (b) A mechanism involving the following steps has been proposed for the reaction: NO(g) + Clig) NOCI 2 (g) NOCI 2 (g) + NO(g) • 2NOCI(g) If this mechanism is correct, what does it imply about the relative rates of these two steps? 14.60 For the reaction X z + Y + Z • XY + XZ, it is found that doubling the concentration of X 2 doubles the reaction rate, tripling the concentration ofY 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. 14.61 The rate law for the reaction is rate = k[H z][NOf Which of the following mechanisms can be ruled out on the basis of the observed rate expression? Mechanis ml H2 + NO -_. H 2 0 + N N + NO • N z + 0 0+ Hz • HzO Mechanism II Hz + 2NO -_. NzO + HzO NzO + Hz • N2 + H 2 0 Mechanism III (slow) (fast) (fast) (slow) (fast) 2NO :;:.==' N 2 0 2 N 2 0 2 + H 2 • N 2 0 + HzO (fast equilibrium) (slow) N 2 0 + Hz • N z + H 2 0 (fast) 14.62 The rate law for the decomposition of ozone to molecular oxygen • IS [0 3 ]2 rate = k [0 2 ] The mechanism proposed for this process is k, 0 3 ' k '0+0 2 -I 0 + 0 3 k 2 • 20 z 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. Section 14.6: Catalysis Review Questions 14.63 How does a catalyst increase the rate of a reaction? 14.64 What are the characteristics of a catalyst? 14.65 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? 584 CHAPTER 14 Chemical Kinetics 14.66 14.67 14.68 14.69 14.70 Distinguish between homogeneous catalysis and he terogeneous catalysis. Are enzyme-catalyzed reactions e xa mples of homogeneous or heterogeneous catalysis? Explain. The concentrations of enzymes in cells are usually quite sma ll. What is the biological significance of this fact? When fruits such as apples and pears are cut, the exposed areas begin to turn brown. This is the result of an enz yme-catalyzed reaction. Often the browning can be prevented or slowed by adding a few drops of lemon juice. What is the chemical basis of this treatment? The first-order rate constant for the dehydration of carbonic acid is about 1 X 10 2 S - I. In view of this rather high rate constant, explain why it is necessary to have the enzyme carbonic anhydrase to enhance the rate of dehydration in the lungs. Problems 14.71 Most reactions, including enzyme-catalyzed reactions, proceed faster at higher temperatures. Howe ver, for a given enzy me , the rate drops off abruptly at a certain temperature. Account for this behavior. 14.72 Consider the following mechanism for the enzyme-catalyzed reactIOn: E+S (fast equilibrium) ES k 2 • E + P (slow) Derive an express ion 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 the forward reaction is equal to the rate of the reverse reaction.) Additional Problems 14.73 List four factors that influence the rate of a reaction. 14.74 Suggest experimental means by which the rates of the following reactions could be followed: (a) CaC0 3 (s) • CaO(s) + CO 2 (g) (b) CI 2 (g) + 2Br-(aq) • Br 2(aq) + 2Cqaq) (c) C 2 H 6 (g) • C 2 H 4 (g) + H 2 (g) (d) C 2 H s I(g) + H 2 0( l) • C 2 H s OH (aq) + H +(aq) + r-(aq) 14.75 'The rate constant for the reaction N0 2 (g) + CO( g) +. NO(g) + CO 2 (g) is 1.64 X 10- 6 1M . s." What is incomplete about this statement? 14.76 In a certain industrial process involving a heterogeneous catalyst, the volume of the catalyst (in the shape of a s phere ) is 10.0 cm 3 Calculate the surface area of the catalyst. If the sphere is broken down into eight sma ller spheres, each having a volume of 1.25 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 41Tr 2, where r is the radius of the sphere.) Based on y our analysis here, explain why it is some times dangerous to work in grain elevators. 14.77 14.78 The following pictures 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. • • • • •• • • • ••• t = 0 s • • t = 20 s • •• t = 40 s The following pictures show the progress of the reaction 2A • A 2 . Determine whether the reaction is first order or second order, and calculate the rate constant. • t=Omin t = 15 min t = 30 min 14.79 Use the data in Sample Problem 14.5 to determine graphically the half-life of the reaction. 14.80 The following data were collected for the reaction between hydrogen and nitric oxide at 700°C: 2H2(g) + 2NO(g) • 2H 2 0(g) + N 2 (g) Experiment [H 2 ] (M) [NO] (M) Initial Rate (MIs) 1 0.010 0.025 24 X 10- 6 2 0.0050 0.025 1.2 X 10- 6 3 0.010 0.0125 0.60 X 10- 6 (a) Determine the order of the reaction. (b) Calculate the rate constant. (c) Suggest a plausible mechanism that is consistent with the rate la w. (Hint: Assume that the oxygen atom is the intermediate. ) 14.81 When methyl phosphate is heated in acid solution, it reacts with water: If the reaction is carried out in water enriched with 18 0, 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? 14.82 The rate of the reaction CH 3 COOH(aq) + C 2 HsOH(aq) shows first-order characteristics- that is, rate = k[CH 3 COOC 2 H s ]- even though this is a second-order reaction (first order in CH 3 COOC 2 H s and first order in H 2 0). Explain. 14.83 Explain why most metals used in catalysis are transition metals. 14.84 The reaction 2A + 3B • C is first order with respect to A and B. When the initial concentrations are [A] = 1.6 X 10- 2 M and [BJ = 2.4 X 10- 3 M, the rate is 4.1 X 10- 4 MIs. Calculate the rate constant of the reaction. 14.85 The bromination of acetone is acid-catalyzed: catalyst H+ + CH 3 COCH 3 + Brz =.:_. CH 3 COCH 2 Br + H + Br- The rate of disappearance of bromine was mea sured for several different concentrations of acetone, bromine, and H+ ions at a certain temperature: [CH 3 COCH 3 ] [ Br 2] [H +] Rate of Disappearance (M) (M) (M) ofBr2 (Mis) (1) 0.30 0.050 0.050 5.7 X lO- s (2) 0.30 0.10 0.050 5.7 X lO- s (3) 0.30 0.050 0.10 1.2 X 10- 4 (4) 0.40 0.050 0.20 3.1 X 10- 4 (5) 0.40 0.050 0.050 7.6 X lO- s (a) What is the rate law for the reaction? (b) Determine the rate constant. (c) The following mechanism has been proposed for the reaction: (fast equilibrium) (s low) Show that the rate law deduced from the mechanism is consistent with that shown in part (a). 14.86 The decomposition of N 2 0 to N2 and O 2 is a first-order reaction. At 730°C the half-life of the reaction is 3.58 X 10 3 min. If the initial pressure of N 2 0 is 2.10 atm at 730°C, calculate the total gas pressure after one half-life. Assume that the volume remains constant. 14.87 The reaction S20~- + 21 - • 2S0~ - + 12 proceeds slowly in aqueous solution, but it can be catalyzed by the Fe 3 + ion. Given that Fe 3 + can oxidize 1- and Fe 2 + can reduce SzO ~-, write a plausible two-step mechani sm for this reaction. Explain why the uncatalyzed reaction is slow. 14.88 What are the units of the rate constant for a third-order reaction? 14.89 The integrated rate law for the zeroth-order reaction A • B is [Al, = [Ala - kt. (a) Sketch the following plots: (i) rate versus [Al, and (ii) [Al, 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 i s, when [Al, = o. 14.90 A flask contains a mixture of compounds A and B. Both compounds decompose by first-order kinetics. The ha lf -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? 14.91 Referring to Sample Problem 14.5, explain how you would mea sure the partial pressure of azomethane experimentally as a function of time. 14.92 The rate law for the reaction 2N0 2 (g) • N 2 0 4 (g) is rate = k[N0 2 f Which of the following changes will change the value of k? (a) The pressure of NO z is doubled. (b) The reaction is run in an organic solvent. (c) The volume of the container is doubled. (d) The temperature is decreased. (e) A catalyst is added to the container. QUESTIONS AND PROBLEMS 585 14.93 The reaction of G 2 with ~ to form 2EG is exothermic, and the reaction of G z with X z 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 sa me graph. 14.94 In the nuclear industry, workers use a rule of thumb that the radioactivity from any sample will be relatively harmless after 10 half-live s. Calculate the fraction of a radioactive sample that remains after this time period. (Hint: Radioactive decays obey first-order kinetics.) 14.95 Briefly comment on the effect of a catalyst on each of the following: (a) activation energy, (b) reaction mechanism, (c) enthalpy of rea ct ion, (d) rate of forward reaction, ( e) rate of reverse reaction. 14.96 When 6 g of granulated Zn is added to a solution of 2 M HCI in a beaker at room temperature, hydrogen gas is generated. For each of the following changes (at constant vo lume 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 °C. 14.97 Strictly speaking, the rate law derived for the reaction in Problem 14.80 applies only to certain concentrations of Hz. The general rate law for the reaction takes the form k)[NO]z[Hzl fa te = '-' =-=-==-~ 1 + kz[Hzl where k) and k2 are constants. Derive rate law expressions under the conditions of very high and very low hydrogen concentrations. Does the result from Problem 14.80 agree with one of the rate expressions here? 14.98 A certain first-order reaction is 35.5 percent complete in 4.90 min at 25°C. What is its rate constant? 14.99 The decomposition of dinitrogen pentoxide has been st udied in carbon tetrachloride solvent (CCI 4 ) at a certain temperature: 2N 2 O S • 4N0 2 + O 2 [NzOs] (M) Initial Rate (Mis) 0.92 0.95 X lO- s l.23 l.20 X lO- s 1.79 l.9 3 X lO - s 2.00 2.10 X lO - s 2.21 2.26 X 10- 0 Determine graphically the rate law for the reaction, and calculate the rate constant. 14.100 The thermal decomposition of NzO s obeys first-order kinetics. At 45°C, a plot of In [N 2 0 sl versus t gives a slope of - 6.18 X 10- 4 min -1 What is the half-life of the reaction? 14.101 When a mixture of methane and bromine is exposed to light, the following reaction occurs slowly: Suggest a reasonable mechanism for this reaction. (Hint: Bromine vapor is deep red; methane is colorless.) 14.102 The rate of the reaction between H2 and 12 to form HI (discussed on page 571) increases with the intensity of visible light. (a) Explain why this fact suppo'rts the two-step mechanism given. (1 z va por is purple.) (b) Explain why the visible light has no effect on the formation of H atoms. 586 CHAPTER 14 Chemical Kinetics 14.103 To prevent brain damage, a standard procedure is to lower the body temperature of someone who has been resuscitated after suffering cardiac arrest. What is the physiochemical ba sis for this procedure? 14.104 In Lewis Carroll's Through the Looking Glass, Alice wonders whether looking-glass milk on the other side of the mirror would be fit to drink. What do you think? 14.105 Consider the following elementary step: 14.106 14.107 14.108 x + 2Y · XY 2 (a) Write a rate law for this reaction. (b) If the initial rate of formation of XY 2 is 3.8 X 10- 3 Mis and the initial concentrations of X and Yare 0.26 M and 0.88 M, respectively, 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 CFCl 3 is first decomposed by UV radiation: CFCI 3 -_. CFCI 2 + Cl The chlorine radical then reacts with ozone as follows: Cl + 0 3 -_. CIO+ O 2 CIO + 0 • Cl + O 2 (a) Write the overall reaction for the last two steps. (b) What are the roles of CI and CIO? (c) Wh y is the fluorine radical not important in this mechanism? (d) One suggestion to reduce the concentration of chlorine radicals is to add hydrocarbons s uch as ethane (C 2 H 6 ) to the stratosphere. How will this wo rk ? (e) Draw potential energy versus reaction progress diagrams for the uncatalyzed and catalyzed (by Cl) destruction of ozone: 0 3 + o • 20 2 , Use the thermodynamic data in Appendix 2 to determine whether the reaction is exothermic or endothermic. Chlorine oxide (ClO), which plays an important role in the depletion of ozone (see Problem 14.106), decays rapidly at room temperature according to the equation From the following data, determine the reaction order and calculate the rate constant of the reaction. Time (s) 0.12 X 10- 3 0.96 X 10- 3 2.24 X 10- 3 3.20 X 10- 3 4.00 X 10- 3 [CIO] (M) 8.49 X 10- 6 7.10 X 10- 6 5.79 X 10- 6 5.20 X 10- 6 4.77 X 10- 6 A compound X undergoes two simultaneous first-order reactions as follows: X • Y with rate constant k, and X • Z with rate constant k 2 . The ratio of k,lk2 at 40°C is 8.0. What is the ratio at 300°C? Assume that the frequency factors of the two reactions are the same. 14.109 Consider a car fitted with a catalytic converter. The first 5 min or so after it is started are the most polluting. Why? 14.110 The following scheme in which A is converted to B, which is then converted to C, is known as a consecutive reaction. A · B C Assuming that both steps are first order, sketch on the same graph the variations of [A] , [B], and [C] with time. 14.111 ( a) What can you deduce about the activation energy of a reaction if its rate constant changes significantly with a small change in temperature? (b) If a bimolecular reaction occurs every time an A and a B molecule collide, what can you say about the orientation factor and activation energy of the reaction? 14.112 The rate law for the following reaction CO(g) + N0 2 (g) +. COig) + NO(g) is rate = k [N0 2 f Suggest a plausible mechanism for the reaction, given that the unstable species N0 3 is an intermediate. 14.113 Radioactive plutonium-239 (t' /2 = 2.44 X 10 5 yr) is used in nuclear reactors and atomic bombs. If there are 5.0 X 10 2 g of the isotope in a small atomic bomb, how long will it take for the substance to decay to 1.0 X 10 2 g, too small an amount for an effective bomb? 14.114 Many reactions involving heterogeneous catalysts are zeroth order; that i s, rate = k. An example is the decomposition of phosphine (PH)) over tungsten (W): It is found that the reaction is independent of [PH 3 ] as long as phosphine's pressure is sufficiently high (> I atm). Explain. 14.115 Thallium(I) is oxidized by cerium(IV) as follows: The elementary steps, in the presence of Mn(II), are as follows: Ce 4 + + Mn 2+ +. Ce 3+ + Mn3+ Ce 4+ + Mn 3+ • Ce 3+ + Mn4+ TI+ + Mn 4+ • TI 3+ + Mn2+ (a) Identify the catalyst, intermediates, and the rate-determining step if the rate law is rate = k[Ce 4 +][Mn 2+ ]. (b) Explain why the reaction is slow without the catalyst. (c) Classify the type of catalysis (homogeneous or heterogeneous). 14.116 Sucrose (C'2H2201l)' commonly called table sugar, undergoes hydrolysis (reaction with water) to produce fructose (C 6 H I2 0 6 ) and glucose (C6H'206): 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 [H 2 0 ] even though water is a reactant. Time (min) o 60.0 96.4 157.5 0.500 0.400 0.350 0.280 14.117 The first-order rate constant for the decomposition of dimethyl ether is 3.2 X 10- 4 s-' at 450° 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. 14.118 At 25°C, the rate constant for the ozone-depleting reaction is 7.9 X 10 - 15 cm 3 /molecule . s. Express the rate constant in units of II M . s. 14.119 Consider the following elementary steps for a consecutive reaction: A k,. B k, . C (a) Write an expression forthe 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. 14.120 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 hango ve r s. Based on your knowledge of enzyme kinetics, explain why binge drinking (that is, consuming too much alcohol too fast) can prove fatal. 14.121 Strontium-90, a radioactive isotope, is a major product of an atomic bomb explosion. It has a half-life of 28.1 y r. (a) Calculate the first-order rate constant for the nuclear decay. (b) Calculate the fraction of 90 Sr that remains after 10 half-live s. (c) Calculate the number of years required for 99.0 percent of 90 Sr to disappear. 14.122 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 t:.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. - '" ~ c 1l o t t 30 kJ/mol 50 kJ/mol 7 20 kJ/ mol 40 kJ/mol - 20 kJ/mol "><::: L p. -40 kJ/mol Reaction progress (a) Reaction progress ( b) Reaction progress (c) 14.123 Consider the following potential energy profile for the A • D reaction. (a) How many elementary steps are there? (b) How many intermediates are fotTlled? (c) Which step is rate determining? (d) Is the overall reaction exothermic or endothermic? >. ~ <) c <) - A '" ~ c <) B ~ 0 C D p. Reaction progress 14.124 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? 14.125 QUESTIONS AND PROBLEMS 587 The activation energy for the decomposition of hydrogen peroxide 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 decomposition to proceed as rapidly as the enzyme-catalyzed decomposition at 20° e. Assume the frequency factor A to be the same in both cases. 14.126 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 cotTesponds to exactly 3.70 X 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. 14.127 To carry out metabolism, oxygen is taken up by hemoglobin (Hb) to fotTll oxyhemoglobin ( Hb0 2 ) according to the simplified equation 14.128 Hb(aq) + 02(aq) k . Hb0 2 (aq) where the second-order rate constant is 2.1 X 10 6 1M' sat 37°C. For an average adult, the concentrations of Hb and O 2 in the blood at the lungs are 8.0 X 10 - 6 M and 1.5 X 10 - 6 M, respectively. (a) Calculate the rate of formation of Hb0 2 . (b) Calculate the rate of consumption of Oz. (c) The rate of formation of HbO z increases to 1.4 X 10- 4 Mis during exercise to meet the demand of the increased metabolism rate. Assuming the Hb concentration to remain the same, what must the oxygen concentration be to sustain this rate of Hb0 2 formation? At a certain elevated temperature, ammonia decomposes on the surface of tungsten metal as follows: From the following plot of the rate of the reaction versus the pressure of NH 3 , describe the mechanism of the reaction. P NH 3 14.129 The following expression shows the dependence of the half-life of a reaction (t1/2) on the initial reactant concentration [Al o : 1 t 1/? ex :- - [Al3 - 1 where n is the order of the reaction. Verify this dependence for zeroth-, first-, and second-order reactions. [...]... gases are expressed as partial pressures (atm) Thus, for the equilibrium we can either write the equilibrium expression as [N02]~q K = =-= -= =[N20 4 ] eq c or as • (PN02 )2 = -P= - Kp NP4 where the subscript "P" in Kp stands for pressure, and PNO2 and PN 20 4 are the equilibrium partial pressures of N02 and N 20 4 , respectively In general, Kc is not equal to Kp because the partial pressures of reactants... ~q[BrCll ~q Kc = 2 [NOBrl eq[Cl2l eq (e) Probably the simplest way to analyze this reaction is to recognize that it is the reverse of the reaction in part (d), multiplied by ~ Its equilibrium expression is the square root of the reciprocal of the expression in part (d): NO(g) Kc = + BrCl(g) :;::,=~ NOBr(g) + i CI 2(g) [NOBr1 [Cl 2l eq ~q 2 2 [NOl eq[BrCll eq Each equilibrium constant will bear the same... • ' (d) AgCl(s) + 2NH 3(aq) • ' Ag(NH3) ~ (aq) + Cl- (aq) Manipulating Equilibrium Expressions In our study of enthalpy, we learned that it is possible to manipUlate chemical equations to solve thermochemistry problems [ ~~ Section 5.3] Recall that when we made a change in a thermochemical equation, we had to make the corresponding change in the I:1H of the reaction When we reversed a reaction, for... higher concentration of N2 (and a lower concentration of O2) than air Just as the bodies of athletes who reside at high elevations naturally produce additional red blood cells to compensate for the lower partial pressure of oxygen in the air, the bodies of athletes who spend their sleeping hours in a hypoxic tent adjust in the same way Those that "live high and train low" are believed to have an advantage... Kc = (0.014)112 = 0.12 (b) Kc = (0.014i = 2.0 X 10-4 (d) Kc = (0.014)(7.2) = 0.10 (e) Kc = (1/0.10)112 = 3.2 I I Practice Problem A The following reactions have the indicated equilibrium constants at a particular temperature: N 2(g) 2NO(g) + 0 2(g) + ° i g) +=,==::::to , 2NO(g) • 2NO z(g) Kc = 4.3 X 10- 25 Kc = 6.4 X 109 Determine the values of the equilibrium constants for the following equations at... solving equilibrium problems 598 CHAPTER 15 Chemical Equilibrium Magnitude of the Equilibrium Constant One of the things the equilibrium constant tells us is the extent to which a reaction proceeds at a particular temperature If we combine stoichiometric amounts of the reactants in a reaction, three outcomes are possible: 1 The reaction will go essentially to completion, and the equilibrium mixture will... ===:' bB(g) += where a and b are the stoichiometric coefficients The equilibrium constant Kc is given by [B] b K = -= :: - [At c and the expression for Kp is K _ (PB)b p (PAt where P A and P B are the partial pressures of A and B Assuming ideal gas behavior, PAV= nART and PA = [A]RT where [A] is the molar concentration of A Likewise, PBV= nBRT PB = PB = nBRT V (nB ) = V RT and [B]RT Substituting the . + 6H z O(g) 14.7 Consider the reaction 2NO(g) + 0 2(g) +. 2N0 2 (g) Suppose that at a particular moment during the reaction nitric oxide (NO) is reacting at the rate of 0.066 MIs (b) At what rate is molecular oxygen reacting? 14.8 Consider the reaction Suppose that at a particular moment during the reaction molecular hydrogen is reacting at the rate of 0.082 MIs (s low) Show that the rate law deduced from the mechanism is consistent with that shown in part (a). 14.86 The decomposition of N 2 0 to N2 and O 2 is a first-order reaction. At