Solved: The effective rate constant for a gaseous reaction that has a

Consult literature sources and list the observed timescales during which the following processes occur: radiative decay of excited electronic states, molecular rotational motion, molecular vibrational motion, proton transfer reactions, the initial event of vision, energy transfer in photosynthesis, the initial electron transfer events in photosynthesis, the helix-to-coil transition in polypeptides, and collisions in liquids.

Write a brief report on a recent research article in which at least one of the following techniques was used to study the kinetics of a chemical reaction: stopped-flow techniques, flash photolysis, chemical quench-flow methods, freeze quench methods, temperature-jump methods, or pressure-jump methods. Your report should be similar in content and size to one of the Impact sections found throughout this text.

Describe the main features, including advantages and disadvantages, of the following experimental methods for determining the rate law of a reaction: the isolation method, the method of initial rates, and fitting data to integrated rate law expressions.

Distinguish between reaction order and molecularity.

Assess the validity of the following statement: the rate-determining step is the slowest step in a reaction mechanism.

Distinguish between a pre-equilibrium approximation and a steady-state approximation.

Distinguish between kinetic and thermodynamic control of a reaction.

Define the terms in and limit the generality of the expression ln k = ln A Ea/RT.

Distinguish between a primary and a secondary kinetic isotope effect. Discuss how kinetic isotope effects in general can provide insight into the mechanism of a reaction.

Discuss the limitations of the generality of the expression k = kakb[A]/(kb + ka[A]) for the effective rate constant of a unimolecular reaction AP with the following mechanism: A + A5A* + A (ka, ka), A* P (kb). Suggest an experimental procedure that may either support or refute the mechanism.

The rate of the reaction A + 2 B 3 C + D was reported as 1.0 mol dm3 s1. State the rates of formation and consumption of the participants.

The rate of the reaction A + 3 B C + 2 D was reported as 1.0 mol dm3 s1. State the rates of formation and consumption of the participants.

The rate of formation of C in the reaction 2 A + B2 C + 3 D is 1.0 mol dm3 s1. State the reaction rate, and the rates of formation or consumption of A, B, and D.

The rate of consumption of B in the reaction A + 3 B C + 2 D is 1.0 mol dm3 s1. State the reaction rate, and the rates of formation or consumption of A, C, and D.

The rate law for the reaction in Exercise 22.1a was found to be v = k[A][B]. What are the units of k? Express the rate law in terms of the rates of formation and consumption of (a) A, (b) C.

The rate law for the reaction in Exercise 22.1b was found to be v = k[A][B]2. What are the units of k? Express the rate law in terms of the rates of formation and consumption of (a) A, (b) C.

The rate law for the reaction in Exercise 22.2a was reported as d[C]/dt = k[A][B][C]. Express the rate law in terms of the reaction rate; what are the units for k in each case?

The rate law for the reaction in Exercise 22.2b was reported as d[C]/dt = k[A][B][C]1. Express the rate law in terms of the reaction rate; what are the units for k in each case?

At 518C, the rate of decomposition of a sample of gaseous acetaldehyde, initially at a pressure of 363 Torr, was 1.07 Torr s1 when 5.0 per cent had reacted and 0.76 Torr s1 when 20.0 per cent had reacted. Determine the order of the reaction.

At 400 K, the rate of decomposition of a gaseous compound initially at a pressure of 12.6 kPa, was 9.71 Pa s1 when 10.0 per cent had reacted and 7.67 Pa s1 when 20.0 per cent had reacted. Determine the order of the reaction.

At 518C, the half-life for the decomposition of a sample of gaseous acetaldehyde (ethanal) initially at 363 Torr was 410 s. When the pressure was 169 Torr, the half-life was 880 s. Determine the order of the reaction.

At 400 K, the half-life for the decomposition of a sample of a gaseous compound initially at 55.5 kPa was 340 s. When the pressure was 28.9 kPa, the half-life was 178 s. Determine the order of the reaction.

The rate constant for the first-order decomposition of N2O5 in the reaction 2 N2O5(g)4 NO2(g) + O2(g) is k = 3.38 105 s1 at 25C. What is the half-life of N2O5? What will be the pressure, initially 500 Torr, at (a) 10 s, (b) 10 min after initiation of the reaction?

The rate constant for the first-order decomposition of a compound A in the reaction 2 A P is k = 2.78 107 s1 at 25C. What is the half-life of A? What will be the pressure, initially 32.1 kPa, at (a) 10 s, (b) 10 min after initiation of the reaction?

A second-order reaction of the type A + BP was carried out in a solution that was initially 0.050 mol dm3 in A and 0.080 mol dm3 in B. After 1.0 h the concentration of A had fallen to 0.020 mol dm3. (a) Calculate the rate constant. (b) What is the half-life of the reactants?

A second-order reaction of the type A + 2 B P was carried out in a solution that was initially 0.075 mol dm3 in A and 0.030 mol dm3 in B. After 1.0 h the concentration of A had fallen to 0.045 mol dm3. (a) Calculate the rate constant. (b) What is the half-life of the reactants?

If the rate laws are expressed with (a) concentrations in moles per decimetre cubed, (b) pressures in kilopascals, what are the units of the secondorder and third-order rate constants?

If the rate laws are expressed with (a) concentrations in molecules per metre cubed, (b) pressures in newtons per metre squared, what are the units of the second-order and third-order rate constants?

The second-order rate constant for the reaction CH3COOC2H5(aq) + OH(aq)CH3CO2 (aq) + CH3CH2OH(aq) is 0.11 dm3 mol1 s1. What is the concentration of ester after (a) 10 s, (b) 10 min when ethyl acetate is added to sodium hydroxide so that the initial concentrations are [NaOH] = 0.050 mol dm3 and [CH3COOC2H5] = 0.100 mol dm3?

The second-order rate constant for the reaction A + 2 B C + D is 0.21 dm3 mol1 s1. What is the concentration of C after (a) 10 s, (b) 10 min when the reactants are mixed with initial concentrations of [A] = 0.025 mol dm3 and [B] = 0.150 mol dm3?

A reaction 2 A P has a second-order rate law with k = 3.50 104 dm3 mol1 s1. Calculate the time required for the concentration of A to change from 0.260 mol dm3 to 0.011 mol dm3.

A reaction 2 A P has a third-order rate law with k = 3.50 104 dm6 mol2 s1. Calculate the time required for the concentration of A to change from 0.077 mol dm3 to 0.021 mol dm3.

Show that t1/2 1/[A]n1 for a reaction that is nth-order in A.

Deduce an expression for the time it takes for the concentration of a substance to fall to one-third its initial value in an nth-order reaction.

The pKa of NH4 + is 9.25 at 25C. The rate constant at 25C for the reaction of NH4 + and OH to form aqueous NH3 is 4.0 1010 dm3 mol1 s1. Calculate the rate constant for proton transfer to NH3. What relaxation time would be observed if a temperature jump were applied to a solution of 0.15 mol dm3 NH3(aq) at 25C?

The equilibrium A 5B + C at 25C is subjected to a temperature jump that slightly increases the concentrations of B and C. The measured relaxation time is 3.0 s. The equilibrium constant for the system is 2.0 1016 at 25C, and the equilibrium concentrations of B and C at 25C are both 2.0 104 mol dm3. Calculate the rate constants for steps (1) and (2).

The rate constant for the decomposition of a certain substance is 2.80 103 dm3 mol1 s1 at 30C and 1.38 102 dm3 mol1 s1 at 50C. Evaluate the Arrhenius parameters of the reaction.

The rate constant for the decomposition of a certain substance is 1.70 102 dm3 mol1 s1 at 24C and 2.01 102 dm3 mol1 s1 at 37C. Evaluate the Arrhenius parameters of the reaction.

The base-catalysed bromination of nitromethane-d3 in water at room temperature (298 K) proceeds 4.3 times more slowly than the bromination of the undeuterated material. Account for this difference. Use kf(C-H) = 450 N m1.

Predict the order of magnitude of the isotope effect on the relative rates of displacement of (a) 1H and 3H, (b) 16O and 18O. Will raising the temperature enhance the difference? Take kf(C-H) = 450 N m1, kf(C-O) = 1750 N m1.

The effective rate constant for a gaseous reaction that has a LindemannHinshelwood mechanism is 2.50 104 s1 at 1.30 kPa and 2.10 105 s1 at 12 Pa. Calculate the rate constant for the activation step in the mechanism.

The effective rate constant for a gaseous reaction that has a LindemannHinshelwood mechanism is 1.7 103 s1 at 1.09 kPa and 2.2 104 s1 at 25 Pa. Calculate the rate constant for the activation step in the mechanism.

The addition of hydrogen halides to alkenes has played a fundamental role in the investigation of organic reaction mechanisms. In one study (M.J. Haugh and D.R. Dalton, J. Amer. Chem. Soc. 97, 5674 (1975)), high pressures of hydrogen chloride (up to 25 atm) and propene (up to 5 atm) were examined over a range of temperatures and the amount of 2-chloropropane formed was determined by NMR. Show that, if the reaction A + BP proceeds for a short time t, the concentration of product follows [P]/[A] = k[A]m1[B]nt if the reaction is mth-order in A and nth-order in B. In a series of runs the ratio of [chloropropane] to [propene] was independent of [propene] but the ratio of [chloropropane] to [HCl] for constant amounts of propene depended on [HCl]. For t 100 h (which is short on the timescale of the reaction) the latter ratio rose from zero to 0.05, 0.03, 0.01 for p(HCl) = 10 atm, 7.5 atm, 5.0 atm, respectively. What are the orders of the reaction with respect to each reactant?

Use mathematical software or an electronic spreadsheet to examine the time dependence of [I] in the reaction mechanism A IP (k1, k2). You may either integrate eqn 22.39 numerically (see Appendix 2) or use eqn 22.40 directly. In all the following calculations, use [A]0 = 1 mol dm3 and a time range of 0 to 5 s. (a) Plot [I] against t for k1 = 10 s1 and k2 = 1 s1. (b) Increase the ratio k2/k1 steadily by decreasing the value of k1 and examine the plot of [I] against t at each turn. What approximation about d[I]/dt becomes increasingly valid?

Show that the following mechanism can account for the rate law of the reaction in Problem 22.11: HCl + HCl5(HCl)2 K1 HCl + CH3CH=CH25complex K2 (HCl)2 + complexCH3CHClCH3 + 2 HCl k (slow) What further tests could you apply to verify this mechanism?

Consider the dimerization \(2 \mathrm{~A} \rightleftharpoons \mathrm{A}_2\), with forward rate constant \(k_{\mathrm{a}}\) and backward rate constant \(k_{\mathrm{b}}\). (a) Derive the following expression for the relaxation time in terms of the total concentration of protein, \([\mathrm{A}]_{{tot }^{\prime}}=[\mathrm{A}]+2\left[\mathrm{~A}_2\right]\): \(\frac{1}{\tau^2}=k_{\mathrm{b}}^2+8 k_{\mathrm{a}} k_{\mathrm{b}}[\mathrm{A}]_{tot}\) (b) Describe the computational procedures that lead to the determination of the rate constants \(k_{\mathrm{a}}\) and \(k_{\mathrm{b}}\) from measurements of \(\tau\) for different values of \([\mathrm{A}]_{tot}\). (c) Use the data provided below and the procedure you outlined in part (b) to calculate the rate constants \(k_{\mathrm{a}}\) and \(k_{\mathrm{b}}\), and the equilibrium constant K for formation of hydrogen-bonded dimers of 2-pyridone: \(\begin{array}{llllll}{[\mathrm{A}]_{\mathrm{tot}} /\left(\mathrm{mol} \mathrm{dm}^{-3}\right)} & 0.500 & 0.352 & 0.251 & 0.151 & 0.101 \\ \tau / \mathrm{ns} & 2.3 & 2.7 & 3.3 & 4.0 & 5.3\end{array}\)

In the experiments described in Problems 22.11 and 22.13 an inverse temperature dependence of the reaction rate was observed, the overall rate of reaction at \(70^{\circ} \mathrm{C}\) being roughly one-third that at \(19^{\circ} \mathrm{C}\). Estimate the apparent activation energy and the activation energy of the rate-determining step given that the enthalpies of the two equilibria are both of the order of \(-14 \mathrm{~kJ} \mathrm{~mol}^{-1}\).

The second-order rate constants for the reaction of oxygen atoms with aromatic hydrocarbons have been measured (R. Atkinson and J.N. Pitts, J. Phys. Chem. 79, 295 (1975)). In the reaction with benzene the rate constants are 1.44 107 dm3 mol1 s1 at 300.3 K, 3.03 107 dm3 mol1 s1 at 341.2 K, and 6.9 107 dm3 mol1 s1 at 392.2 K. Find the pre-exponential factor and activation energy of the reaction.

In Problem 22.10 the isomerization of cyclopropane over a limited pressure range was examined. If the Lindemann mechanism of first-order reactions is to be tested we also need data at low pressures. These have been obtained (H.O. Pritchard, R.G. Sowden, and A.F. Trotman-Dickenson, Proc. R. Soc. A217, 563 (1953)): p/Torr 84.1 11.0 2.89 0.569 0.120 0.067 104 keff/s1 2.98 2.23 1.54 0.857 0.392 0.303 Test the Lindemann theory with these data.

P.W. Seakins, M.J. Pilling, L.T. Niiranen, D. Gutman, and L.N. Krasnoperov (J. Phys. Chem. 96, 9847 (1992)) measured the forward and reverse rate constants for the gas-phase reaction \(\mathrm{C}_2 \mathrm{H}_5(\mathrm{~g})+\mathrm{HBr}(\mathrm{g}) \rightarrow \mathrm{C}_2 \mathrm{H}_6(\mathrm{~g}) +\operatorname{Br}(\mathrm{g})\) and used their findings to compute thermodynamic parameters for \(\mathrm{C}_2 \mathrm{H}_5\). The reaction is bimolecular in both directions with Arrhenius parameters \(A=1.0 \times 10^9 \mathrm{dm}^3 \mathrm{~mol}^{-1} \mathrm{~s}^{-1}, E_{\mathrm{a}}=-4.2 \mathrm{~kJ} \mathrm{~mol}^{-1}\) for the forward reaction and \(k^{\prime}=1.4 \times 10^{11} \mathrm{dm}^3 \mathrm{~mol}^{-1} \mathrm{~s}^{-1}, E_{\mathrm{a}}=53.3 \mathrm{~kJ} \mathrm{~mol}^{-1}\) for the reverse reaction. Compute \(\Delta_{\mathrm{f}} H^{\ominus}, S_{\mathrm{m}}^{\ominus}\), and \(\Delta_{\mathrm{f}} G^{\ominus}\) of \(\mathrm{C}_2 \mathrm{H}_5\) at \(298 \mathrm{~K}\).

Two products are formed in reactions in which there is kinetic control of the ratio of products. The activation energy for the reaction leading to Product 1 is greater than that leading to Product 2. Will the ratio of product concentrations [P1]/[P2] increase or decrease if the temperature is raised?

The reaction mechanism A25A + A (fast) A + BP (slow) involves an intermediate A. Deduce the rate law for the reaction.

The equilibrium A 5B is first-order in both directions. Derive an expression for the concentration of A as a function of time when the initial molar concentrations of A and B are [A]0 and [B]0. What is the final composition of the system?

Derive an integrated expression for a second-order rate law v = k[A][B] for a reaction of stoichiometry 2 A + 3 B P.

Derive the integrated form of a third-order rate law v = k[A]2[B] in which the stoichiometry is 2 A + BP and the reactants are initially present in (a) their stoichiometric proportions, (b) with B present initially in twice the amount.

Set up the rate equations for the reaction mechanism: A B C Show that the mechanism is equivalent to A C under specified circumstances.

Show that the ratio t1/2 /t3/4, where t1/2 is the half-life and t3/4 is the time for the concentration of A to decrease to 34 of its initial value (implying that t3/4 < t1/2) can be written as a function of n alone, and can therefore be used as a rapid assessment of the order of a reaction.

Derive an equation for the steady-state rate of the sequence of reactions A5B5C5D, with [A] maintained at a fixed value and the product D removed as soon as it is formed.

For a certain second-order reaction A + BProducts, the rate of reaction, v, may be written v= =k([A]0 x)([B]0 + x) where x is the decrease in concentration of A or B as a result of reaction. Find an expression for the maximum rate and the conditions under which it applies. Draw a graph of v against x, and noting that v and x cannot be negative, identify the portion of the curve that corresponds to reality.

Consider the dimerization A A2 with forward rate constant ka and backward rate constant kb. Show that the relaxation time is: =

The half-life for the (first-order) radioactive decay of 14C is 5730 y (it emits rays with an energy of 0.16 MeV). An archaeological sample contained wood that had only 72 per cent of the 14C found in living trees. What is its age?

One of the hazards of nuclear explosions is the generation of 90Sr and its subsequent incorporation in place of calcium in bones. This nuclide emits rays of energy 0.55 MeV, and has a half-life of 28.1 y. Suppose 1.00 g was absorbed by a newly born child. How much will remain after (a) 18 y, (b) 70 y if none is lost metabolically?

Pharmacokinetics is the study of the rates of absorption and elimination of drugs by organisms. In most cases, elimination is slower than absorption and is a more important determinant of availability of a drug for binding to its target. A drug can be eliminated by many mechanisms, such as metabolism in the liver, intestine, or kidney followed by excretion of breakdown products through urine or faeces. As an example of pharmacokinetic analysis, consider the elimination of beta adrenergic blocking agents (beta blockers), drugs used in the treatment of hypertension. After intravenous administration of a beta blocker, the blood plasma of a patient was analysed for remaining drug and the data are shown below, where c is the drug concentration measured at a time t after the injection. t/min 30 60 120 150 240 360 480 c/(ng cm3) 699 622 413 292 152 60 24 (a) Is removal of the drug a first- or second-order process? (b) Calculate the rate constant and half-life of the process. Comment. An essential aspect of drug development is the optimization of the half-life of elimination, which needs to be long enough to allow the drug to find and act on its target organ but not so long that harmful side-effects become important.

The absorption and elimination of a drug in the body may be modelled with a mechanism consisting of two consecutive reactions: \(\begin{array}{cllll} A & \rightarrow & B & \rightarrow & C \\ \text { drug at site of } & & drug dispersed & & eliminated \\ administration & & in blood & & drug \end{array}\) where the rate constants of absorption \((\mathrm{A} \rightarrow \mathrm{B})\) and elimination are, respectively, \(k_1\) and \(k_2\). (a) Consider a case in which absorption is so fast that it may be regarded as instantaneous so that a dose of \(\mathrm{A}\) at an initial concentration \([\mathrm{A}]_0\) immediately leads to a drug concentration in blood of \([\mathrm{B}]_0\). Also, assume that elimination follows first-order kinetics. Show that, after the administration of n equal doses separated by a time interval \(\tau\), the peak concentration of drug B in the blood, \([\mathrm{P}]_m\), rises beyond the value of \([\mathrm{B}]_0\) and eventually reaches a constant, maximum peak value given by \([\mathrm{P}]_{\infty}=[\mathrm{B}]_0\left(1-\mathrm{e}^{-k_2 \tau}\right)^{-1}\) where \([\mathrm{P}]_n\) is the (peak) concentration of \(\mathrm{B}\) immediately after administration of the nth dose and \([\mathrm{P}]_{\infty}\) is the value at very large n. Also, write a mathematical expression for the residual concentration of \(\mathrm{B},[\mathrm{R}]_n\), which we define to be the concentration of drug \(\mathrm{B}\) immediately before the administration of the (n+1)th dose. \([\mathrm{R}]_n\) is always smaller than \([\mathrm{P}]_n\) on account of drug elimination during the period \(\tau\) between drug administrations. Show that \([\mathrm{P}]_{\infty}-[\mathrm{R}]_{\infty}=[\mathrm{B}]_0\). (b) Consider a drug for which \(k_2=0.0289 \mathrm{~h}^{-1}\). (i) Calculate the value of \(\tau\) required to achieve \([\mathrm{P}]_{\infty} /[\mathrm{B}]_0=10\). Prepare a graph that plots both \([\mathrm{P}]_n /[\mathrm{B}]_0\) and \([\mathrm{R}]_n /[\mathrm{B}]_0\) against n. (ii) How many doses must be administered to achieve \([\mathrm{P}]_n\) value that is 75 per cent of the maximum value? What time has passed during the administration of these doses? (iii) What actions can be taken to reduce the variation \([\mathrm{P}]_{\infty}-[\mathrm{R}]_{\infty}\) while maintaining the same value of \([\mathrm{P}]_{\infty}\)? (c) Now consider the administration of a single dose \([\mathrm{A}]_0\) for which absorption follows first-order kinetics and elimination follows zero-order kinetics. Show that with the initial concentration \([\mathrm{B}]_0=0\), the concentration of drug in the blood is given by \([\mathrm{B}]=[\mathrm{A}]_0\left(1-\mathrm{e}^{-k_1 t}\right)-k_2 t\) Plot \([\mathrm{B}] /[\mathrm{A}]_0\) against t for the case \(k_1=10 \mathrm{~h}^{-1}, k_2=4.0 \times 10^{-3} \mathrm{mmol} \mathrm{dm}^{-3} \mathrm{~h}^{-1}\), and \([\mathrm{A}]_0=0.1 \mathrm{mmol} \mathrm{dm}^{-3}\). Comment on the shape of the curve. (d) Using the model from part (c), set \(\mathrm{d}[\mathrm{B}] / \mathrm{d} t=0\) and show that the maximum value of [B] occurs at the time \(t_{\max }=\frac{1}{k_1} \ln \left(\frac{k_1[\mathrm{~A}]_0}{k_2}\right)\). Also, show that the maximum concentration of drug in blood is given by \([\mathrm{B}]_{\max }=[\mathrm{A}]_0-k_2 / k_1-k_2 t_{\max }\).

Consider a mechanism for the helixcoil transition in which nucleation occurs in the middle of the chain: hhhh . . . 5hchh . . . hchh . . . 5cccc . . . We saw in Impact I22.1 that this type of nucleation is relatively slow, so neither step may be rate-determining. (a) Set up the rate equations for this mechanism. (b) Apply the steady-state approximation and show that, under these circumstances, the mechanism is equivalent to hhhh . . . cccc . . . . (c) Use your knowledge of experimental techniques and your results from parts (a) and (b) to support or refute the following statement: It is very difficult to obtain experimental evidence for intermediates in protein folding by performing simple rate measurements and one must resort to special flow, relaxation, or trapping techniques to detect intermediates directly.

Propose a set of experiments in which analysis of the line-shapes of NMR transitions (Section 15.7) can be used to monitor fast events in protein folding and unfolding. What are the disadvantages and disadvantages of this NMR method over methods that use electronic and vibrational spectroscopy?

Consider the following mechanism for renaturation of a double helix from its strands A and B: A + B5unstable helix (fast) Unstable helix stable double helix (slow) Derive the rate equation for the formation of the double helix and express the rate constant of the renaturation reaction in terms of the rate constants of the individual steps.

Prebiotic reactions are reactions that might have occurred under the conditions prevalent on the Earth before the first living creatures emerged and which can lead to analogues of molecules necessary for life as we now know it. To qualify, a reaction must proceed with favourable rates and equilibria. M.P. Robertson and S.I. Miller (Science 268, 702(1995)) have studied the prebiotic synthesis of 5-substituted uracils, among them 5-hydroxymethyluracil (HMU). Amino acid analogues can be formed from HMU under prebiotic conditions by reaction with various nucleophiles, such as H2S, HCN, indole, imidazole, etc. For the synthesis of HMU (the uracil analogue of serine) from uracil and formaldehyde (HCHO), the rate of addition is given by log {k/(dm3 mol1 s1)} = 11.75 5488/(T/K) (at pH = 7), and log K = 1.36 + 1794/(T/K). For this reaction, calculate the rates and equilibrium constants over a range of temperatures corresponding to possible prebiotic conditions, such as 050C, and plot them against temperature. Also, calculate the activation energy and the standard reaction Gibbs energy and enthalpy at 25C. Prebiotic conditions are not likely to be standard conditions. Speculate about how the actual values of the reaction Gibbs energy and enthalpy might differ from the standard values. Do you expect that the reaction would still be favourable?

Methane is a by-product of a number of natural processes (such as digestion of cellulose in ruminant animals, anaerobic decomposition of organic waste matter) and industrial processes (such as food production and fossil fuel use). Reaction with the hydroxyl radical \(\mathrm{OH}\) is the main path by which \(\mathrm{CH}_4\) is removed from the lower atmosphere. T. Gierczak, R.K. Talukdar, S.C. Herndon, G.L. Vaghjiani, and A.R. Ravishankara (J. Phys. Chem. A 101, 3125 (1997)) measured the rate constants for the elementary bimolecular gas-phase reaction of methane with the hydroxyl radical over a range of temperatures of importance to atmospheric chemistry. Deduce the Arrhenius parameters A and \(E_{\mathrm{a}}\) from the following measurements: \(\begin{array}{llllllll}T / \mathrm{K} & 295 & 223 & 218 & 213 & 206 & 200 & 195 \\ k /\left(10^6 \mathrm{dm}^3 \mathrm{~mol}^{-1} \mathrm{~s}^{-1}\right) & 3.55 & 0.494 & 0.452 & 0.379 & 0.295 & 0.241 & 0.217\end{array}\)

As we saw in Problem 22.37, reaction with the hydroxyl radical OH is the main path by which CH4, a by-product of many natural and industrial processes, is removed from the lower atmosphere. T. Gierczak, R.K. Talukdar, S.C. Herndon, G.L. Vaghjiani, and A.R. Ravishankara (J. Phys. Chem. A 101, 3125 (1997)) measured the rate constants for the bimolecular gas-phase reaction CH4(g) + OH(g)CH3(g) + H2O(g) and found A = 1.13 109 dm3 mol1 s1 and Ea = 14.1 kJ mol1 for the Arrhenius parameters. (a) Estimate the rate of consumption of CH4. Take the average OH concentration to be 1.5 1021 mol dm3, that of CH4 to be 4.0 108 mol dm3, and the temperature to be 10C. (b) Estimate the global annual mass of CH4 consumed by this reaction (which is slightly less than the amount introduced to the atmosphere) given an effective volume for the Earths lower atmosphere of 4 1021 dm3.

T. Gierczak, R.K. Talukdar, S.C. Herndon, G.L. Vaghjiani, and A.R. Ravishankara (J. Phys. Chem. A 101, 3125 (1997)) measured the rate constants for the bimolecular gas-phase reaction of methane with the hydroxyl radical in several isotopic variations. From their data, the following Arrhenius parameters can be obtained: A/(dm3 mol1 s1) Ea/(kJ mol1) CH4 + OHCH3 + H2O 1.13 109 14.1 CD4 + OHCD3 + DOH 6.0 108 17.5 CH4 + ODCH3 + DOH 1.01 109 13.6 Compute the rate constants at 298 K, and interpret the kinetic isotope effects.

The oxidation of HSO3 by O2 in aqueous solution is a reaction of importance to the processes of acid rain formation and flue gas desulfurization. R.E. Connick, Y.-X. Zhang, S. Lee, R. Adamic, and P. Chieng (Inorg. Chem. 34, 4543 (1995)) report that the reaction 2 HSO3 + O2 2 SO4 2 + 2 H+ follows the rate law v = k[HSO3 ]2[H+]2. Given a pH of 5.6 and an oxygen molar concentration of 2.4 104 mol dm3 (both presumed constant), an initial HSO3 molar concentration of 5 105 mol dm3, and a rate constant of 3.6 106 dm9 mol3 s1, what is the initial rate of reaction? How long would it take for HSO3 to reach half its initial concentration?