Chapter 9

The oxidation of ammonia in air is catalysed by platinum metal \\[ 4 \mathrm{NH}_{3}(\mathrm{g})+5 \mathrm{O}_{2}(\mathrm{g}) \rightarrow 4 \mathrm{NO}(\mathrm{g})+6 \mathrm{H}_{2} \mathrm{O}(\mathrm{g}) \\] Write an expression for the rate of reaction in terms of differentials for the consumption of the reactants and formation of the products. (Sections \(9.2 \text { and } 9.4)\)

Write the rate equation for the following elementary reactions and give the molecularity for each reaction. (Sections 9.4 and 9.8) (a) \(\mathrm{Cr}+\mathrm{O}_{3} \rightarrow \mathrm{ClO}^{*}+\mathrm{O}_{2}\) (b) \(\mathrm{CH}_{3} \mathrm{N}_{2} \mathrm{CH}_{3} \rightarrow 2 \mathrm{CH}_{3}^{*}+\mathrm{N}_{2}\) (c) \(2 \mathrm{Cl}^{\prime} \rightarrow \mathrm{Cl}_{2}\) (d) \(\mathrm{NO}_{2}^{*}+\mathrm{F}_{2} \rightarrow \mathrm{NO}_{2} \mathrm{F}+\mathrm{F}\)

For the complex reaction of \(\mathrm{NO}\) and \(\mathrm{H}_{2}\) \\[ 2 \mathrm{NO}(\mathrm{g})+2 \mathrm{H}_{2}(\mathrm{g}) \rightarrow \mathrm{N}_{2}(\mathrm{g})+2 \mathrm{H}_{2} \mathrm{O}(\mathrm{g}) \\] the rate equation is given by rate of reaction \(\left.=k[\mathrm{NO}]^{2}\left[\mathrm{H}_{2}\right] \text { (Section } 9.5\right)\) (a) What are the orders of the reaction with respect to NO and \(\mathrm{H}_{2}\) ? (b) What is the overall order of the reaction? (c) What will happen to the rate of reaction if: (i) \(\left[H_{2}\right]\) is doubled (ii) \(\left[\mathrm{H}_{2}\right]\) is halved; (iii) \([\mathrm{NO}]\) is doubled (iv) \([\mathrm{NO}]\) is increased by a factor of three?

The gas phase decomposition of dinitrogen pentoxide \(\left(\mathrm{N}_{2} \mathrm{O}_{5}\right)\) \\[ \mathrm{N}_{2} \mathrm{O}_{5} \rightarrow \mathrm{NO}_{2}^{*}+\mathrm{NO}_{3}^{*} \\] was studied in a large stainless steel cell, at \(294 \mathrm{K}\) and a pressure of 1 bar. The concentration of \(\mathrm{N}_{2} \mathrm{O}_{5}\) was monitored using infrared spectroscopy. The following results were obtained. (Section \(9.4)\) $$\begin{array}{lllllll} t / \mathrm{s} & 0 & 10 & 20 & 30 & 40 & 50 \\ {\left[\mathrm{N}_{2} \mathrm{O}_{8}\right] / 10^{-9} \mathrm{moldm}^{-3}} & 34.0 & 27.0 & 19.5 & 15.0 & 11.5 & 8.7 \\ t / \mathrm{s} & 60 & 70 & 80 & 90 & 100 & \\ {\left[\mathrm{N}_{2} \mathrm{O}_{8}\right] / 10^{-9} \mathrm{moldm}^{-3}} & 6.6 & 5.1 & 3.9 & 2.9 & 2.2 & \end{array}$$ Show that the reaction is first order and find a value for the rate constant at \(294 \mathrm{K}\)

From the data provided in the table, deduce the rate equation and the value of the rate constant for the following reaction. (Section \(9.4)\) \\[ \begin{aligned} \mathrm{CH}_{3} \mathrm{COCH}_{3}(\mathrm{aq})+\mathrm{Br}_{2}(\mathrm{aq})+\mathrm{H}^{+}(\mathrm{aq}) \rightarrow & \mathrm{CH}_{3} \mathrm{COCH}_{2} \mathrm{Br}(\mathrm{aq}) \\ &+2 \mathrm{H}^{+}(\mathrm{aq})+\mathrm{Br}(\mathrm{aq}) \end{aligned} \\] $$\begin{array}{llll} \hline & \text { Initial } & \text { Initial } & \text { Initial } & \text { Initial } \\ & \text { concentration } & \text { concentration } & \text { concentration } & \text { rate of } \\ & \begin{array}{l} \text { of } \mathrm{CH}_{3} \mathrm{COCH}_{3} \\ \text { /moldm }^{-3} \end{array} & \begin{array}{l} \text { of } \mathrm{Br}_{2} \\ \text { /moldm }^{-3} \end{array} & \begin{array}{l} \text { of } \mathrm{H}^{+} \\ \text {/moldm }^{-3} \end{array} & \begin{array}{l} \text { reaction } \\ \text { /moldm }^{-3} \mathrm{s}^{-1} \end{array} \\ \hline 1 & 1.00 & 1.00 & 1.00 & 4.0 \times 10^{-3} \\ 2 & 2.00 & 1.00 & 1.00 & 8.0 \times 10^{3} \\ 3 & 2.00 & 2.00 & 1.00 & 8.0 \times 10^{3} \\ 4 & 1.00 & 1.00 & 2.00 & 8.0 \times 10^{-3} \end{array}$$

For a particular first order reaction, half of the reactant is used up after 15 s. What fraction of the reactant will remain after 1 \(\min ?(\text { Section } 9.4)\)

Cyclobutane decomposes to form ethane according to: \\[ \mathrm{C}_{4} \mathrm{H}_{8}(\mathrm{g}) \rightarrow 2 \mathrm{C}_{2} \mathrm{H}_{4}(\mathrm{g}) \\] A quantity of cyclobutane was sealed in a container and exerted a pressure of \(53.30 \mathrm{kPa}\) at \(700 \mathrm{K}\). The pressure changed during the reaction as follows. At the end of the reaction, the pressure was \(106.60 \mathrm{kPa}\) $$\begin{array}{ll} \hline \text { Time/s } & \text { Total pressure/kPa } \\ \hline 0 & 53.30 \\ 2000 & 64.53 \\ 4000 & 73.59 \\ 6000 & 80.53 \\ 8000 & 85.99 \\ 10000 & 90.39 \end{array}$$ Assuming the gases behave ideally, show graphically that the reaction is first order with respect to cyclobutane. Calculate the rate constant and the half life for the reaction. (Section \(9.4)\)

The rate constant for the decomposition of a compound in solution is \(2.0 \times 10^{-4} \mathrm{s}^{-1}\). If the initial concentration is \(0.02 \mathrm{moldm}^{-3},\) what will the concentration of the compound be after \(10 \text { min? (Section } 9.4)\)

The reaction of methyl radicals to form ethane was investigated in a laser flash photolysis experiment at \(300 \mathrm{K}\) \\[ \mathrm{CH}_{3}^{*}+\mathrm{CH}_{3}^{*} \rightarrow \mathrm{C}_{2} \mathrm{H}_{8} \\] The rate constant for this reaction at \(300 \mathrm{K}\) is \(3.7 \times 10^{10} \mathrm{dm}^{3}\) \(\mathrm{mol}^{-1} \mathrm{s}^{-1}\). The concentration of methyl radicals, \(\left[\mathrm{CH}_{3}^{\circ}\right],\) at time \(t=0\) was \(1.70 \times 10^{-8} \mathrm{moldm}^{-3}\). Calculate a value for \(\left[\mathrm{CH}_{3}\right]\) at \(\left.t=1.00 \times 10^{-3} \mathrm{s} \text { . (Section } 9.4\right)\)

The acid-catalysed hydrolysis of sucrose shows first order kinetics. The half life for the reaction at room temperature was found to be 190 min. Calculate the rate constant for the reaction under these conditions. (Section \(9.4)\)

The reaction between \(\mathrm{H}_{2} \mathrm{PO}_{4}^{-}\) and \(\mathrm{OH}^{-}\) was investigated at \(298 \mathrm{K}\) using the initial rate method \\[ \mathrm{H}_{2} \mathrm{PO}_{4}^{-}(\mathrm{aq})+\mathrm{OH}^{-}(\mathrm{aq}) \rightarrow \mathrm{HPO}_{4}^{2-}(\mathrm{aq})+\mathrm{H}_{2} \mathrm{O}(\mathrm{aq}) \\] The following results were obtained. (Section 9.5 ) $$\begin{array}{llll} & \begin{array}{l} \text { Initial rate } / 10^{-3} \\ \text {moldm }^{-3} \text {min }^{-1} \end{array} & \begin{array}{l} {[\mathrm{OH}]_{0} / 10^{-3}} \\ \text {moldm }^{-3} \end{array} & \begin{array}{l} {\left[\mathrm{H}_{2} \mathrm{PO}_{4}\right]_{0} / 10^{-3}} \\ \mathrm{mol} \mathrm{dm}^{-3} \end{array} \\ \hline \text { Experiment 1 } & 2.0 & 0.40 & 3.0 \\ \text { Experiment 2 } & 3.7 & 0.55 & 3.0 \\ \text { Experiment 3 } & 7.1 & 0.75 & 3.0 \end{array}$$ (a) Plot a log-log graph to determine the order of reaction with respect to \(\mathrm{OH}^{-}(\mathrm{aq})\) (b) What further experiments would you need to do to find the order with respect to \(\mathrm{H}_{2} \mathrm{PO}_{4}-?\)

The addition of bromine to propene is an elementary reaction with a rate constant, \(k\) \\[ \mathrm{CH}_{2}=\mathrm{CHCH}_{3}+\mathrm{Br}_{2} \rightarrow \mathrm{CH}_{2} \mathrm{BrCHBrCH}_{3} \\] Kinetic studies were carried out at \(298 \mathrm{K}\) using excess \(\mathrm{Br}_{2}\). For \(\left[\mathrm{Br}_{2}\right]_{0}=0.20 \mathrm{mol} \mathrm{dm}^{-3},\) the pseudo-first order rate constant, \(k^{\prime}\) for the reaction was found to be \(900 \mathrm{s}^{-1}\). What is the value of \(k\) at \(298 \mathrm{K} ? \text { (Section } 9.4)\)

The decomposition of ammonia on a platinum surface at \(856^{\circ} \mathrm{C}\) \\[ 2 \mathrm{NH}_{3}(\mathrm{g}) \rightarrow \mathrm{N}_{2}(\mathrm{g})+3 \mathrm{H}_{2}(\mathrm{g}) \\] shows the following dependence of the concentration of ammonia gas on time. $$\begin{array}{llllllll} t / s & 0 & 200 & 400 & 600 & 800 & 1000 & 1200 \\ {\left[\mathrm{NH}_{3} / 110^{-3} \mathrm{moldm}^{-3}\right.} & 2.10 & 1.85 & 1.47 & 1.23 & 0.86 & 0.57 & 0.34 \end{array}$$ Find the order of the reaction and a value for the rate constant at \(856^{\circ} \mathrm{C}\). Suggest an explanation for the order you obtain. (Section \(9.5)\)

Rate constants at a series of temperatures were obtained for the decomposition of azomethane \\[ \mathrm{CH}_{3} \mathrm{N}_{2} \mathrm{CH}_{3} \rightarrow 2 \mathrm{CH}_{3}^{*}+\mathrm{N}_{2} \\] $$\begin{array}{llllll} \hline T / \mathrm{K} & 523 & 541 & 560 & 576 & 593 \\ k / 10^{-6} \mathrm{s}^{-1} & 1.8 & 15 & 60 & 160 & 950 \end{array}$$ Use the data in the table to find the activation energy, \(E_{a}\), for the reaction. (Section \(9.7)\)

Calculate the activation energy for a reaction in which the rate constant is \(\left.5 \text { times faster at } 50^{\circ} \mathrm{C} \text { than at } 20^{\circ} \mathrm{C} \text { . (Section } 9.7\right)\)

The mechanism for the formation of a DNA double helix from two strands \(A \text { and } B \text { is as follows. (Section } 9.6)\) (a) Experiments show that the overall reaction is first order with respect to strand A and first order with respect to strand B. Write the equation for the overall reaction. (b) Write the rate equation for the overall reaction. (c) Assuming the processes involved are elementary reactions, derive an expression for the rate constant for the overall reaction in terms of the rate constants for the individual steps.

The reaction of methane with hydrogen atoms is an elementary process. \\[ \mathrm{CH}_{4}+\mathrm{H}^{*} \rightleftharpoons \mathrm{CH}_{3}^{*}+\mathrm{H}_{2} \\] The rate constant for the forward reaction at \(1000 \mathrm{K}\) is \(1.6 \times 10^{8} \mathrm{dm}^{3} \mathrm{mol}^{-1} \mathrm{s}^{-1}\). The equilibrium constant, \(K_{\mathrm{c}},\) at the same temperature is \(19.8 .\) Calculate the rate constant for the reverse reaction at \(1000 \mathrm{K} \text { . (Section } 9.6)\)

After intravenous injection of a drug to treat hypertension (high blood pressure), the blood plasma of the patient was analysed for the remaining drug at various times after the injection. (Section 9.5 ) $$\begin{array}{lllllllll} t / \min & 50 & 100 & 150 & 200 & 250 & 300 & 400 & 500 \\ {[\mathrm{drug}] / 10^{-9} \mathrm{g} \mathrm{cm}^{-3}} & 650 & 445 & 304 & 208 & 142 & 97 & 45 & 21 \end{array}$$ (a) Is the removal of the drug in the body a first or a second order process? (b) Calculate the rate constant, \(k\), and the half life, \(t_{1 / 2}\), for the process. (c) An essential part of drug development is achieving an optimum value of \(t_{1 / 2}\) for effective operation and elimination of the drug from the bloodstream. What would be the possible problems if \(t_{1 / 2}\) were too short or too long?

The dissociation of propane in which a C-C bond breaks to form a methyl radical and an ethyl radical is a unimolecular reaction. \\[ \mathrm{C}_{3} \mathrm{H}_{8} \rightarrow \mathrm{CH}_{3}^{*}+\mathrm{C}_{2} \mathrm{H}_{5}^{*} \\] The rate of formation of \(\mathrm{CH}_{3}^{*}\) (and of \(\mathrm{C}_{2} \mathrm{H}_{5}\) ') is given by \(\frac{\mathrm{d}\left[\mathrm{CH} \mathrm{b}^{*}\right]}{\mathrm{d} t}=\mathrm{k}_{\text {overd }}\left[\mathrm{C}_{3} \mathrm{H}_{8}\right],\) where \(k_{\text {overd }}\) is the unimolecular rate constant for the reaction. (Sections 9.6 and 9.8 ) (a) Explain what is meant by the term 'unimolecular? The propane molecules obtain sufficient energy to dissociate by colliding with other molecules, \(M\), where \(M\) may be an unreactive gas such as nitrogen. The mechanism for this process can be written as where \(\mathrm{C}_{3} \mathrm{H}_{\mathrm{g}}^{*}\) is a propane molecule in a high energy state, which has sufficient energy to dissociate. (b) By applying the steady state approximation to \(\mathrm{C}_{3} \mathrm{H}_{8}^{*}\), derive an expression for \(\left[\mathrm{C}_{3} \mathrm{H}_{4}^{\prime}\right] .\) since the rate of formation of \(\mathrm{CH}_{3}^{*}\) is equal to \(k_{2}\left[\mathrm{C}_{3} \mathrm{H}_{8}^{*}\right],\) show that \\[ k_{\text {overal }}=\frac{k_{1} k_{2}[M]}{k_{-1}[M]+k_{2}} \\] (c) Show that: (i) when \([\mathrm{M}]\) is very large, \(k_{\text {overal }}=\frac{k_{1} k_{2}}{k_{-1}}\) (ii) when \([\mathrm{M}]\) is very small, \(k_{\text {oved }}=k_{1}[\mathrm{M}]\) (d) What are the rate-determining stages in the mechanism under each of the conditions in (c)(1) and (c)(ii)? Sketch a graph to show the dependence of \(k_{\text {overall }}\) on \([\mathrm{M}]\) (e) It is always better to find a linear expression to analyse experimental data. Show that \\[ \frac{1}{k_{\text {overal }}}=\frac{k_{-1}}{k_{\sqrt{2}}}+\frac{1}{k_{1}[M]} \\] A plot of \(1 / k_{\text {werall }}(y-\text { axis ) against } 1 /[\mathrm{M}] \text { ( } x\) -axis) is a straight line. What are the gradient and the intercept on the \(y\) -axis at \(1 /[\mathrm{M}]=0 ?\)

At low substrate concentrations, the initial rate of an enzyme catalysed reaction was found to be directly proportional to the initial substrate concentration, [S]o, and directly proportional to the initial enzyme concentration, \([\mathrm{E}]_{0} .\) (Section 9.9 ) (a) Outline the series of experiments that led to these results. (b) Write a rate equation for the reaction under these conditions and explain the observed kinetics in terms of the mechanism for the reaction. What is the rate-determining step under these conditions? At much higher substrate concentrations, the initial rate was found to be constant and independent of the initial substrate concentrations. The initial enzyme concentration was the same in each experiment. (c) Write a rate equation for the reaction under these conditions and explain the observed kinetics in terms of the mechanism for the reaction. What is the rate-determining step under these conditions?