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Chapter 13: Problem 113

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

Short Answer

Expert verified
The rate constant in units of \(1 / M \cdot \mathrm{s}\) is \(4.75 \times 10^{-6} \mathrm{M}^{-1} \, \mathrm{s}^{-1}\)

Step by step solution

01

Understand the given information

The reaction under consideration is \(\mathrm{O}(g)+\mathrm{O}_{3}(g) \longrightarrow 2 \mathrm{O}_{2}(g)\) , for which we are given the rate constant \(k\) as \(7.9 \times 10^{-15} \mathrm{~cm}^{3} molecule^{-1}s^{-1}\). The aim is to express this rate constant in new units: \(\mathrm{M}^{-1} \, \mathrm{s}^{-1}\).
02

Convert from cm³ to L

To start, we need to convert the volume from cm³ to L. We know that 1L = 10^3 cm³. So \(k=7.9 \times 10^{-15} cm^{3} molecule^{-1}s^{-1} =7.9 \times 10^{-18} L molecule^{-1}s^{-1}\)
03

Convert from molecule^-1 to M^-1

Next, convert from \(molecule^{-1}\) to \(\mathrm{M}^{-1}\). We use Avogadro's number for this, which states that \(1M = 6.022 \times 10^{23} molecules/mole\). So \(K = 7.9 \times 10^{-18} L molecule^{-1}s^{-1} \times \frac{6.022 \times 10^{23} molecules/mole}{1M}\)
04

Finalize calculation

When we do this multiplication, the units of K become L/M.s and K equals \(4.75 \times 10^{-6} \mathrm{M}^{-1} \, \mathrm{s}^{-1}\) (after rounding to 4 significant figures).

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Most popular questions from this chapter

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

Chlorine oxide (ClO), which plays an important role in the depletion of ozone (see Problem 13.101 ), decays rapidly at room temperature according to the equation $$ 2 \mathrm{ClO}(g) \longrightarrow \mathrm{Cl}_{2}(g)+\mathrm{O}_{2}(g) $$ From the following data, determine the reaction order and calculate the rate constant of the reaction. $$ \begin{array}{ll} \hline \text { Time (s) } & {[\mathrm{ClO}](M)} \\ \hline 0.12 \times 10^{-3} & 8.49 \times 10^{-6} \\ 0.96 \times 10^{-3} & 7.10 \times 10^{-6} \\ 2.24 \times 10^{-3} & 5.79 \times 10^{-6} \\ 3.20 \times 10^{-3} & 5.20 \times 10^{-6} \\ 4.00 \times 10^{-3} & 4.77 \times 10^{-6} \\ \hline \end{array} $$

Consider the reaction $$ \mathrm{A}+\mathrm{B} \longrightarrow \text { products } $$ From the following data obtained at a certain temperature, determine the order of the reaction and calculate the rate constant. $$ \begin{array}{ccc} \hline[\mathrm{A}](M) & {[\mathrm{B}](M)} & \text { Rate }(M / \mathrm{s}) \\ \hline 1.50 & 1.50 & 3.20 \times 10^{-1} \\ 1.50 & 2.50 & 3.20 \times 10^{-1} \\ 3.00 & 1.50 & 6.40 \times 10^{-1} \\ \hline \end{array} $$

A protein molecule, \(\mathrm{P}\), of molar mass \(\mathscr{A}\) 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: $$ \begin{array}{rlr} \mathrm{P} & \stackrel{k}{\longrightarrow} \mathrm{P}^{*}(\text { denatured }) & \text { slow } \\ 2 \mathrm{P}^{*} \longrightarrow \mathrm{P}_{2} & \text { fast } \end{array} $$ where the asterisk denotes a denatured protein molecule. Derive an expression for the average molar mass (of \(\mathrm{P}\) and \(\mathrm{P}_{2}\) ), \(, \overline{\mathscr{M}}\), in terms of the initial protein concentration \([\mathrm{P}]_{0}\) and the concentration at time \(t,\) \([\mathrm{P}]_{,}\) and \(\mathscr{A} .\) Describe how you would determine \(k\) from molar mass measurements.

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