The bromination of acetone is acid-catalyzed: \(\mathrm{CH}_{3} \mathrm{COCH}_{3}+\mathrm{Br}_{2} \frac{\mathrm{H}^{+}}{\text {cually }} \mathrm{CH}_{3} \mathrm{COCH}_{2} \mathrm{Br}+\mathrm{H}^{+}+\mathrm{Br}^{-}\) The rate of disappearance of bromine was measured for several different concentrations of acetone, bromine, and \(\mathrm{H}^{+}\) ions at a certain temperature: $$ \begin{array}{lclll} \hline & &{\text { Rate of }} \\ & & & & \text { Disappearance } \\ & {\left[\mathrm{CH}_{3} \mathrm{COCH}_{3}\right]} & {\left[\mathrm{Br}_{2}\right]} & {\left[\mathrm{H}^{+}\right]} & \text {of } \mathrm{Br}_{2}(M / \mathrm{s}) \\ \hline(1) & 0.30 & 0.050 & 0.050 & 5.7 \times 10^{-5} \\ (2) & 0.30 & 0.10 & 0.050 & 5.7 \times 10^{-5} \\ (3) & 0.30 & 0.050 & 0.20 & 1.2 \times 10^{-4} \\ (4) & 0.40 & 0.050 & 0.20 & 3.1 \times 10^{-4} \\ (5) & 0.40 & 0.050 & 0.050 & 7.6 \times 10^{-5} \\ \hline \end{array} $$ (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).

Step by step solution

01

Determine the order of reaction with respect to each reactant

Using the data in experiment 1 and 2, where the concentration of \(CH_3COCH_3\) and \(H^+\) are unchanged and only the concentration of \(Br_2\) is changed. This yields the same rate, implying that the rate of the reaction is zero order with respect to \(Br_2\). Next, comparing the rates of experiments 1 and 3, where the concentrations of \(CH_3COCH_3\) and \(Br_2\) are kept the same and only the concentration of \(H^+\) is changed. The rate becomes double when the concentration of \(H^+\) is quadrupled. Therefore, it is first order with respect to \(H^+\). Lastly, comparing the rates of experiments 3 and 4, \(H^+\) and \(Br_2\) are held constant while the concentration of \(CH_3COCH_3\) is increased. This causes more than a doubling of the rate, suggesting the reaction is also first order with respect to \(CH_3COCH_3\). Hence the rate-law for the reaction is: \(rate = k[CH_3COCH_3][H^+]\)

02

Calculation of the rate constant

Using the rate law and the experiment 1 conditions, the rate constant \(k\) can be determined: Using 5.7 x 10^{-5} M/s = \(k\) x 0.300 M x 0.050 M, therefore, \(k\) = 3.8 L/mol.s

03

Evaluating the proposed reaction mechanism

The proposed mechanism needs to be consistent with the deduced rate law. As the reaction involves \(H^+\) ions, acid-catalyzed reactions generally have the catalyst in the slow (rate-determining) step. The rate law suggests the reaction is first order in both \(CH_3COCH_3\) and \(H^+\) and zero order in \(Br_2\), indicating that both \(CH_3COCH_3\) and \(H^+\) may be involved in the slow step of the reaction, while \(Br_2\) is not. Hence, this proposed mechanism is consistent with the deduced rate law.

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