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The bromination of acetone is acid-catalyzed: $$ \mathrm{CH}_{3} \mathrm{COCH}_{3}+\mathrm{Br}_{2} \underset{\text { catalyst }}{\mathrm{H}^{+}} \mathrm{CH}_{3} \mathrm{COCH}_{2} \mathrm{Br}+\mathrm{H}^{+}+\mathrm{Br}^{-} $$ (a) What is the rate law for the reaction? (b) Determine the rate constant. The rate of disappearance of bromine was measured for several different concentrations of acetone, bromine, and \(\mathrm{H}^{+}\) ions at a certain temperature: $$ \begin{array}{lcllc} & & & & \text { Rate of } \\ & & & & \text { Disappearance } \\ & {\left[\mathrm{CH}_{3} \mathrm{COCH}_{3}\right]} & {\left[\mathrm{Br}_{2}\right]} & {\left[\mathrm{H}^{+}\right]} & \text {of } \mathrm{Br}_{2}(M / \mathrm{s}) \\ \hline \text { (a) } & 0.30 & 0.050 & 0.050 & 5.7 \times 10^{-5} \\ \text {(b) } & 0.30 & 0.10 & 0.050 & 5.7 \times 10^{-5} \\ \text {(c) } & 0.30 & 0.050 & 0.10 & 1.2 \times 10^{-4} \\ \text {(d) } & 0.40 & 0.050 & 0.20 & 3.1 \times 10^{-4} \\ \text {(e) } & 0.40 & 0.050 & 0.050 & 7.6 \times 10^{-5} \end{array} $$

Step by step solution

01

Identify Variables

The rate law for the reaction is derived from the general form rate = k[A]^m[B]^n[C]^p where [A], [B], and [C] are the concentrations of the reactants, and k is the rate constant. In this case, A = CH鈧僀OCH鈧�, B = Br鈧�, and C = H鈦�.

02

Determine Order Respect to Acetone

Compare experiments (a) and (e), where [Br鈧俔 and [H鈦篯 are constant. The concentrations of acetone change from 0.30 M to 0.40 M, and the rate changes from 5.7 脳 10鈦烩伒 to 7.6 脳 10鈦烩伒 M/s. This indicates the order of reaction with respect to acetone: \[\text{Order of CH}_3\text{COCH}_3 = \frac{\log(\frac{7.6 \times 10^{-5}}{5.7 \times 10^{-5}})}{\log(\frac{0.40}{0.30})} = 1\] The reaction is first order with respect to acetone.

03

Determine Order Respect to Bromine

Compare experiments (a) and (b), where [CH鈧僀OCH鈧僝 and [H鈦篯 are constant. The concentration of Br鈧� doubles, yet the rate remains the same, suggesting that the rate does not depend on Br鈧� concentration. Therefore, the reaction is zero order with respect to Br鈧�.

04

Determine Order Respect to H鈦�

Compare experiments (a) and (c) as [CH鈧僀OCH鈧僝 and [Br鈧俔 remain constant. The concentration of H鈦� doubles, leading to the rate doubling from 5.7脳10鈦烩伒 to 1.2脳10鈦烩伌 M/s. The rate law, therefore, becomes first order in H鈦�:\[\text{Order of H}^+ = \frac{\log(\frac{1.2 \times 10^{-4}}{5.7 \times 10^{-5}})}{\log(\frac{0.10}{0.05})} = 1\]

05

Write the Rate Law

From the determined orders, the rate law for the reaction is:\[\text{Rate} = k[\text{CH}_3\text{COCH}_3]^1[\text{Br}_2]^0[\text{H}^+]^1 = k[\text{CH}_3\text{COCH}_3][\text{H}^+]\]

06

Calculate the Rate Constant k

Using data from experiment (a), where [CH鈧僀OCH鈧僝 = 0.30 M, [H鈦篯 = 0.050 M, and Rate = 5.7 脳 10鈦烩伒 M/s, substitute into the rate equation:\[5.7 \times 10^{-5} = k \times 0.30 \times 0.050\]Solving for k,\[k = \frac{5.7 \times 10^{-5}}{0.30 \times 0.050} = 3.8 \times 10^{-3} \text{ M}^{-1}\text{s}^{-1}\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Rate Law

In chemical kinetics, the rate law expresses the relationship between the rate of a chemical reaction and the concentration of its reactants. Specifically, the rate law for a reaction is a mathematical equation that demonstrates how the rate depends on the concentration of each reactant.
The general format of the rate law is typically written as: \[ \text{Rate} = k[\text{A}]^m[\text{B}]^n \]where \([\text{A]}\) and \([\text{B]}\) are the molar concentrations of reactants, \(k\) is the rate constant, and \(m\) and \(n\) are the orders of the reaction with respect to \([\text{A]}\) and \([\text{B]}\).

• Each term in the equation indicates how its corresponding reactant affects the rate of reaction.
• Reactants not appearing in the rate equation, such as \([\text{Br}_2]\) in this instance, indicate a zero order dependence, implying they do not affect the rate.

The bromination of acetone is found to follow the rate law:\[ \text{Rate} = k[\text{CH}_3\text{COCH}_3]^1[\text{H}^+]^1 \]indicating that the rate depends linearly on both acetone and \(\text{H}^+\) concentrations.

Reaction Order

The reaction order provides insight into how the concentration of each reactant influences the reaction rate. It's determined experimentally and can vary for different reactants in the same reaction.
For the bromination of acetone, we need to determine the order with respect to each reactant using experimental data:

• Acetone \([\text{CH}_3\text{COCH}_3]\): By observing the changes in rate with varying concentration of acetone, a first-order dependency is established. This means the rate of reaction directly doubles when the concentration of acetone doubles.
• Bromine \([\text{Br}_2]\): The rate remains constant despite changes in \([\text{Br}_2]\). Hence, it is zero order with respect to bromine, indicating no impact on rate by changing its concentration.
• Hydrogen ions \([\text{H}^+]\): The reaction is first order with respect to \([\text{H}^+]\), as the reaction rate doubles when the concentration of \([\text{H}^+]\) doubles.

Overall, the reaction shows combined orders of first order in acetone and \([\text{H}^+]\), and zero order in bromine.

Rate Constant

The rate constant \(k\) is a unique value for each reaction at a given temperature. It acts as a proportionality constant in the rate law equation, connecting the rate of reaction to the product of the concentrations of reactants raised to their respective powers.
For the bromination of acetone, the rate constant can be computed from experimental data using the established rate law:\[ \text{Rate} = k[\text{CH}_3\text{COCH}_3][\text{H}^+] \]Substituting the values from experiment (a), where the rate is \(5.7 \times 10^{-5}\ \text{M/s}\), \([\text{CH}_3\text{COCH}_3]\ = 0.30\ \text{M}\), and \([\text{H}^+]\ = 0.050\ \text{M}\), we solve for \(k\) using:\[ 5.7 \times 10^{-5} = k \times 0.30 \times 0.050 \]Therefore, \[ k = \frac{5.7 \times 10^{-5}}{0.30 \times 0.050} = 3.8 \times 10^{-3} \ \text{M}^{-1}\text{s}^{-1} \] This value illustrates how fast the bromination proceeds at specified conditions.

Bromination of Acetone

The bromination of acetone is a fascinating and classic example of an acid-catalyzed reaction. In this scenario, acetone reacts with bromine in the presence of an acid catalyst, such as \(\text{H}^+\) ions, resulting in the formation of brominated acetone and hydrobromic acid.
The equation for this reaction is:\[ \mathrm{CH}_{3}\mathrm{COCH}_{3} + \mathrm{Br}_2 \underset{\text{catalyst}}{\mathrm{H}^+} \rightarrow \mathrm{CH}_{3}\mathrm{COCH}_{2}\mathrm{Br} + \mathrm{H}^+ + \mathrm{Br}^- \]

• This reaction is particularly notable because it illustrates the role of a catalyst in facilitating the reaction without being consumed in the process.
• The reaction exhibits unique kinetics due to its dependence on acid concentration, making it first order in both acetone and \(\text{H}^+\) ions, and zero order in bromine.

Hence, studying this reaction provides valuable insights into how reaction kinetics and catalysis work in chemical processes.