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The oxidation of 2 -butanone \(\left(\mathrm{CH}_{3} \mathrm{COC}_{2} \mathrm{H}_{5}\right)\) by the cerium(IV) ion in aqueous solution to form acetic acid \(\left(\mathrm{CH}_{3} \mathrm{CO}_{2} \mathrm{H}\right)\) occurs according to the following balanced equation: \(\mathrm{CH}_{3} \mathrm{COC}_{2} \mathrm{H}_{3}(a q)+6 \mathrm{Ce}^{4+}(a q)+3 \mathrm{H}_{2} \mathrm{O}(l) \longrightarrow\) \(2 \mathrm{CH}_{3} \mathrm{CO}_{2} \mathrm{H}(a q)+6 \mathrm{Ce}^{3+}(a q)+6 \mathrm{H}^{+}(a q)\) (a) If acetic acid appears at an average rate of \(5.0 \times 10^{-8} \mathrm{M} / \mathrm{s}\) what is \(\Delta\left[\mathrm{H}^{-}\right] / \Delta t\) during the same time interval? (b) What is the average rate of consumption of \(\mathrm{Ce}^{4+}\) during the same time interval?

Step by step solution

01

Identify the Stoichiometry

Based on the balanced chemical equation provided: \[\text{CH}_3\text{COC}_2\text{H}_3 (aq) + 6\text{Ce}^{4+}(aq) + 3\text{H}_2\text{O}(l) \rightarrow 2\text{CH}_3\text{CO}_2\text{H}(aq) + 6\text{Ce}^{3+}(aq) + 6\text{H}^+(aq) \] It's established that 2 moles of acetic acid are produced for every 6 moles of \(\text{H}^{+}\).

02

Relate Rates of Change

The rate of appearance of acetic acid is given as \(5.0 \times 10^{-8}\, \text{M/s}\). Since 2 moles of acetic acid produce 6 moles of \(\text{H}^{+}\), the rate of change of \([\text{H}^+]\) should be 3 times that of acetic acid. Hence, \[\Delta[\text{H}^+]/\Delta t = 3 \times (5.0 \times 10^{-8} \text{ M/s}) \].

03

Calculate \(\Delta [\text{H}^+] / \Delta t\)

By multiplying the rate of appearance of acetic acid by the ratio of \([\text{H}^+]\) to acetic acid from the reaction equation: \[\Delta[\text{H}^+]/\Delta t = 3 \times 5.0 \times 10^{-8} = 1.5 \times 10^{-7} \text{ M/s}\]

04

Determine Rate of Ce(IV) Consumption

In the reaction, 2 moles of acetic acid correspond to the consumption of 6 moles of \(\text{Ce}^{4+}\), or a mole ratio of 1:3. Therefore, the rate of consumption of \(\text{Ce}^{4+}\) is 3 times the rate of acetic acid appearance: \[\text{Rate of } \text{Ce}^{4+} = 3 \times (5.0 \times 10^{-8} \text{ M/s}) \]

05

Calculate Average Rate of Ce(IV) Consumption

By performing the multiplication based on stoichiometry: \[\text{Rate of } \text{Ce}^{4+} = 3 \times 5.0 \times 10^{-8} = 1.5 \times 10^{-7} \text{ M/s}\]

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Key Concepts

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

Reaction Kinetics

Reaction kinetics explores how quickly reactions occur and the factors affecting their speed. In the oxidation of 2-butanone, understanding kinetics helps us determine how fast acetic acid forms and cerium(IV) ion gets consumed. The kinetics of this reaction is affected by concentrations, temperature, and pressure. Reactants like cerium(IV) are crucial to the process, and their consumption rate helps us understand efficiency in converting chemical energy. Studying the speed of this reaction can pave the way for optimizing conditions in industrial applications involving the formation of acetic acid. Kinetics forms the basis for calculating different rates in reactions.

Stoichiometry

Stoichiometry is like a recipe for chemical reactions. It tells us how much of each reactant is needed to get the desired amount of product. In our reaction, stoichiometry shows that 6 moles of cerium(IV) ion react with 1 mole of 2-butanone to yield 2 moles of acetic acid. This relationship helps us predict

• How reactants convert to products
• Required proportions of each substance
• Expected yields of chemical products

It forms the backbone of quantitative chemical reactions and guides chemists in balancing equations, which is crucial in laboratory settings.

Chemical Equations

Chemical equations describe what happens in a chemical reaction using symbols and formulas. They show reactants transforming into products through chemical changes. The balanced equation for our reaction is: \[ \text{CH}_3\text{COC}_2\text{H}_3 (aq) + 6\text{Ce}^{4+}(aq) + 3\text{H}_2\text{O}(l) \rightarrow 2\text{CH}_3\text{CO}_2\text{H}(aq) + 6\text{Ce}^{3+}(aq) + 6\text{H}^+(aq)\]This equation indicates:

• The substances involved including states of matter
• The stoichiometric relationships between reactants and products
• Mass conservation, as the number of each type of atom is equal on both sides

Writing balanced chemical equations is a fundamental skill in chemistry.

Rate of Reaction

The rate of reaction measures how fast products form or reactants are used up. For the oxidation of 2-butanone, we looked at the rate of acetic acid formation, given as \(5.0 \times 10^{-8} \text{ M/s}\). This rate provides insights into:

• How quickly a reaction proceeds towards completion
• The role of catalysts and other variables in speeding up the reaction
• Determining when equilibrium is reached in reversible reactions

By understanding these rates, chemists can control reaction conditions to optimize processes in fields like chemical manufacturing and pharmaceuticals.