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Chapter 14: Problem 26

Gaseous azomethane, \(\mathrm{CH}_{3} \mathrm{N}=\mathrm{NCH}_{3,}\) decomposes in a first-order reaction when heated: $$ \mathrm{CH}_{3} \mathrm{N}=\mathrm{NCH}_{3}(\mathrm{g}) \rightarrow \mathrm{N}_{2}(\mathrm{g})+\mathrm{C}_{2} \mathrm{H}_{6}(\mathrm{g}) $$ The rate constant for this reaction at \(600 \mathrm{K}\) is \(0.0216 \mathrm{min}^{-1} .\) If the initial quantity of azomethane in the flask is \(2.00 \mathrm{g}\), how much remains after 0.0500 hour? What mass of \(\mathrm{N}_{2}\) is formed in this time?

Short Answer

Expert verified
1.86 g of azomethane remains, and 0.0645 g of N2 is formed.

Step by step solution

01

Understand First-Order Kinetics

In a first-order reaction, the rate of reaction is directly proportional to the concentration of one reactant. The integrated rate law for a first-order reaction is given by the formula: \( [A]_t = [A]_0 \times e^{-kt} \), where \([A]_0\) is the initial concentration, \([A]_t\) is the concentration at time \(t\), \(k\) is the rate constant, and \(t\) is the time.
02

Convert Mass to Moles

The initial mass of azomethane is \(2.00 \text{ g}\). The molar mass of azomethane (\(\text{CH}_3\text{N} = \text{NCH}_3\)) is 58.10 g/mol. Calculate the initial moles, \([A]_0\), using \([A]_0 = \frac{2.00 \text{ g}}{58.10 \text{ g/mol}} = 0.0344 \text{ mol}\).
03

Calculate Time in Minutes

Convert the time from hours to minutes for consistency with the units of the rate constant. \(0.0500 \text{ hour} \times 60 \text{ min/hour} = 3.00 \text{ minutes}\).
04

Calculate Remaining Moles

Use the first-order integrated rate law to calculate the amount of azomethane remaining after 3.00 minutes: \([A]_t = 0.0344 \text{ mol} \times e^{-0.0216 \text{ min}^{-1} \times 3.00 \text{ min}} = 0.0344 \text{ mol} \times e^{-0.0648} \approx 0.0321 \text{ mol}\).
05

Convert Remaining Moles to Mass

Calculate the remaining mass of azomethane: \(\text{mass} = 0.0321 \text{ mol} \times 58.10 \text{ g/mol} = 1.86 \text{ g} \).
06

Calculate Moles of N2 Formed

Since the stoichiometry of the decomposition is 1:1, the moles of \(\text{N}_2\) formed will be equal to the initial moles of azomethane minus the remaining moles: \(0.0344 \text{ mol} - 0.0321 \text{ mol} = 0.0023 \text{ mol} \text{ N}_2\).
07

Convert Moles of N2 to Mass

Calculate the mass of \(\text{N}_2\) formed using its molar mass (28.02 g/mol): \(\text{mass of } \text{N}_2 = 0.0023 \text{ mol} \times 28.02 \text{ g/mol} = 0.0645 \text{ g} \).

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Key Concepts

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

Rate Law
In the study of first-order kinetics, the rate law plays a crucial role. The rate law is an expression that relates the rate of a chemical reaction to the concentration of its reactants. For a first-order reaction like the decomposition of azomethane, the rate of reaction is directly proportional to the concentration of azomethane itself. This can be mathematically expressed as \( ext{Rate} = k[A] \), where \( k \) is the rate constant, and \( [A] \) represents the concentration of azomethane. This linear relationship ensures that as the concentration of azomethane decreases, so does the rate of the reaction, maintaining a consistent reaction speed relative to concentration changes.
Reaction Rate
The reaction rate is a measure of how quickly reactants are converted into products in a chemical process. It's typically expressed in terms of the change of concentration of a reactant or product over time. For the decomposition of azomethane, the reaction rate can be observed by measuring how fast the azomethane converts into nitrogen gas (\( ext{N}_2 \)) and ethane (\( ext{C}_2 ext{H}_6 \)). This reaction, being first-order, implies that the rate depends on the concentration of azomethane alone. As time progresses, the concentration diminishes logarithmically, reflected by the integrated rate equation \( [A]_t = [A]_0 e^{-kt} \). This equation allows you to calculate either the concentration at a particular time or the time required to reach a certain concentration.
Stoichiometry
Stoichiometry is the branch of chemistry that deals with the quantitative relationships that exist in chemical formulas and reactions. In this exercise, the stoichiometry of the decomposition reaction \( ext{CH}_3 ext{N}= ext{NCH}_3 ightarrow ext{N}_2 + ext{C}_2 ext{H}_6 \) is straightforward since each molecule of azomethane decomposes into one molecule of nitrogen and one molecule of ethane. This 1:1 ratio simplifies calculations, as every mole of azomethane consumed results in an equivalent mole of \( ext{N}_2 \) produced. This stoichiometric relationship is crucial when using the mole method for finding the final quantities of particles post-reaction.
Molar Mass
Molar mass is an essential concept in chemistry, serving as the bridge between the mass of a substance and the amount in moles. It is the mass of one mole of a substance, usually expressed in g/mol. For azomethane, it is calculated as 58.10 g/mol, which allows for the conversion of its initial mass to moles. This conversion is critical since reaction rates and stoichiometric calculations are often mole-based. Molar mass also applies to products like \( ext{N}_2 \), which has a molar mass of 28.02 g/mol, and is employed in calculating the mass of \( ext{N}_2 \) generated in the reaction. Understanding molar mass ensures accurate translations between working quantities in laboratory settings and theoretical predictions.

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Most popular questions from this chapter

Chlorine atoms contribute to the destruction of the Earth's ozone layer by the following sequence of reactions: $$ \begin{array}{l} \mathrm{Cl}+\mathrm{O}_{3} \rightarrow \mathrm{ClO}+\mathrm{O}_{2} \\ \mathrm{ClO}+\mathrm{O} \rightarrow \mathrm{Cl}+\mathrm{O}_{2} \end{array} $$ where the O atoms in the second step come from the decomposition of ozone by sunlight: $$ \mathrm{O}_{3}(\mathrm{g}) \rightarrow \mathrm{O}(\mathrm{g})+\mathrm{O}_{2}(\mathrm{g}) $$ What is the net equation on summing these three equations? Why does this lead to ozone loss in the stratosphere? What is the role played by Cl in this sequence of reactions? What name is given to species such as ClO?

Draw a reaction coordinate diagram for an exothermic reaction that occurs in a single step. Identify the activation energy and the net energy change for the reaction on this diagram. Draw a second diagram that represents the same reaction in the presence of a catalyst, assuming a single-step reaction is involved here also. Identify the activation energy of this reaction and the energy change. Is the activation energy in the two drawings different? Does the energy evolved in the two reactions differ?

Nitramide, \(\mathrm{NO}_{2} \mathrm{NH}_{2}\), decomposes slowly in aqueous solution according to the following reaction: $$ \mathrm{NO}_{2} \mathrm{NH}_{2}(\mathrm{aq}) \rightarrow \mathrm{N}_{2} \mathrm{O}(\mathrm{g})+\mathrm{H}_{2} \mathrm{O}(\ell) $$ The reaction follows the experimental rate law $$ \text { Rate }=\frac{k\left[\mathrm{NO}_{2} \mathrm{NH}_{2}\right]}{\left[\mathrm{H}_{3} \mathrm{O}^{+}\right]} $$ (a) What is the apparent order of the reaction in a pH buffered solution? (In a pH buffered solution, the concentration of \(\mathrm{H}_{3} \mathrm{O}^{+}\) is a constant.) (b) Which of the following mechanisms is the most appropriate for the interpretation of this rate law? Explain. (Note that when writing the expression for \(K,\) the equilibrium constant, \(\left.\left[\mathrm{H}_{2} \mathrm{O}\right] \text { is not involved. See Chapter } 15 .\right)\) Mechanism 1 \(\mathrm{NO}_{2} \mathrm{NH}_{2} \stackrel{k_{1}}{\longrightarrow} \mathrm{N}_{2} \mathrm{O}+\mathrm{H}_{2} \mathrm{O}\) Mechanism 2 $$ \mathrm{NO}_{2} \mathrm{NH}_{2}+\mathrm{H}_{3} \mathrm{O}^{+} \frac{k_{2}}{\overleftarrow{k_{2}^{\prime}}} \mathrm{NO}_{2} \mathrm{NH}_{3}^{+}+\mathrm{H}_{2} \mathrm{O} $$ \(\mathrm{NO}_{2} \mathrm{NH}_{3}^{+} \stackrel{k_{3}}{\longrightarrow} \mathrm{N}_{2} \mathrm{O}+\mathrm{H}_{3} \mathrm{O}^{+} \quad\) (rate-limiting step) Mechanism 3 \(\mathrm{NO}_{2} \mathrm{NH}_{2}+\mathrm{H}_{2} \mathrm{O} \frac{k_{4}}{k_{4}^{\prime}} \mathrm{NO}_{2} \mathrm{NH}^{-}+\underset{(\text { rapid equilibrium })}{\mathrm{H}_{3} \mathrm{O}^{+}}\) \(\mathrm{NO}_{2} \mathrm{NH}^{-} \stackrel{k_{5}}{\longrightarrow} \mathrm{N}_{2} \mathrm{O}+\mathrm{OH}^{-} \quad\) (rate-limiting step) \(\mathrm{H}_{3} \mathrm{O}^{+}+\mathrm{OH}^{-} \stackrel{k_{6}}{\longrightarrow} 2 \mathrm{H}_{2} \mathrm{O} \quad\) (very fast reaction) (c) Show the relationship between the experimentally observed rate constant, \(k\), and the rate constants in the selected mechanism. (d) Based on the experimental rate law, will the reaction rate increase or decrease if the pH of the solution is increased?

The radioactive isotope \(^{64} \mathrm{Cu}\) is used in the form of copper(II) acetate to study Wilson's disease. The isotope has a half-life of 12.70 hours. What fraction of radioactive copper(II) acetate remains after 64 hours?

The decomposition of \(\mathrm{SO}_{2} \mathrm{Cl}_{2}\) $$ \mathrm{sO}_{2} \mathrm{Cl}_{2}(\mathrm{g}) \rightarrow \mathrm{SO}_{2}(\mathrm{g})+\mathrm{Cl}_{2}(\mathrm{g}) $$ is first-order in \(\mathrm{SO}_{2} \mathrm{Cl}_{2}\), and the reaction has a halflife of 245 minutes at 600 K. If you begin with \(3.6 \times 10^{-3}\) mol of \(\mathrm{SO}_{2} \mathrm{Cl}_{2}\) in a \(1.0-\mathrm{L}\) flask, how long will it take for the amount of \(\mathrm{SO}_{2} \mathrm{Cl}_{2}\) to decrease to \(2.00 \times 10^{-4}\) mol?
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