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Chapter 13: Problem 39

Relate the rate of decomposition of \(\mathrm{NO}_{2}\) to the rate of formation of \(\mathrm{O}_{2}\) for the following reaction: $$2 \mathrm{NO}_{2}(g) \longrightarrow 2 \mathrm{NO}(g)+\mathrm{O}_{2}(g)$$

Short Answer

Expert verified
\(-\frac{d[\mathrm{NO}_{2}]}{dt} = 2 \frac{d[\mathrm{O}_{2}]}{dt}\), meaning the decomposition rate of \(\mathrm{NO}_{2}\) is twice the formation rate of \(\mathrm{O}_{2}\).

Step by step solution

01

Understand the Reaction

The reaction given is \(2 \mathrm{NO}_{2}(g) \rightarrow 2 \mathrm{NO}(g)+\mathrm{O}_{2}(g)\). From this, we need to relate the rates of decomposition and formation for the involved substances.
02

Write the Rate Expression for NO2 Decomposition

Since \(2 \mathrm{NO}_{2}\) decomposes into other products, the rate at which \(\mathrm{NO}_{2}\) decomposes can be represented by \(-\frac{1}{2} \frac{d[\mathrm{NO}_{2}]}{dt}\). The negative sign indicates it's being consumed.
03

Write the Rate Expression for O2 Formation

The oxygen molecule \(\mathrm{O}_{2}\) is formed in this reaction. Its rate of formation is given by \(\frac{d[\mathrm{O}_{2}]}{dt}\), with no coefficient adjustment since the coefficient is 1.
04

Relate the Rates

Connect the rate of decomposition of \(\mathrm{NO}_{2}\) to the rate of formation of \(\mathrm{O}_{2}\) using the stoichiometry of the reaction. For every 2 moles of \(\mathrm{NO}_{2}\) that decompose, 1 mole of \(\mathrm{O}_{2}\) is formed, leading to the relationship: \(-\frac{1}{2} \frac{d[\mathrm{NO}_{2}]}{dt} = \frac{1}{1} \frac{d[\mathrm{O}_{2}]}{dt}\).
05

Simplify the Equation

Simplify the relation to \(-\frac{d[\mathrm{NO}_{2}]}{dt} = 2 \frac{d[\mathrm{O}_{2}]}{dt}\) to clearly see how the rates are related.

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Key Concepts

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

Chemical Kinetics
Chemical kinetics explores the speed or rate at which chemical reactions occur and the factors that affect these rates. For our reaction involving the decomposition of nitrogen dioxide ( NO鈧� ), understanding kinetic principles helps us determine how quickly the reaction converts reactants to products. This is crucial because:
  • It helps predict how long a reaction will take, which is useful in industrial processes.
  • It allows chemists to adjust conditions to optimize reactions, such as changing temperatures or concentrations.
  • Kinetics can also help in understanding reaction mechanisms, which are the step-by-step pathways by which a reaction occurs.
In a broader sense, chemical kinetics provides insight into both the microscopic world of molecular interactions and the macroscopic reality of observable reaction speeds. It's like looking behind the curtain at how substances transform from one into another.
Stoichiometry
Stoichiometry refers to the quantitative relationships between reactants and products in a chemical reaction, derived from the balanced chemical equation. For the decomposition of NO鈧� , it plays a key role:
  • The coefficients in the equation, 2 NO鈧� 鈫� 2 NO + O鈧� , tell us how molecules relate to each other numerically.
  • It helps us establish the rate relationship between different substances. For instance, according to stoichiometry, for every 2 moles of NO鈧� that decompose, 1 mole of O鈧� is formed.
  • This relationship is used to write rate expressions, which describe how the concentrations of reactants and products change with time.
Stoichiometry is like the "recipe" of the chemical reaction world, ensuring that the proportions are just right to understand how much product can be formed from certain amounts of reactants.
Decomposition Reactions
Decomposition reactions are a type of chemical reaction where one compound breaks down into two or more products. In our example, nitrogen dioxide ( NO鈧� ) breaks down into nitrogen monoxide ( NO ) and oxygen gas ( O鈧� ). These reactions are characterized by:
  • A single compound transforming into more than one product, as seen in our equation.
  • They often require energy input, like heat or light, to break down the bonds in the original compound.
  • The simplicity and predictability of these reactions make them common in various applications, such as in the decomposition of ozone in the atmosphere or even in some explosive reactions.
Understanding decomposition reactions is fundamental in areas like environmental science and engineering, as they play a significant role in both nature and industrial processes.

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Most popular questions from this chapter

In the presence of excess thiocyanate ion, \(\mathrm{SCN}^{-}\), the following reaction is first order in iron(III) ion, \(\mathrm{Fe}^{3+}\); the rate constant is \(1.27 / \mathrm{s}\). $$\mathrm{Fe}^{3+}(a q)+\mathrm{SCN}^{-}(a q) \longrightarrow \mathrm{Fe}(\mathrm{SCN})^{2+}(a q)$$ What is the half-life in seconds? How many seconds would be required for the initial concentration of \(\mathrm{Fe}^{3+}\) to decrease to each of the following values: \(25.0 \%\) left, \(12.5 \%\) left, \(6.25 \%\) left, \(3.125 \%\) left? What is the relationship between these times and the half-life?

Write a rate equation, showing the dependence of rate on reactant concentrations, for each of the following elementary reactions. a. \(\mathrm{O}_{3} \longrightarrow \mathrm{O}_{2}+\mathrm{O}\) b. \(\mathrm{NOCl}_{2}+\overrightarrow{\mathrm{NO}} \longrightarrow 2 \mathrm{NOCl}\)

A second-order reaction has a rate constant of \(8.7 \times\) \(10^{-4} /(M \cdot \mathrm{s})\) at \(30^{\circ} \mathrm{C}\). At \(40^{\circ} \mathrm{C}\), the rate constant is \(1.8 \times\) \(10^{-3} /(M \cdot \mathrm{s}) .\) What are the activation energy and frequency factor for this reaction? Predict the value of the rate constant at \(45^{\circ} \mathrm{C}\).

Cyclobutane, \(\mathrm{C}_{4} \mathrm{H}_{8}\), consisting of molecules in which four carbon atoms form a ring, decomposes, when heated, to give ethylene. The reaction is first order. In an experiment, the initial concentration of cyclobutane was \(0.00150 \mathrm{M}\). After heating at \(450^{\circ} \mathrm{C}\) for \(455 \mathrm{~s}\), this was reduced to \(0.00119 \mathrm{M}\). What was the concentration of cyclobutane after a total of \(750 \mathrm{~s}\) ?

Sketch a potential-energy diagram for the reaction of nitric oxide with ozone. $$\mathrm{NO}(g)+\mathrm{O}_{3}(g) \longrightarrow \mathrm{NO}_{2}(g)+\mathrm{O}_{2}(g)$$ The activation energy for the forward reaction is \(10 \mathrm{~kJ}\); the \(\Delta H^{\circ}\) is \(-200 \mathrm{~kJ} .\) What is the activation energy for the reverse reaction? Label your diagram appropriately.
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