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Ch.14 - Chemical Kinetics
Tro - Chemistry: A Molecular Approach 4th Edition
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Tutti i libri di testoTro 4th EditionCh.14 - Chemical KineticsProblema 86
Capitolo 14, Problema 86

Consider the reaction: 2 O3(g) → 3 O2( g) The rate law for this reaction is: Rate = k [O3]2 [O2] Suppose that a 1.0-L reaction vessel initially contains 1.0 mol of O3 and 1.0 mol of O2. What fraction of the O3 will have reacted when the rate falls to one-half of its initial value?

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Identify the initial rate of the reaction using the rate law: \( \text{Rate}_{\text{initial}} = k [\text{O}_3]^2 [\text{O}_2] \).
Determine the expression for the rate when it falls to half of its initial value: \( \text{Rate}_{\text{half}} = \frac{1}{2} \times \text{Rate}_{\text{initial}} \).
Set up the equation for the rate at half its initial value: \( \frac{1}{2} \times k [\text{O}_3]^2 [\text{O}_2] = k [\text{O}_3']^2 [\text{O}_2'] \), where \([\text{O}_3']\) and \([\text{O}_2']\) are the concentrations when the rate is halved.
Assume \( x \) moles of \( \text{O}_3 \) have reacted, then \([\text{O}_3'] = 1.0 - x\) and \([\text{O}_2'] = 1.0 + \frac{3}{2}x\) because for every 2 moles of \( \text{O}_3 \) that react, 3 moles of \( \text{O}_2 \) are produced.
Substitute \([\text{O}_3']\) and \([\text{O}_2']\) into the rate equation and solve for \( x \) to find the fraction of \( \text{O}_3 \) that has reacted.

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Rate Law

The rate law expresses the relationship between the rate of a chemical reaction and the concentration of its reactants. It is typically formulated as Rate = k [A]^m [B]^n, where k is the rate constant, and m and n are the orders of the reaction with respect to reactants A and B. Understanding the rate law is crucial for predicting how changes in concentration affect the reaction rate.
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Rate Law Fundamentals

Reaction Order

The reaction order is the sum of the exponents in the rate law and indicates how the rate of reaction depends on the concentration of reactants. In the given reaction, the order with respect to O<sub>3</sub> is 2, meaning that the rate is proportional to the square of its concentration. This concept helps in determining how the concentration of reactants influences the speed of the reaction.
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Average Bond Order

Half-life of a Reaction

The half-life of a reaction is the time required for the concentration of a reactant to decrease to half of its initial value. For reactions of different orders, the half-life can vary significantly. In this context, understanding how the rate changes as the concentration of O<sub>3</sub> decreases is essential for calculating the fraction that has reacted when the rate falls to half its initial value.
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First-Order Half-Life
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Dinitrogen pentoxide decomposes in the gas phase to form nitrogen dioxide and oxygen gas. The reaction is first order in dinitrogen pentoxide and has a half-life of 2.81 h at 25 °C. If a 1.5-L reaction vessel initially contains 745 torr of N2O5 at 25 °C, what partial pressure of O2 is present in the vessel after 215 minutes?

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