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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Identify the reaction: \( 2 \text{N}_2\text{O}_5 (g) \rightarrow 4 \text{NO}_2 (g) + \text{O}_2 (g) \).

Since the reaction is first order, use the first-order kinetics equation: \( [A] = [A]_0 e^{-kt} \), where \( k \) is the rate constant.

Calculate the rate constant \( k \) using the half-life formula for first-order reactions: \( t_{1/2} = \frac{0.693}{k} \).

Convert the time from minutes to hours to match the half-life units: \( 215 \text{ minutes} = 3.583 \text{ hours} \).

Use the integrated rate law to find the remaining \( \text{N}_2\text{O}_5 \) and then determine the amount of \( \text{O}_2 \) produced, considering the stoichiometry of the reaction.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

First-Order Reactions

First-order reactions are chemical reactions where the rate is directly proportional to the concentration of one reactant. In this case, the decomposition of dinitrogen pentoxide (N2O5) follows first-order kinetics, meaning that as N2O5 is consumed, the rate of reaction decreases. The half-life of a first-order reaction is constant and can be used to determine the concentration of reactants over time.

Half-Life

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 first-order reactions, the half-life is independent of the initial concentration. In this problem, the half-life of 2.81 hours allows us to calculate how much N2O5 remains after a specific time, which is crucial for determining the amount of products formed, including oxygen gas (O2).

Partial Pressure

Partial pressure is the pressure exerted by a single component of a gas mixture. According to Dalton's Law of Partial Pressures, the total pressure of a gas mixture is the sum of the partial pressures of its individual gases. In this scenario, calculating the partial pressure of O2 produced from the decomposition of N2O5 involves understanding the stoichiometry of the reaction and the initial conditions of the system.

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Textbook Question

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?

Textbook Question

Iodine atoms combine to form I₂ in liquid hexane solvent with a rate constant of 1.5⨉10¹⁰ L/mols. The reaction is second order in I. Since the reaction occurs so quickly, the only way to study the reaction is to create iodine atoms almost instantaneously, usually by photochemical decomposition of I₂. Suppose a flash of light creates an initial [I] concentration of 0.0100 M. How long will it take for 95% of the newly created iodine atoms to recombine to form I₂?