The half-life for the first-order decomposition of N2O4 is 1.3 * 10-5 s. N2O41g2S 2 NO21g2 If N2O4 is introduced into an evacuated flask at a pressure of 17.0 mm Hg, how many seconds are required for the pressure of NO2 to reach 1.3 mm Hg?

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Identify that the decomposition of \( \text{N}_2\text{O}_4 \) is a first-order reaction, which means the rate of reaction depends on the concentration of \( \text{N}_2\text{O}_4 \).

Use the first-order kinetics formula: \( t = \frac{1}{k} \ln \left( \frac{[A]_0}{[A]} \right) \), where \( t \) is the time, \( k \) is the rate constant, \( [A]_0 \) is the initial concentration, and \( [A] \) is the concentration at time \( t \).

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

Determine the initial pressure of \( \text{N}_2\text{O}_4 \) and the pressure of \( \text{N}_2\text{O}_4 \) at the time when \( \text{NO}_2 \) reaches 1.3 mm Hg. Since \( \text{N}_2\text{O}_4 \) decomposes to form \( 2 \text{NO}_2 \), use stoichiometry to relate the pressures.

Substitute the values into the first-order kinetics formula to solve for \( t \), the time required for the pressure of \( \text{NO}_2 \) to reach 1.3 mm Hg.

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Key Concepts

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

First-Order Kinetics

First-order kinetics refers to a reaction rate that is directly proportional to the concentration of one reactant. In this case, the decomposition of N2O4 follows first-order kinetics, meaning that the rate of reaction decreases exponentially over time. The half-life of a first-order reaction is constant and can be used to determine the time required for a specific change in concentration or pressure.

Half-Life

Half-life 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, allowing for straightforward calculations of how long it takes for a reactant to reach a certain concentration. In this problem, the half-life of N2O4 is given, which is essential for determining the time needed for NO2 to reach a specific pressure.

Pressure and Gas Laws

The behavior of gases can be described by various gas laws, which relate pressure, volume, temperature, and the number of moles. In this scenario, the pressure of NO2 produced from the decomposition of N2O4 is crucial for solving the problem. Understanding how the pressure of a gas changes with the decomposition of a reactant allows us to calculate the time required for the pressure of NO2 to reach a specified value.

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