The decomposition of XY is second order in XY and has a rate constant of 7.02 * 10^-3 M^-1 s^-1 at a certain temperature. b. How long will it take for the concentration of XY to decrease to 12.5% of its initial concentration when the initial concentration is 0.100 M? When the initial concentration is 0.200 M? c. If the initial concentration of XY is 0.150 M, how long will it take for the concentration to decrease to 0.062 M? d. If the initial concentration of XY is 0.050 M, what is the concentration of XY after 5.0 * 10^1 s? After 5.50 * 10^2 s?

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Identify that the reaction is second order, which means the rate law is given by \( \text{Rate} = k[XY]^2 \).

Use the integrated rate law for a second-order reaction: \( \frac{1}{[XY]} = kt + \frac{1}{[XY]_0} \), where \([XY]_0\) is the initial concentration and \([XY]\) is the concentration at time \(t\).

For part b, substitute \([XY]_0 = 0.100 \text{ M}\) and \([XY] = 0.0125 \text{ M}\) into the integrated rate law to solve for \(t\). Repeat the process for \([XY]_0 = 0.200 \text{ M}\).

For part c, substitute \([XY]_0 = 0.150 \text{ M}\) and \([XY] = 0.062 \text{ M}\) into the integrated rate law to solve for \(t\).

For part d, use the integrated rate law with \([XY]_0 = 0.050 \text{ M}\) to find \([XY]\) after \(t = 50 \text{ s}\) and \(t = 550 \text{ s}\).

Key Concepts

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

Second-Order Reactions

A second-order reaction is one where the rate of reaction is proportional to the square of the concentration of one reactant or to the product of the concentrations of two reactants. The rate law for a second-order reaction can be expressed as rate = k[XY]^2 or rate = k[XY1][XY2]. Understanding this concept is crucial for calculating reaction times and concentrations at various points in the reaction.

Integrated Rate Law

The integrated rate law for a second-order reaction relates the concentration of the reactant to time. It is given by the equation 1/[XY] = 1/[XY]₀ + kt, where [XY]₀ is the initial concentration, k is the rate constant, and t is time. This equation allows us to determine how long it takes for the concentration of a reactant to reach a certain value.

Half-Life of a Second-Order Reaction

The half-life of a second-order reaction is dependent on the initial concentration of the reactant and is given by the formula t₁/₂ = 1/(k[XY]₀). Unlike first-order reactions, the half-life increases as the initial concentration decreases. This concept is important for understanding how quickly a reactant will deplete over time and is essential for solving the time-related questions in the problem.

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