The reaction AS C is first order in the reactant A and is known to go to completion. The product C is colored and absorbs light strongly at 550 nm, while the reactant and intermediates are colorless. A solution of A was prepared, and the absorbance of C at 550 nm was measured as a function of time. (Note that the absorbance of C is directly proportional to its concentration.) Use the following data to determine the half-life of the reaction.

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Step 1: Understand that for a first-order reaction, the rate of reaction is directly proportional to the concentration of the reactant. The integrated rate law for a first-order reaction is given by \( \ln[A]_t = -kt + \ln[A]_0 \), where \([A]_t\) is the concentration of A at time t, \([A]_0\) is the initial concentration, and k is the rate constant.

Step 2: Recognize that the absorbance of C at 550 nm is directly proportional to its concentration. Therefore, you can use the absorbance data to determine the concentration of C at different times.

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Step 3: Use the relationship between absorbance and concentration (Beer-Lambert Law: \( A = \varepsilon c l \), where A is absorbance, \( \varepsilon \) is the molar absorptivity, c is concentration, and l is the path length) to convert absorbance data into concentration data for C.

Step 4: Since the reaction goes to completion, the concentration of C at any time t is equal to the initial concentration of A minus the concentration of A at time t. Use this relationship to express \([A]_t\) in terms of the concentration of C.

Step 5: Calculate the half-life of the reaction using the first-order half-life formula \( t_{1/2} = \frac{0.693}{k} \). To find k, use the slope of the plot of \( \ln[A]_t \) versus time, which is equal to -k.

Key Concepts

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

First-Order Reactions

A first-order reaction is one where the rate of reaction is directly proportional to the concentration of one reactant. This means that as the concentration of reactant A decreases, the rate at which it reacts also decreases. The mathematical representation of a first-order reaction is given by the equation: rate = k[A], where k is the rate constant. Understanding this concept is crucial for analyzing the kinetics of the reaction and determining the half-life.

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Absorbance and Concentration Relationship

The absorbance of a solution is related to the concentration of the absorbing species through Beer-Lambert Law, which states that absorbance (A) is equal to the product of the molar absorptivity (ε), the path length (l), and the concentration (c) of the solution: A = εlc. In this case, since the product C is colored and absorbs light at 550 nm, measuring its absorbance over time allows for the determination of its concentration, which is essential for calculating the half-life of the reaction.

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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 first-order reactions, the half-life is constant and can be calculated using the formula t1/2 = 0.693/k, where k is the rate constant. This concept is important for understanding the kinetics of the reaction and allows for the prediction of how long it will take for the reactant A to diminish significantly during the course of the reaction.

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Related Practice

Textbook Question

The following experimental data were obtained in a studyof the reaction 2 HI1g2S H21g2 + I21g2. Predict the concentrationof HI that would give a rate of 1.0 * 10-5 M>sat 650 K.