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Ch.14 - Chemical Kinetics
All textbooksBrown 15th EditionCh.14 - Chemical KineticsProblem 102
Chapter 14, Problem 102

A colored dye compound decomposes to give a colorless product. The original dye absorbs at 608 nm and has an extinction coefficient of 4.7 * 10^4 M^-1 cm^-1 at that wavelength. You perform the decomposition reaction in a 1-cm cuvette in a spectrometer and obtain the following data: Time (min) Absorbance at 608 nm 0 1.254 30 0.941 60 0.752 90 0.672 120 0.545. From these data, determine the rate law for the reaction 'dye → product' and determine the rate constant.

Verified step by step guidance
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Step 1: Use the Beer-Lambert Law, A = εcl, to determine the concentration of the dye at each time point. Here, A is the absorbance, ε is the extinction coefficient (4.7 * 10^4 M^-1 cm^-1), c is the concentration, and l is the path length of the cuvette (1 cm). Rearrange the equation to solve for concentration: c = A / (εl).
Step 2: Calculate the concentration of the dye at each time point using the absorbance values provided. For example, at time 0 min, the concentration c = 1.254 / (4.7 * 10^4 * 1). Repeat this calculation for each time point.
Step 3: Plot the concentration of the dye versus time to visually inspect the data. This will help determine the order of the reaction. If the plot of concentration vs. time is linear, the reaction is zero-order. If the plot of ln(concentration) vs. time is linear, the reaction is first-order. If the plot of 1/concentration vs. time is linear, the reaction is second-order.
Step 4: Based on the plot that gives a straight line, determine the order of the reaction. Use the appropriate integrated rate law to calculate the rate constant (k). For a first-order reaction, use ln([A]_t/[A]_0) = -kt. For a second-order reaction, use 1/[A]_t - 1/[A]_0 = kt.
Step 5: Calculate the rate constant (k) using the slope of the linear plot from Step 3. The slope of the plot corresponds to -k for a first-order reaction or k for a second-order reaction. Ensure the units of k are consistent with the order of the reaction.

Key Concepts

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

Beer-Lambert Law

The Beer-Lambert Law relates the absorbance of light by a substance to its concentration and path length. It states that absorbance (A) is directly proportional to the concentration (c) of the absorbing species and the path length (l) of the light through the sample: A = εcl, where ε is the molar absorptivity. This law is fundamental for analyzing spectroscopic data, allowing us to determine concentration changes over time.
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Hess's Law

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 determining how the concentration of the dye changes over time and how it influences the reaction rate.
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Rate Law Fundamentals

Order of Reaction

The order of a reaction indicates how the rate is affected by the concentration of reactants. It can be determined experimentally by analyzing how changes in concentration influence the rate of reaction. For example, a first-order reaction shows a linear relationship between the natural logarithm of concentration and time, while a second-order reaction shows a quadratic relationship. Identifying the order is essential for deriving the rate law and calculating the rate constant.
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Related Practice
Textbook Question

The rate of a first-order reaction is followed by spectroscopy, monitoring the absorbance of a colored reactant at 520 nm. The reaction occurs in a 1.00-cm sample cell, and the only colored species in the reaction has an extinction coefficient of 5.60 × 103 M-1 cm-1 at 520 nm.

(a) Calculate the initial concentration of the colored reactant if the absorbance is 0.605 at the beginning of the reaction.

Textbook Question

The rate of a first-order reaction is followed by spectroscopy, monitoring the absorbance of a colored reactant at 520 nm. The reaction occurs in a 1.00-cm sample cell, and the only colored species in the reaction has an extinction coefficient of 5.60 × 103 M-1 cm-1 at 520 nm.

(c) Calculate the half-life of the reaction.

Textbook Question

The rate of a first-order reaction is followed by spectroscopy, monitoring the absorbance of a colored reactant at 520 nm. The reaction occurs in a 1.00-cm sample cell, and the only colored species in the reaction has an extinction coefficient of 5.60 × 103 M-1 cm-1 at 520 nm.

(d) How long does it take for the absorbance to fall to 0.100?

Textbook Question

At 28 C, raw milk sours in 4.0 h but takes 48 h to sour in a refrigerator at 5 C. Estimate the activation energy in kJ>mol for the reaction that leads to the souring of milk.