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.

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Identify the relationship between absorbance, concentration, and path length using Beer's Law: \( A = \varepsilon c l \), where \( A \) is absorbance, \( \varepsilon \) is the molar extinction coefficient, \( c \) is the concentration, and \( l \) is the path length.

Substitute the given values into Beer's Law: \( A = 0.605 \), \( \varepsilon = 5.60 \times 10^3 \text{ M}^{-1} \text{ cm}^{-1} \), and \( l = 1.00 \text{ cm} \).

Rearrange the equation to solve for the concentration \( c \): \( c = \frac{A}{\varepsilon l} \).

Substitute the known values into the rearranged equation: \( c = \frac{0.605}{5.60 \times 10^3 \times 1.00} \).

Calculate the initial concentration of the colored reactant using the values substituted into the equation.

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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 to the properties of the material through which the light is traveling. It states that absorbance (A) is directly proportional to the concentration (c) of the absorbing species, the path length (l) of the sample, and the molar absorptivity (ε) of the substance. The equation is A = εcl, which is essential for calculating the concentration of a colored reactant from its absorbance.

First-Order Reaction Kinetics

In a first-order reaction, the rate of reaction is directly proportional to the concentration of one reactant. This means that as the concentration decreases, the rate of reaction also decreases. Understanding this concept is crucial for analyzing how the concentration of the colored reactant changes over time, which is monitored through absorbance measurements.

Extinction Coefficient

The extinction coefficient (ε) is a measure of how strongly a chemical species absorbs light at a given wavelength. It is expressed in units of M^-1 cm^-1 and is specific to each substance and wavelength. In this question, the extinction coefficient is used in conjunction with the Beer-Lambert Law to calculate the initial concentration of the colored reactant based on the measured absorbance.

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Textbook Question

Americium-241 is used in smoke detectors. It has a first-order rate constant for radioactive decay of k = 1.6 * 10-3 yr-1. By contrast, iodine-125, which is used to test for thyroid functioning, has a rate constant for radioactive decay of k = 0.011 day-1. (a) What are the half-lives of these two isotopes? (b) Which one decays at a faster rate?

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Textbook Question

Americium-241 is used in smoke detectors. It has a first-order rate constant for radioactive decay of k = 1.6 * 10-3 yr-1. By contrast, iodine-125, which is used to test for thyroid functioning, has a rate constant for radioactive decay of k = 0.011 day-1. (c) How much of a 1.00-mg sample of each isotope remains after three half-lives? (d) How much of a 1.00-mg sample of each isotope remains after 4 days?

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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.

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Textbook Question

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

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