Ch.14 - Chemical Kinetics

Problem 101

Chapter 14, Problem 101

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 * 10^3 M^-1 cm^-1 at 520 nm. If the absorbance falls to 0.250 at 30.0 min, calculate the rate constant in units of s^-1.

Verified step by step guidance

Start by using Beer's Law to relate absorbance (A) to concentration (C) of the colored species. Beer's Law is given by:

A = ε C l

, where

ε

is the extinction coefficient,

C

is the concentration, and

l

is the path length of the sample cell.

Given that the absorbance (A) is 0.250, the extinction coefficient (

ε

) is

5.60 \times 10^{3}

M^-1 cm^-1, and the path length (

l

) is 1.00 cm, solve for the concentration (

C

) using the equation:

C = \frac{A}{ε l}

For a first-order reaction, the rate constant (

k

) can be determined using the integrated rate law:

A = A o \times e^{- k t}

, where

A

is the absorbance at time

t

A o

is the initial absorbance, and

k

is the rate constant.

Rearrange the integrated rate law to solve for the rate constant (

k

k = \frac{- \ln (\frac{A}{A})}{t}

. Substitute the values for

A

A o

, and

t

(converted to seconds) into the equation.

Calculate the rate constant (

k

) using the rearranged equation. Ensure that the time is converted from minutes to seconds by multiplying by 60, as the rate constant is required in units of s^-1.

Key Concepts

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

First-Order Reactions

First-order reactions are chemical reactions where the rate is directly proportional to the concentration of one reactant. This means that as the concentration of the reactant decreases, the rate of the reaction also decreases. The rate law for a first-order reaction can be expressed as rate = k[A], where k is the rate constant and [A] is the concentration of the reactant.

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 equal to the product of the extinction coefficient (ε), the concentration (c) of the absorbing species, and the path length (l) of the sample: A = εcl. This law is crucial for quantifying the concentration of colored species in solution using spectroscopy.

Rate Constant (k)

The rate constant (k) is a proportionality factor in the rate law that provides insight into the speed of a reaction. For first-order reactions, the rate constant has units of s^-1, indicating the reaction's dependence on time. The value of k can be determined from experimental data, such as changes in absorbance over time, allowing for the calculation of reaction kinetics.

Recommended video:

Guided course

01:14

Equilibrium Constant K

Related Practice

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?

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?

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

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