Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 12: Problem 49

A certain first-order reaction is \(45.0 \%\) complete in \(65 \mathrm{~s}\). What are the values of the rate constant and the half-life for this process?

Short Answer

Expert verified
The rate constant (k) for this first-order reaction is approximately \(0.0101 \thinspace s^{-1}\), and the half-life (t₁/₂) is approximately 68.6 seconds.

Step by step solution

01

Determine the fraction remaining after 45.0% completion

To determine the fraction remaining after the reaction is 45.0% complete, we need to subtract the completion percentage from 100%. This will give us the remaining percentage of the reactants present after the reaction. Fraction remaining = 1 - 0.450 = 0.550
02

Apply the integrated rate law for first-order reactions

The integrated rate law for a first-order reaction is: \[\ln{\frac{[A]_0}{[A]}} = kt\] Now, we know that the fraction remaining is 0.550, so: \[\ln{\frac{[A]_0}{0.550[A]_0}} = kt\] Since we need to find the rate constant, and the initial concentration will cancel out anyway, we can rewrite the equation as: \[\ln{\frac{1}{0.550}} = k \times 65\]
03

Calculate the rate constant (k)

Now we can solve for k: \[k = \frac{\ln{\frac{1}{0.550}}}{65} \approx 0.0101 \thinspace s^{-1}\] The rate constant k for this first-order reaction is approximately 0.0101 s⁻¹.
04

Calculate the half-life (t₁/₂)

Now that we have the rate constant, we can calculate the half-life using the equation: \[t_{1/2} = \frac{0.693}{k}\] Plug in the value of k: \[t_{1/2} = \frac{0.693}{0.0101} \approx 68.6 \thinspace s\] The half-life for this first-order reaction is approximately 68.6 seconds. In conclusion, the rate constant for this process is approximately 0.0101 s⁻¹, and the half-life is approximately 68.6 seconds.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Sulfuryl chloride undergoes first-order decomposition at \(320 .{ }^{\circ} \mathrm{C}\) with a half-life of \(8.75 \mathrm{~h}\). $$ \mathrm{SO}_{2} \mathrm{Cl}_{2}(g) \longrightarrow \mathrm{SO}_{2}(g)+\mathrm{Cl}_{2}(g) $$ What is the value of the rate constant, \(k\), in \(\mathrm{s}^{-1}\) ? If the initial pressure of \(\mathrm{SO}_{2} \mathrm{Cl}_{2}\) is 791 torr and the decomposition occurs in a \(1.25-\mathrm{L}\) container, how many molecules of \(\mathrm{SO}_{2} \mathrm{Cl}_{2}\) remain after \(12.5 \mathrm{~h}\) ?

One mechanism for the destruction of ozone in the upper atmosphere is $$ \begin{aligned} &\mathrm{O}_{3}(g)+\mathrm{NO}(g) \longrightarrow \mathrm{NO}_{2}(g)+\mathrm{O}_{2}(g) \\ &\mathrm{NO}_{2}(g)+\mathrm{O}(g) \longrightarrow \mathrm{NO}(g)+\mathrm{O}_{2}(g) \end{aligned} $$ Overall reaction \(\mathrm{O}_{3}(g)+\mathrm{O}(g) \longrightarrow 2 \mathrm{O}_{2}(g)\) a. Which species is a catalyst? b. Which species is an intermediate? c. \(E_{a}\) for the uncatalyzed reaction $$ \mathrm{O}_{3}(g)+\mathrm{O}(g) \longrightarrow 2 \mathrm{O}_{2}(g) $$ is \(14.0 \mathrm{~kJ} . E_{a}\) for the same reaction when catalyzed is \(11.9 \mathrm{~kJ}\). What is the ratio of the rate constant for the catalyzed reaction to that for the uncatalyzed reaction at \(25^{\circ} \mathrm{C}\) ? Assume that the frequency factor \(A\) is the same for each reaction.

Would the slope of \(a \ln (k)\) versus \(1 / T\) plot (with temperature in kelvin) for a catalyzed reaction be more or less negative than the slope of the \(\ln (k)\) versus \(1 / T\) plot for the uncatalyzed reaction? Explain. Assume both rate laws are first-order overall.

For the reaction $$ \mathrm{O}_{2}(g)+2 \mathrm{NO}(g) \longrightarrow 2 \mathrm{NO}_{2}(g) $$ the observed rate law is $$ \text { Rate }=k[\mathrm{NO}]^{2}\left[\mathrm{O}_{2}\right] $$ Which of the changes listed below would affect the value of the rate constant \(k ?\) a. increasing the partial pressure of oxygen gas b. changing the temperature c. using an appropriate catalyst

A certain reaction has the following general form: $$ \mathrm{aA} \longrightarrow \mathrm{bB} $$ At a particular temperature and \([\mathrm{A}]_{0}=2.80 \times 10^{-3} M\), concentration versus time data were collected for this reaction, and a plot of \(1 /[\mathrm{A}]\) versus time resulted in a straight line with a slope value of \(+3.60 \times 10^{-2} \mathrm{~L} / \mathrm{mol} \cdot \mathrm{s}\) a. Determine the rate law, the integrated rate law, and the value of the rate constant for this reaction. b. Calculate the half-life for this reaction. c. How much time is required for the concentration of \(\mathrm{A}\) to decrease to \(7.00 \times 10^{-4} M ?\)
See all solutions

Recommended explanations on Chemistry Textbooks

Chemical Analysis

Read Explanation

Making Measurements

Read Explanation

Chemistry Branches

Read Explanation

Inorganic Chemistry

Read Explanation

Chemical Reactions

Read Explanation

Ionic and Molecular Compounds

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.

Sign-up for free