Chemistry
10th Edition
ISBN:9781305957404
Author:Steven S. Zumdahl, Susan A. Zumdahl, Donald J. DeCoste
Publisher:Steven S. Zumdahl, Susan A. Zumdahl, Donald J. DeCoste
Chapter1: Chemical Foundations
Section: Chapter Questions
Problem 1RQ: Define and explain the differences between the following terms. a. law and theory b. theory and...
Related questions
Question
![**Problem Statement:**
The Cs-131 nuclide has a half-life of 30 years. After 120 years, 3.8 grams remain. What is the original mass of the Cs-131 sample in grams?
**Solution:**
To solve this problem, we need to use the concept of half-life in radioactive decay. The half-life is the time required for half of the radioactive substance to decay.
The process can be broken down into the following steps:
1. **Calculate the Number of Half-Lives:**
Half-life (T₁/₂) of Cs-131 = 30 years.
Total time elapsed = 120 years.
Number of half-lives = Total time elapsed / Half-life
\[
\text{Number of half-lives} = \frac{120 \text{ years}}{30 \text{ years/half-life}} = 4 \text{ half-lives}
\]
2. **Determine the Original Mass:**
Let \( M_0 \) be the original mass of the Cs-131 sample.
The remaining mass after \( n \) half-lives can be calculated using the formula:
\[
M = M_0 \left(\frac{1}{2}\right)^n
\]
Here, \( M \) is the remaining mass (3.8 grams), and \( n \) is the number of half-lives (4).
Rearrange the formula to solve for \( M_0 \):
\[
M_0 = M \left(\frac{1}{2}\right)^{-n} = 3.8 \text{ grams} \left(\frac{1}{2}\right)^{-4}
\]
Simplify the expression:
\[
M_0 = 3.8 \text{ grams} \cdot 2^4 = 3.8 \text{ grams} \cdot 16 = 60.8 \text{ grams}
\]
**Answer:**
The original mass of the Cs-131 sample is 60.8 grams.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fbc3b72c7-eb99-406f-83fe-82fc41b2a8b5%2Fcc560a43-da06-462a-a12b-f98af4626131%2Ffr4bh2i_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
The Cs-131 nuclide has a half-life of 30 years. After 120 years, 3.8 grams remain. What is the original mass of the Cs-131 sample in grams?
**Solution:**
To solve this problem, we need to use the concept of half-life in radioactive decay. The half-life is the time required for half of the radioactive substance to decay.
The process can be broken down into the following steps:
1. **Calculate the Number of Half-Lives:**
Half-life (T₁/₂) of Cs-131 = 30 years.
Total time elapsed = 120 years.
Number of half-lives = Total time elapsed / Half-life
\[
\text{Number of half-lives} = \frac{120 \text{ years}}{30 \text{ years/half-life}} = 4 \text{ half-lives}
\]
2. **Determine the Original Mass:**
Let \( M_0 \) be the original mass of the Cs-131 sample.
The remaining mass after \( n \) half-lives can be calculated using the formula:
\[
M = M_0 \left(\frac{1}{2}\right)^n
\]
Here, \( M \) is the remaining mass (3.8 grams), and \( n \) is the number of half-lives (4).
Rearrange the formula to solve for \( M_0 \):
\[
M_0 = M \left(\frac{1}{2}\right)^{-n} = 3.8 \text{ grams} \left(\frac{1}{2}\right)^{-4}
\]
Simplify the expression:
\[
M_0 = 3.8 \text{ grams} \cdot 2^4 = 3.8 \text{ grams} \cdot 16 = 60.8 \text{ grams}
\]
**Answer:**
The original mass of the Cs-131 sample is 60.8 grams.
Expert Solution
![](/static/compass_v2/shared-icons/check-mark.png)
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 2 steps with 2 images
![Blurred answer](/static/compass_v2/solution-images/blurred-answer.jpg)
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, chemistry and related others by exploring similar questions and additional content below.Recommended textbooks for you
![Chemistry](https://www.bartleby.com/isbn_cover_images/9781305957404/9781305957404_smallCoverImage.gif)
Chemistry
Chemistry
ISBN:
9781305957404
Author:
Steven S. Zumdahl, Susan A. Zumdahl, Donald J. DeCoste
Publisher:
Cengage Learning
![Chemistry](https://www.bartleby.com/isbn_cover_images/9781259911156/9781259911156_smallCoverImage.gif)
Chemistry
Chemistry
ISBN:
9781259911156
Author:
Raymond Chang Dr., Jason Overby Professor
Publisher:
McGraw-Hill Education
![Principles of Instrumental Analysis](https://www.bartleby.com/isbn_cover_images/9781305577213/9781305577213_smallCoverImage.gif)
Principles of Instrumental Analysis
Chemistry
ISBN:
9781305577213
Author:
Douglas A. Skoog, F. James Holler, Stanley R. Crouch
Publisher:
Cengage Learning
![Chemistry](https://www.bartleby.com/isbn_cover_images/9781305957404/9781305957404_smallCoverImage.gif)
Chemistry
Chemistry
ISBN:
9781305957404
Author:
Steven S. Zumdahl, Susan A. Zumdahl, Donald J. DeCoste
Publisher:
Cengage Learning
![Chemistry](https://www.bartleby.com/isbn_cover_images/9781259911156/9781259911156_smallCoverImage.gif)
Chemistry
Chemistry
ISBN:
9781259911156
Author:
Raymond Chang Dr., Jason Overby Professor
Publisher:
McGraw-Hill Education
![Principles of Instrumental Analysis](https://www.bartleby.com/isbn_cover_images/9781305577213/9781305577213_smallCoverImage.gif)
Principles of Instrumental Analysis
Chemistry
ISBN:
9781305577213
Author:
Douglas A. Skoog, F. James Holler, Stanley R. Crouch
Publisher:
Cengage Learning
![Organic Chemistry](https://www.bartleby.com/isbn_cover_images/9780078021558/9780078021558_smallCoverImage.gif)
Organic Chemistry
Chemistry
ISBN:
9780078021558
Author:
Janice Gorzynski Smith Dr.
Publisher:
McGraw-Hill Education
![Chemistry: Principles and Reactions](https://www.bartleby.com/isbn_cover_images/9781305079373/9781305079373_smallCoverImage.gif)
Chemistry: Principles and Reactions
Chemistry
ISBN:
9781305079373
Author:
William L. Masterton, Cecile N. Hurley
Publisher:
Cengage Learning
![Elementary Principles of Chemical Processes, Bind…](https://www.bartleby.com/isbn_cover_images/9781118431221/9781118431221_smallCoverImage.gif)
Elementary Principles of Chemical Processes, Bind…
Chemistry
ISBN:
9781118431221
Author:
Richard M. Felder, Ronald W. Rousseau, Lisa G. Bullard
Publisher:
WILEY