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...
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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.
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