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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![**Radioactive Decay Problem:**
A radioactive isotope has a half-life of 12.0 seconds. How much of a 100 gram sample is left after 36 seconds?
**Solution:**
To determine how much of the sample is left after a certain period, we need to calculate how many half-lives have passed.
1. **Calculate the number of half-lives:**
\[
\text{Number of half-lives} = \frac{\text{total time}}{\text{half-life}}
\]
Given:
- Total time = 36 seconds
- Half-life = 12.0 seconds
\[
\frac{36 \text{ seconds}}{12 \text{ seconds}} = 3 \text{ half-lives}
\]
2. **Calculate the remaining amount of the isotope:**
Each half-life reduces the amount of the substance by half.
- After 1 half-life:
\[
\frac{100 \text{ grams}}{2} = 50 \text{ grams}
\]
- After 2 half-lives:
\[
\frac{50 \text{ grams}}{2} = 25 \text{ grams}
\]
- After 3 half-lives:
\[
\frac{25 \text{ grams}}{2} = 12.5 \text{ grams}
\]
**Conclusion:**
After 36 seconds, 12.5 grams of the 100 gram sample of the radioactive isotope will remain.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F11cb493f-74d8-445b-bfd0-ead085c73a2d%2F5cf7ba20-20f7-4584-83ee-03624f8953ad%2F88tsth_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Radioactive Decay Problem:**
A radioactive isotope has a half-life of 12.0 seconds. How much of a 100 gram sample is left after 36 seconds?
**Solution:**
To determine how much of the sample is left after a certain period, we need to calculate how many half-lives have passed.
1. **Calculate the number of half-lives:**
\[
\text{Number of half-lives} = \frac{\text{total time}}{\text{half-life}}
\]
Given:
- Total time = 36 seconds
- Half-life = 12.0 seconds
\[
\frac{36 \text{ seconds}}{12 \text{ seconds}} = 3 \text{ half-lives}
\]
2. **Calculate the remaining amount of the isotope:**
Each half-life reduces the amount of the substance by half.
- After 1 half-life:
\[
\frac{100 \text{ grams}}{2} = 50 \text{ grams}
\]
- After 2 half-lives:
\[
\frac{50 \text{ grams}}{2} = 25 \text{ grams}
\]
- After 3 half-lives:
\[
\frac{25 \text{ grams}}{2} = 12.5 \text{ grams}
\]
**Conclusion:**
After 36 seconds, 12.5 grams of the 100 gram sample of the radioactive isotope will remain.
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