Consider a radioisotope "X" with a half-life of 46.1 years. Determine (to the nearest 0.1 mg) how much of a 1000 mg sample of "X" remains after a period of 322.7 years (show all work).
Consider a radioisotope "X" with a half-life of 46.1 years. Determine (to the nearest 0.1 mg) how much of a 1000 mg sample of "X" remains after a period of 322.7 years (show all work).
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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![**Problem Statement:**
Consider a radioisotope "X" with a half-life of 46.1 years. Determine (to the nearest 0.1 mg) how much of a 1000 mg sample of "X" remains after a period of 322.7 years (show all work).
**Solution Explanation:**
To solve this problem, we will use the concept of half-life, which is the time required for half of a sample of a radioactive substance to decay.
**Step-by-Step Calculation:**
1. **Calculate the Number of Half-lives:**
\[
\text{Number of half-lives} = \frac{\text{Total time elapsed}}{\text{Half-life of the isotope}} = \frac{322.7 \text{ years}}{46.1 \text{ years/half-life}}
\]
2. **Determine Remaining Amount:**
Use the formula for exponential decay:
\[
\text{Remaining amount} = \text{Initial amount} \times \left(\frac{1}{2}\right)^{\text{Number of half-lives}}
\]
Substituting in the values:
- Initial amount = 1000 mg
- Calculated number of half-lives from step 1
3. **Find the Answer:**
Calculate the remaining mass of the sample after the given time period and round to the nearest 0.1 mg.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fbccf7821-1fc8-44d5-87c9-3888afd6c6a5%2F2b5dc07f-d97c-460d-a18a-62a53ebe130a%2F537keba_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Consider a radioisotope "X" with a half-life of 46.1 years. Determine (to the nearest 0.1 mg) how much of a 1000 mg sample of "X" remains after a period of 322.7 years (show all work).
**Solution Explanation:**
To solve this problem, we will use the concept of half-life, which is the time required for half of a sample of a radioactive substance to decay.
**Step-by-Step Calculation:**
1. **Calculate the Number of Half-lives:**
\[
\text{Number of half-lives} = \frac{\text{Total time elapsed}}{\text{Half-life of the isotope}} = \frac{322.7 \text{ years}}{46.1 \text{ years/half-life}}
\]
2. **Determine Remaining Amount:**
Use the formula for exponential decay:
\[
\text{Remaining amount} = \text{Initial amount} \times \left(\frac{1}{2}\right)^{\text{Number of half-lives}}
\]
Substituting in the values:
- Initial amount = 1000 mg
- Calculated number of half-lives from step 1
3. **Find the Answer:**
Calculate the remaining mass of the sample after the given time period and round to the nearest 0.1 mg.
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