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
![### Example Half-Life Calculation
In this educational example, we explore the concept of half-life using Titanium-51.
#### Problem Statement:
Titanium-51 decays with a half-life of 6.00 minutes. You have a bottle of Titanium-51 that initially weighs 452 grams. How much of the sample is left after half an hour?
#### Given:
- **Initial mass of Titanium-51:** 452 grams
- **Half-life of Titanium-51:** 6.00 minutes
- **Total elapsed time:** 30 minutes (half an hour)
#### Calculation Steps:
1. **Determine the number of half-lives that occur in 30 minutes:**
\[
\text{Number of half-lives} = \frac{\text{Total time}}{\text{Half-life time}} = \frac{30 \text{ minutes}}{6 \text{ minutes/half-life}} = 5 \text{ half-lives}
\]
2. **Calculate the remaining mass after each half-life:**
Each half-life reduces the remaining amount of the sample by half.
\[
\text{Remaining mass} = \text{Initial mass} \times \left(\frac{1}{2}\right)^{\text{Number of half-lives}}
\]
\[
\text{Remaining mass} = 452 \text{ grams} \times \left(\frac{1}{2}\right)^5
\]
\[
\text{Remaining mass} = 452 \text{ grams} \times \frac{1}{32}
\]
\[
\text{Remaining mass} = 452 \text{ grams} \div 32
\]
\[
\text{Remaining mass} \approx 14.1 \text{ grams}
\]
#### Answer Choices:
- 5.00 g
- **90.4 g**
- **14.1 g** (correct answer)
- 75.3 g
Thus, your final answer should be: **14.1 grams** of Titanium-51 remains after half an hour.
### Explanation:
In this problem, we utilized the concept of radioactive decay and the formula for calculating the remaining mass of a sample after a given number of half-lives. Understanding these calculations is fundamental in the field of nuclear chemistry and has practical applications in medicine, archaeology](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3e1f2a85-1db3-48f4-acfb-e624a8f47344%2Fa2c70ca6-099b-43b7-9150-a81810bc0524%2Fidawcbj_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Example Half-Life Calculation
In this educational example, we explore the concept of half-life using Titanium-51.
#### Problem Statement:
Titanium-51 decays with a half-life of 6.00 minutes. You have a bottle of Titanium-51 that initially weighs 452 grams. How much of the sample is left after half an hour?
#### Given:
- **Initial mass of Titanium-51:** 452 grams
- **Half-life of Titanium-51:** 6.00 minutes
- **Total elapsed time:** 30 minutes (half an hour)
#### Calculation Steps:
1. **Determine the number of half-lives that occur in 30 minutes:**
\[
\text{Number of half-lives} = \frac{\text{Total time}}{\text{Half-life time}} = \frac{30 \text{ minutes}}{6 \text{ minutes/half-life}} = 5 \text{ half-lives}
\]
2. **Calculate the remaining mass after each half-life:**
Each half-life reduces the remaining amount of the sample by half.
\[
\text{Remaining mass} = \text{Initial mass} \times \left(\frac{1}{2}\right)^{\text{Number of half-lives}}
\]
\[
\text{Remaining mass} = 452 \text{ grams} \times \left(\frac{1}{2}\right)^5
\]
\[
\text{Remaining mass} = 452 \text{ grams} \times \frac{1}{32}
\]
\[
\text{Remaining mass} = 452 \text{ grams} \div 32
\]
\[
\text{Remaining mass} \approx 14.1 \text{ grams}
\]
#### Answer Choices:
- 5.00 g
- **90.4 g**
- **14.1 g** (correct answer)
- 75.3 g
Thus, your final answer should be: **14.1 grams** of Titanium-51 remains after half an hour.
### Explanation:
In this problem, we utilized the concept of radioactive decay and the formula for calculating the remaining mass of a sample after a given number of half-lives. Understanding these calculations is fundamental in the field of nuclear chemistry and has practical applications in medicine, archaeology
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