Gold (Au) has a synthetic isotope that is relatively unstable. After 25.5 minutes, a 128-gram sample has decayed to 2 g. What is the half-life of this isotope? O 1.5 minutes O 2.25 minutes O 8.5 minutes O 4.25 minutes
Gold (Au) has a synthetic isotope that is relatively unstable. After 25.5 minutes, a 128-gram sample has decayed to 2 g. What is the half-life of this isotope? O 1.5 minutes O 2.25 minutes O 8.5 minutes O 4.25 minutes
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter5: Inverse, Exponential, And Logarithmic Functions
Section: Chapter Questions
Problem 44RE
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![**Understanding Half-Life Calculation using Synthetic Isotopes**
Gold (Au) has a synthetic isotope that is relatively unstable. After 25.5 minutes, a 128-gram sample has decayed to 2 grams.
**Question:** What is the half-life of this isotope?
**Multiple Choice Options:**
- 1.5 minutes
- 2.25 minutes
- 8.5 minutes
- 4.25 minutes
**Explanation:**
To solve for the half-life of an isotope, we use the concept of exponential decay. The half-life is the time it takes for a substance to reduce to half its initial amount. For this isotope, we need to determine the number of half-lives it has undergone to decay from 128 grams to 2 grams in 25.5 minutes.
1. Start with the initial mass of the isotope: 128 grams.
2. After one half-life, the mass would be: \( \frac{128}{2} = 64 \) grams.
3. Continuing this process, we get:
- Second half-life: \( \frac{64}{2} = 32 \) grams
- Third half-life: \( \frac{32}{2} = 16 \) grams
- Fourth half-life: \( \frac{16}{2} = 8 \) grams
- Fifth half-life: \( \frac{8}{2} = 4 \) grams
- Sixth half-life: \( \frac{4}{2} = 2 \) grams
Since it took 25.5 minutes to decay through 6 half-lives, we divide the total time by the number of half-lives to find the duration of one half-life:
\[ \text{Half-life} = \frac{25.5 \text{ minutes}}{6} = 4.25 \text{ minutes} \]
Thus, based on the calculation, the half-life of this synthetic isotope is 4.25 minutes. The correct answer is the last option:
- 4.25 minutes](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6f2ab5f1-0d3c-425f-bc10-ec4282dce5ef%2Ff89b5b49-4232-42da-ba4b-9f42fe8b3810%2Fu6c0zdx_processed.png&w=3840&q=75)
Transcribed Image Text:**Understanding Half-Life Calculation using Synthetic Isotopes**
Gold (Au) has a synthetic isotope that is relatively unstable. After 25.5 minutes, a 128-gram sample has decayed to 2 grams.
**Question:** What is the half-life of this isotope?
**Multiple Choice Options:**
- 1.5 minutes
- 2.25 minutes
- 8.5 minutes
- 4.25 minutes
**Explanation:**
To solve for the half-life of an isotope, we use the concept of exponential decay. The half-life is the time it takes for a substance to reduce to half its initial amount. For this isotope, we need to determine the number of half-lives it has undergone to decay from 128 grams to 2 grams in 25.5 minutes.
1. Start with the initial mass of the isotope: 128 grams.
2. After one half-life, the mass would be: \( \frac{128}{2} = 64 \) grams.
3. Continuing this process, we get:
- Second half-life: \( \frac{64}{2} = 32 \) grams
- Third half-life: \( \frac{32}{2} = 16 \) grams
- Fourth half-life: \( \frac{16}{2} = 8 \) grams
- Fifth half-life: \( \frac{8}{2} = 4 \) grams
- Sixth half-life: \( \frac{4}{2} = 2 \) grams
Since it took 25.5 minutes to decay through 6 half-lives, we divide the total time by the number of half-lives to find the duration of one half-life:
\[ \text{Half-life} = \frac{25.5 \text{ minutes}}{6} = 4.25 \text{ minutes} \]
Thus, based on the calculation, the half-life of this synthetic isotope is 4.25 minutes. The correct answer is the last option:
- 4.25 minutes
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