Putting it all together! Radioisotopes are often used in diagnostic imaging for detecting disease. The isotope 64Cu (copper-64), which has a half-life of 12.7 hours, is used to study diseases affecting copper metabolism such as Wilson's disease. What percentage of the original activity in the sample remains after 38.0 hours? Show your work, include units, and pay attention to significant figures. Hint: Remember that this process follows first order kinetics and 0.693 t1/2 that half-life is given by: k =
Putting it all together! Radioisotopes are often used in diagnostic imaging for detecting disease. The isotope 64Cu (copper-64), which has a half-life of 12.7 hours, is used to study diseases affecting copper metabolism such as Wilson's disease. What percentage of the original activity in the sample remains after 38.0 hours? Show your work, include units, and pay attention to significant figures. Hint: Remember that this process follows first order kinetics and 0.693 t1/2 that half-life is given by: k =
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...
Related questions
Question
100%
Please help
![### Transcription and Explanation for Educational Website
---
**Topic: Application of Radioisotopes in Medical Diagnostics**
**11. Putting it all together!** Radioisotopes are often used in diagnostic imaging for detecting disease. The isotope \(^{64}\text{Cu}\) (copper-64), which has a half-life of 12.7 hours, is used to study diseases affecting copper metabolism such as Wilson’s disease. What percentage of the original activity in the sample remains after 38.0 hours? **Show your work, include units, and pay attention to significant figures.**
**Hint:** Remember that this process follows first-order kinetics and that half-life is given by:
\[
k = \frac{0.693}{t_{1/2}}
\]
### Detailed Explanation
**Concepts:**
1. **Radioisotopes:** Radioisotopes are unstable isotopes of an element that decay and emit radiation. They are used in medical diagnostics for imaging and studying metabolic processes.
2. **Half-Life (\(t_{1/2}\))**: The time required for half of the radioactive nuclei in a sample to decay. For \(^{64}\text{Cu}\), this is 12.7 hours.
3. **First-Order Kinetics:** Radioactive decay processes are often first-order, meaning the rate is directly proportional to the number of undecayed nuclei present.
4. **Decay Constant (\(k\))**: This is a constant used to describe the rate of decay:
\[
k = \frac{0.693}{t_{1/2}}
\]
**Problem Solving:**
To find out the remaining percentage of \(^{64}\text{Cu}\) after 38.0 hours:
1. **Calculate \(k\):**
\(t_{1/2} = 12.7 \, \text{hours}\)
\[
k = \frac{0.693}{12.7 \, \text{hours}}
\]
2. **Use the formula for first-order decay:**
The fraction of the remaining activity (N) at time \(t\) relative to the initial activity \(N_0\) is given by:
\[
N = N_0 \times e^{-kt}
\]
Where \(t = 38.0 \, \text{hours}\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1205cd08-0e41-4ad8-bd1c-6b97372ee2c1%2F4be9ae79-d06e-4fe7-bda3-13771cf4bd95%2F5qnoetj_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Transcription and Explanation for Educational Website
---
**Topic: Application of Radioisotopes in Medical Diagnostics**
**11. Putting it all together!** Radioisotopes are often used in diagnostic imaging for detecting disease. The isotope \(^{64}\text{Cu}\) (copper-64), which has a half-life of 12.7 hours, is used to study diseases affecting copper metabolism such as Wilson’s disease. What percentage of the original activity in the sample remains after 38.0 hours? **Show your work, include units, and pay attention to significant figures.**
**Hint:** Remember that this process follows first-order kinetics and that half-life is given by:
\[
k = \frac{0.693}{t_{1/2}}
\]
### Detailed Explanation
**Concepts:**
1. **Radioisotopes:** Radioisotopes are unstable isotopes of an element that decay and emit radiation. They are used in medical diagnostics for imaging and studying metabolic processes.
2. **Half-Life (\(t_{1/2}\))**: The time required for half of the radioactive nuclei in a sample to decay. For \(^{64}\text{Cu}\), this is 12.7 hours.
3. **First-Order Kinetics:** Radioactive decay processes are often first-order, meaning the rate is directly proportional to the number of undecayed nuclei present.
4. **Decay Constant (\(k\))**: This is a constant used to describe the rate of decay:
\[
k = \frac{0.693}{t_{1/2}}
\]
**Problem Solving:**
To find out the remaining percentage of \(^{64}\text{Cu}\) after 38.0 hours:
1. **Calculate \(k\):**
\(t_{1/2} = 12.7 \, \text{hours}\)
\[
k = \frac{0.693}{12.7 \, \text{hours}}
\]
2. **Use the formula for first-order decay:**
The fraction of the remaining activity (N) at time \(t\) relative to the initial activity \(N_0\) is given by:
\[
N = N_0 \times e^{-kt}
\]
Where \(t = 38.0 \, \text{hours}\).
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 3 steps
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, chemistry and related others by exploring similar questions and additional content below.Recommended textbooks for you
Chemistry
Chemistry
ISBN:
9781305957404
Author:
Steven S. Zumdahl, Susan A. Zumdahl, Donald J. DeCoste
Publisher:
Cengage Learning
Chemistry
Chemistry
ISBN:
9781259911156
Author:
Raymond Chang Dr., Jason Overby Professor
Publisher:
McGraw-Hill Education
Principles of Instrumental Analysis
Chemistry
ISBN:
9781305577213
Author:
Douglas A. Skoog, F. James Holler, Stanley R. Crouch
Publisher:
Cengage Learning
Chemistry
Chemistry
ISBN:
9781305957404
Author:
Steven S. Zumdahl, Susan A. Zumdahl, Donald J. DeCoste
Publisher:
Cengage Learning
Chemistry
Chemistry
ISBN:
9781259911156
Author:
Raymond Chang Dr., Jason Overby Professor
Publisher:
McGraw-Hill Education
Principles of Instrumental Analysis
Chemistry
ISBN:
9781305577213
Author:
Douglas A. Skoog, F. James Holler, Stanley R. Crouch
Publisher:
Cengage Learning
Organic Chemistry
Chemistry
ISBN:
9780078021558
Author:
Janice Gorzynski Smith Dr.
Publisher:
McGraw-Hill Education
Chemistry: Principles and Reactions
Chemistry
ISBN:
9781305079373
Author:
William L. Masterton, Cecile N. Hurley
Publisher:
Cengage Learning
Elementary Principles of Chemical Processes, Bind…
Chemistry
ISBN:
9781118431221
Author:
Richard M. Felder, Ronald W. Rousseau, Lisa G. Bullard
Publisher:
WILEY