For all of the remaining questions, show or describe your work. 3. The mass of a sample of a mystery element is measured periodically as it decays. If 0.78 % of the sample remains after 100 days, what is the half-life of the element?
For all of the remaining questions, show or describe your work. 3. The mass of a sample of a mystery element is measured periodically as it decays. If 0.78 % of the sample remains after 100 days, what is the half-life of the element?
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
![**Problem 3:**
For all of the remaining questions, show or describe your work.
3. The mass of a sample of a mystery element is measured periodically as it decays. If 0.78% of the sample remains after 100 days, what is the half-life of the element?
---
**Explanation for Students:**
This problem asks you to determine the half-life of an unknown element based on its decay over time. The concept of half-life is critical in understanding radioactive decay, as it represents the time required for half of the substance to decay.
**Calculation Steps:**
1. **Initial Setup:**
- Let the initial mass be \( M_0 \).
- After 100 days, 0.78% of \( M_0 \) remains.
2. **Decay Formula:**
- Use the exponential decay formula:
\[ N(t) = N_0 \left( \frac{1}{2} \right)^{t / T_{1/2}} \]
where:
- \( N(t) \) is the remaining amount after time \( t \),
- \( N_0 \) is the initial amount,
- \( T_{1/2} \) is the half-life of the substance.
3. **Substitute Known Values:**
- \( 0.0078 \times M_0 = M_0 \left( \frac{1}{2} \right)^{100 / T_{1/2}} \)
4. **Solve for Half-Life \( T_{1/2} \):**
- Simplify the equation to solve for \( T_{1/2} \).
This exercise is an application of mathematical modeling using exponential functions to describe physical phenomena. It requires algebraic manipulation and understanding of logarithms to solve for the unknown half-life.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F39adedda-6069-4c18-8ba4-c25003d3e364%2Fbcfe5fbb-9409-406e-a378-ce72077502ff%2F7r1na8s_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem 3:**
For all of the remaining questions, show or describe your work.
3. The mass of a sample of a mystery element is measured periodically as it decays. If 0.78% of the sample remains after 100 days, what is the half-life of the element?
---
**Explanation for Students:**
This problem asks you to determine the half-life of an unknown element based on its decay over time. The concept of half-life is critical in understanding radioactive decay, as it represents the time required for half of the substance to decay.
**Calculation Steps:**
1. **Initial Setup:**
- Let the initial mass be \( M_0 \).
- After 100 days, 0.78% of \( M_0 \) remains.
2. **Decay Formula:**
- Use the exponential decay formula:
\[ N(t) = N_0 \left( \frac{1}{2} \right)^{t / T_{1/2}} \]
where:
- \( N(t) \) is the remaining amount after time \( t \),
- \( N_0 \) is the initial amount,
- \( T_{1/2} \) is the half-life of the substance.
3. **Substitute Known Values:**
- \( 0.0078 \times M_0 = M_0 \left( \frac{1}{2} \right)^{100 / T_{1/2}} \)
4. **Solve for Half-Life \( T_{1/2} \):**
- Simplify the equation to solve for \( T_{1/2} \).
This exercise is an application of mathematical modeling using exponential functions to describe physical phenomena. It requires algebraic manipulation and understanding of logarithms to solve for the unknown half-life.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 2 steps with 2 images
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