If a substance is radioactive, this means that the nucleus is unstable and will therefore decay by any number of processes (alpha decay, beta decay, etc.). The decay of radioactive elements follows first-order kinetics. Therefore, the rate of decay can be described by the same integrated rate equations and half-life equations that are used to describe the rate of first-order chemical reactions: =-kt and In A₁ Ao 0.693 k t1/2 = where Ao is the initial amount or activity, A, is the amount or activity at time t, and k is the rate constant. By manipulation of these equations (substituting 0.693/t1/2 for k in the integrated rate equation), we can arrive at the following formula: fraction remaining = = (0.5) A₁ Ao where n is the number of half-lives. The equation relating the number of half-lives to time t is t 11/2 where t1/2 is the length of one half-life. n= ▾ Part A You are using a Geiger counter to measure the activity of a radioactive substance over the course of several minutes. If the reading of 400. counts has diminished to 100. counts after 74.5 minutes, what is the half-life of this substance? Express your answer with the appropriate units. ▸ View Available Hint(s) t1/2 = Value Submit Part B μÀ Ao = Submit HA 3 An unknown radioactive substance has a half-life of 3.20 hours. If 48.9 g of the substance is currently present, what mass Ao was present 8.00 hours ago? Express your answer with the appropriate units. ▸ View Available Hint(s) Value → Units P) ? Units www. ?
If a substance is radioactive, this means that the nucleus is unstable and will therefore decay by any number of processes (alpha decay, beta decay, etc.). The decay of radioactive elements follows first-order kinetics. Therefore, the rate of decay can be described by the same integrated rate equations and half-life equations that are used to describe the rate of first-order chemical reactions: =-kt and In A₁ Ao 0.693 k t1/2 = where Ao is the initial amount or activity, A, is the amount or activity at time t, and k is the rate constant. By manipulation of these equations (substituting 0.693/t1/2 for k in the integrated rate equation), we can arrive at the following formula: fraction remaining = = (0.5) A₁ Ao where n is the number of half-lives. The equation relating the number of half-lives to time t is t 11/2 where t1/2 is the length of one half-life. n= ▾ Part A You are using a Geiger counter to measure the activity of a radioactive substance over the course of several minutes. If the reading of 400. counts has diminished to 100. counts after 74.5 minutes, what is the half-life of this substance? Express your answer with the appropriate units. ▸ View Available Hint(s) t1/2 = Value Submit Part B μÀ Ao = Submit HA 3 An unknown radioactive substance has a half-life of 3.20 hours. If 48.9 g of the substance is currently present, what mass Ao was present 8.00 hours ago? Express your answer with the appropriate units. ▸ View Available Hint(s) Value → Units P) ? Units www. ?
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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![**Radioactive Decay and Half-Life**
**Understanding Radioactive Decay**
When a substance is radioactive, it indicates that its nucleus is unstable and will decay by any of several processes, such as alpha decay or beta decay. The decay of radioactive elements follows first-order kinetics. This implies that the rate of decay can be expressed using the integrated rate equations commonly used to describe first-order chemical reactions:
\[
\ln \frac{A_t}{A_0} = -kt
\]
Additionally, the equation for the half-life is:
\[
t_{1/2} = \frac{0.693}{k}
\]
Here:
- \(A_0\) is the initial amount or activity,
- \(A_t\) is the amount or activity at time \(t\),
- \(k\) is the rate constant.
By manipulating these equations (substituting \(0.693/t_{1/2}\) for \(k\) in the integrated rate equation), we derive:
\[
\text{fraction remaining} = \frac{A_t}{A_0} = (0.5)^n
\]
where \(n\) is the number of half-lives. The equation relating the number of half-lives to time \(t\) is:
\[
n = \frac{t}{t_{1/2}}
\]
where \(t_{1/2}\) is the duration of one half-life.
---
**Exercises**
**Part A**
You are using a Geiger counter to measure the activity of a radioactive substance over several minutes. If the reading of 400 counts has diminished to 100 counts after 74.5 minutes, what is the half-life of this substance?
Express your answer with the appropriate units.
\[ t_{1/2} = \text{Value} \, \text{Units} \]
**Part B**
An unknown radioactive substance has a half-life of 3.20 hours. If 48.9 g of the substance is currently present, what mass \(A_0\) was present 8.00 hours ago?
Express your answer with the appropriate units.
\[ A_0 = \text{Value} \, \text{Units} \]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1205cd08-0e41-4ad8-bd1c-6b97372ee2c1%2F89b3907c-eb40-4f6e-8957-0f61669545b1%2Fufjcofj_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Radioactive Decay and Half-Life**
**Understanding Radioactive Decay**
When a substance is radioactive, it indicates that its nucleus is unstable and will decay by any of several processes, such as alpha decay or beta decay. The decay of radioactive elements follows first-order kinetics. This implies that the rate of decay can be expressed using the integrated rate equations commonly used to describe first-order chemical reactions:
\[
\ln \frac{A_t}{A_0} = -kt
\]
Additionally, the equation for the half-life is:
\[
t_{1/2} = \frac{0.693}{k}
\]
Here:
- \(A_0\) is the initial amount or activity,
- \(A_t\) is the amount or activity at time \(t\),
- \(k\) is the rate constant.
By manipulating these equations (substituting \(0.693/t_{1/2}\) for \(k\) in the integrated rate equation), we derive:
\[
\text{fraction remaining} = \frac{A_t}{A_0} = (0.5)^n
\]
where \(n\) is the number of half-lives. The equation relating the number of half-lives to time \(t\) is:
\[
n = \frac{t}{t_{1/2}}
\]
where \(t_{1/2}\) is the duration of one half-life.
---
**Exercises**
**Part A**
You are using a Geiger counter to measure the activity of a radioactive substance over several minutes. If the reading of 400 counts has diminished to 100 counts after 74.5 minutes, what is the half-life of this substance?
Express your answer with the appropriate units.
\[ t_{1/2} = \text{Value} \, \text{Units} \]
**Part B**
An unknown radioactive substance has a half-life of 3.20 hours. If 48.9 g of the substance is currently present, what mass \(A_0\) was present 8.00 hours ago?
Express your answer with the appropriate units.
\[ A_0 = \text{Value} \, \text{Units} \]
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