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: and In A-kt Ao t1/2 = 0.693 k 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)" where n is the number of half-lives. The equation relating the number of half-lives to time t is V where t1/2 is the length of one half-life. n= Part C Americium-241 is used in some smoke detectors. It is an alpha emitter with a half-life of 432 years. How long will it take in years for 45.0 % of an Am-241 sample to decay? Express your answer with the appropriate units. View Available Hint(s) Submit Part D O t= Value HA O D A fossil was analyzed and determined to have a carbon-14 level that is 20 % that of living organisms. The half-life of C-14 is 5730 years. How old is the fossil? Express your answer with the appropriate units. ► View Available Hint(s) HÅ Units Value ? Units Review | Constants | Periodic Table ?
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: and In A-kt Ao t1/2 = 0.693 k 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)" where n is the number of half-lives. The equation relating the number of half-lives to time t is V where t1/2 is the length of one half-life. n= Part C Americium-241 is used in some smoke detectors. It is an alpha emitter with a half-life of 432 years. How long will it take in years for 45.0 % of an Am-241 sample to decay? Express your answer with the appropriate units. View Available Hint(s) Submit Part D O t= Value HA O D A fossil was analyzed and determined to have a carbon-14 level that is 20 % that of living organisms. The half-life of C-14 is 5730 years. How old is the fossil? Express your answer with the appropriate units. ► View Available Hint(s) HÅ Units Value ? Units Review | Constants | Periodic Table ?
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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![### Understanding Radioactive Decay
If a substance is radioactive, its nucleus is unstable and will decay through various processes, such as alpha decay and beta decay. The decay of radioactive elements follows first-order kinetics, which can be described using integrated rate equations and half-life equations typical for first-order chemical reactions:
\[ \ln \frac{A_t}{A_0} = -kt \]
\[ t_{1/2} = \frac{0.693}{k} \]
Where:
- \( A_0 \) 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 can derive the following formula:
\[ \text{fraction remaining} = \frac{A_t}{A_0} = (0.5)^n \]
Where:
- \( n \) is the number of half-lives, calculated as \( n = \frac{t}{t_{1/2}} \).
- \( t_{1/2} \) is the half-life period.
### Example Calculations
#### Part C
Americium-241 is an alpha emitter with a half-life of 432 years. The question poses: How long will it take for 45.0% of an Am-241 sample to decay?
To solve this, express the answer with appropriate units using the formula provided above. You would first determine the fraction remaining and use the equations to solve for \( t \).
#### Part D
A fossil was analyzed and found to have a carbon-14 level that is 20% of that in living organisms. Given that the half-life of C-14 is 5730 years, calculate the age of the fossil.
Again, express the answer with appropriate units by applying the decay equations.
This exercise demonstrates the real-world application of first-order kinetics to determine the age of substances undergoing radioactive decay.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F096b6bb3-2f11-49d8-9074-81efda173389%2F26acb137-7807-4955-919a-1cbd58f0f645%2Fjxofj88_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Understanding Radioactive Decay
If a substance is radioactive, its nucleus is unstable and will decay through various processes, such as alpha decay and beta decay. The decay of radioactive elements follows first-order kinetics, which can be described using integrated rate equations and half-life equations typical for first-order chemical reactions:
\[ \ln \frac{A_t}{A_0} = -kt \]
\[ t_{1/2} = \frac{0.693}{k} \]
Where:
- \( A_0 \) 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 can derive the following formula:
\[ \text{fraction remaining} = \frac{A_t}{A_0} = (0.5)^n \]
Where:
- \( n \) is the number of half-lives, calculated as \( n = \frac{t}{t_{1/2}} \).
- \( t_{1/2} \) is the half-life period.
### Example Calculations
#### Part C
Americium-241 is an alpha emitter with a half-life of 432 years. The question poses: How long will it take for 45.0% of an Am-241 sample to decay?
To solve this, express the answer with appropriate units using the formula provided above. You would first determine the fraction remaining and use the equations to solve for \( t \).
#### Part D
A fossil was analyzed and found to have a carbon-14 level that is 20% of that in living organisms. Given that the half-life of C-14 is 5730 years, calculate the age of the fossil.
Again, express the answer with appropriate units by applying the decay equations.
This exercise demonstrates the real-world application of first-order kinetics to determine the age of substances undergoing radioactive decay.
Expert Solution
Step 1
For part C:
The half-life for Am-241 = 432 years
We have to find the time required for the 45.0% decomposition of Am-241.
For part D:
The half-life for C-14 = 5730 years
We have to find the age of the fossil if the fossil have a C-14 level of 20% of living organisms.
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