Using your radioactive isotope half-life and decay constant (you solved for in question 4), solve the following scenario. A bone fragment was discovered on the grounds of VC, and the fragment was found to contain 12 grams of your isotope. In its original state, this same item would contain 315 grams of your isotope. How old is this fragment? Solve for time. The video gives an example starting at 10:18. Show all algebraic steps to solve this problem for t (time). Give the exact answer and the decimal approximation rounded to 2 decimal places. Be sure to label the unit of time (days, years, minutes, etc...)
Using your radioactive isotope half-life and decay constant (you solved for in question 4), solve the following scenario. A bone fragment was discovered on the grounds of VC, and the fragment was found to contain 12 grams of your isotope. In its original state, this same item would contain 315 grams of your isotope. How old is this fragment? Solve for time. The video gives an example starting at 10:18. Show all algebraic steps to solve this problem for t (time). Give the exact answer and the decimal approximation rounded to 2 decimal places. Be sure to label the unit of time (days, years, minutes, etc...)
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 Isotope Decay Calculation
**Problem Statement:**
Using your radioactive isotope half-life and decay constant (you solved for in question 4), solve the following scenario:
A bone fragment was discovered on the grounds of VC, and the fragment was found to contain 12 grams of your isotope. In its original state, this same item would contain 315 grams of your isotope. How old is this fragment? Solve for time.
The video gives an example starting at 10:18. Show all algebraic steps to solve this problem for t (time). **Give the exact answer and the decimal approximation rounded to 2 decimal places.** Be sure to label the unit of time (days, years, minutes, etc.).
**Detailed Explanation:**
The above problem involves solving for the age of a bone fragment using principles of radioactive decay. Given the initial amount of isotope and the amount remaining, you will use the decay constant or half-life to determine the time that has elapsed. This involves using logarithmic functions and understanding the exponential decay model.
For reference, recall the general decay formula:
\[ N(t) = N_0 e^{-\lambda t} \]
Where:
- \( N(t) \) is the remaining amount of substance at time \( t \).
- \( N_0 \) is the initial amount of substance.
- \( \lambda \) is the decay constant.
- \( t \) is the time elapsed.
Follow the step-by-step procedures provided in the video for solving these types of problems accurately.
**Algebraic Steps Example** *(based on the given values in the problem)*:
1. **Identify the given values:**
- \( N_0 = 315 \) grams
- \( N(t) = 12 \) grams
2. **Use the decay formula:**
\[ N(t) = N_0 e^{-\lambda t} \]
3. **Insert the values:**
\[ 12 = 315 e^{-\lambda t} \]
4. **Isolate the exponential term:**
\[ \frac{12}{315} = e^{-\lambda t} \]
5. **Simplify the fraction:**
\[ \frac{12}{315} = \frac{4}{105} \]
6. **Take the natural logarithm on both sides to solve for \( t \):**
\[ \ln\left](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F55d38b4a-bd6a-41ed-9171-a3a35226d6fd%2Fd58e8741-6170-4ea4-97b5-67dda9ae414d%2Fyc5ngzb_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Radioactive Isotope Decay Calculation
**Problem Statement:**
Using your radioactive isotope half-life and decay constant (you solved for in question 4), solve the following scenario:
A bone fragment was discovered on the grounds of VC, and the fragment was found to contain 12 grams of your isotope. In its original state, this same item would contain 315 grams of your isotope. How old is this fragment? Solve for time.
The video gives an example starting at 10:18. Show all algebraic steps to solve this problem for t (time). **Give the exact answer and the decimal approximation rounded to 2 decimal places.** Be sure to label the unit of time (days, years, minutes, etc.).
**Detailed Explanation:**
The above problem involves solving for the age of a bone fragment using principles of radioactive decay. Given the initial amount of isotope and the amount remaining, you will use the decay constant or half-life to determine the time that has elapsed. This involves using logarithmic functions and understanding the exponential decay model.
For reference, recall the general decay formula:
\[ N(t) = N_0 e^{-\lambda t} \]
Where:
- \( N(t) \) is the remaining amount of substance at time \( t \).
- \( N_0 \) is the initial amount of substance.
- \( \lambda \) is the decay constant.
- \( t \) is the time elapsed.
Follow the step-by-step procedures provided in the video for solving these types of problems accurately.
**Algebraic Steps Example** *(based on the given values in the problem)*:
1. **Identify the given values:**
- \( N_0 = 315 \) grams
- \( N(t) = 12 \) grams
2. **Use the decay formula:**
\[ N(t) = N_0 e^{-\lambda t} \]
3. **Insert the values:**
\[ 12 = 315 e^{-\lambda t} \]
4. **Isolate the exponential term:**
\[ \frac{12}{315} = e^{-\lambda t} \]
5. **Simplify the fraction:**
\[ \frac{12}{315} = \frac{4}{105} \]
6. **Take the natural logarithm on both sides to solve for \( t \):**
\[ \ln\left
![**Radioisotope Activity Assignment**
**Step 3: Radioisotope and Half-Life Assignment**
Each of you will be assigned a radioisotope. Look at the directions in the assignment in Canvas. Write that radioisotope and its half-life out here:
- **Radioisotope:** Sodium-24
- **Half-Life:** 15 hours
**Step 4: Determining the Decay Constant**
Using the formula illustrated in the video (around 4:30-6:30), find the decay constant for your radioisotope. Show all of your work, illustrating the algebraic process at which your decay constant was found. Give the exact value in natural logs and then give the approximate value (to the nearest 6 decimal places).
The formula you will use is:
\[ N = N_0 e^{-\lambda t} \]
Where:
- \( N \) is the quantity that still remains after time \( t \),
- \( N_0 \) is the initial quantity,
- \( \lambda \) is the decay constant,
- \( t \) is time.
Note the important conversion in the calculation:
\[ e^{-\lambda \cdot?} = 1 \]
Please ensure accuracy in your calculations for academic rigor.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F55d38b4a-bd6a-41ed-9171-a3a35226d6fd%2Fd58e8741-6170-4ea4-97b5-67dda9ae414d%2Fgqpptjr_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Radioisotope Activity Assignment**
**Step 3: Radioisotope and Half-Life Assignment**
Each of you will be assigned a radioisotope. Look at the directions in the assignment in Canvas. Write that radioisotope and its half-life out here:
- **Radioisotope:** Sodium-24
- **Half-Life:** 15 hours
**Step 4: Determining the Decay Constant**
Using the formula illustrated in the video (around 4:30-6:30), find the decay constant for your radioisotope. Show all of your work, illustrating the algebraic process at which your decay constant was found. Give the exact value in natural logs and then give the approximate value (to the nearest 6 decimal places).
The formula you will use is:
\[ N = N_0 e^{-\lambda t} \]
Where:
- \( N \) is the quantity that still remains after time \( t \),
- \( N_0 \) is the initial quantity,
- \( \lambda \) is the decay constant,
- \( t \) is time.
Note the important conversion in the calculation:
\[ e^{-\lambda \cdot?} = 1 \]
Please ensure accuracy in your calculations for academic rigor.
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