Answer :
Let's break down the problem into two main parts and solve each step-by-step:
### Part 1: Uranium-238 Decay
Uranium-238 has a half-life of 4500 million years. We want to determine how long it takes for the sample to decay to 15% of the original amount.
1. Half-life Concept:
The half-life formula for radioactive decay is:
[tex]\[ N(t) = N_0 \left( \frac{1}{2} \right)^{t/T} \][/tex]
where:
- [tex]\( N(t) \)[/tex] is the amount remaining after time [tex]\( t \)[/tex],
- [tex]\( N_0 \)[/tex] is the initial amount,
- [tex]\( T \)[/tex] is the half-life period,
- [tex]\( t \)[/tex] is the time elapsed.
2. Given Data:
- Half-life [tex]\( T = 4500 \times 10^6 \)[/tex] years (which we need to convert to hours for consistency).
- Target remaining fraction [tex]\( N(t) / N_0 = 0.15 \)[/tex] or 15%.
3. Conversion:
- 1 year = 365.25 days (taking into account leap years),
- 1 day = 24 hours,
- Therefore, [tex]\( 4500 \times 10^6 \)[/tex] years = [tex]\( 4500 \times 10^6 \times 365.25 \times 24 \)[/tex] hours = 39,447,000,000,000 hours.
4. Calculating Time, [tex]\( t \)[/tex]:
We use the decay formula:
[tex]\[ 0.15 = \left( \frac{1}{2} \right)^{t/T} \][/tex]
Taking natural logarithms on both sides:
[tex]\[ \ln(0.15) = \frac{t}{T} \ln(0.5) \][/tex]
Solving for [tex]\( t \)[/tex]:
[tex]\[ t = \frac{\ln(0.15)}{\ln(0.5)} \times T \][/tex]
[tex]\[ t \approx 107,965,081,793,074.33 \text{ hours} \][/tex]
### Part 2: Substance with 2-Hour Half-life
5. Given Data:
- Half-life [tex]\( T = 2 \)[/tex] hours,
- Total time [tex]\( t = 6 \)[/tex] hours.
6. Calculating Fraction Remaining:
Using the half-life formula:
[tex]\[ \text{Fraction remaining} = \left( \frac{1}{2} \right)^{t/T} \][/tex]
Substituting the given values:
[tex]\[ \text{Fraction remaining} = \left( \frac{1}{2} \right)^{6/2} = \left( \frac{1}{2} \right)^{3} = \frac{1}{8} = 0.125 \][/tex]
### Experimental Demonstration
Objective: Demonstrate the concept of radioactive decay using a substance with a known half-life.
Materials Needed:
- A Geiger-Müller counter for measuring radiation,
- A sample of a radioactive substance with a short half-life (e.g., Barium-137m),
- Stopwatch,
- Graph paper and markers.
Procedure:
1. Setup: Place the Geiger-Müller counter near the radioactive source.
2. Measurement:
- Turn on the counter and start the stopwatch simultaneously.
- Record the count rate (number of decays per unit time) at regular intervals (e.g., every 30 seconds).
3. Data Collection: Continue measuring until you have sufficient data points showing a decrease in the count rate over time.
4. Analysis:
- Plot the count rate against time on graph paper.
- Identify the time interval in which the count rate halves (this is the half-life of the substance).
5. Conclusion: The plotted graph should show an exponential decay curve, demonstrating the concept of half-life.
Diagram:
```
+-------------------+
| Setup |
+-------------------+
| |
| [Counter] |
| | |
| | |
| [Radioactive] |
| [Sample] |
| |
| [Stopwatch]|
| |
+-------------------+
```
In summary:
- For Uranium-238, it takes approximately 107,965,081,793,074.33 hours to decay to 15% of its original amount.
- A substance with a 2-hour half-life will have 12.5% of its original amount remaining after 6 hours.
- The described experiment and setup visually and practically demonstrate the concept of radioactive decay.
### Part 1: Uranium-238 Decay
Uranium-238 has a half-life of 4500 million years. We want to determine how long it takes for the sample to decay to 15% of the original amount.
1. Half-life Concept:
The half-life formula for radioactive decay is:
[tex]\[ N(t) = N_0 \left( \frac{1}{2} \right)^{t/T} \][/tex]
where:
- [tex]\( N(t) \)[/tex] is the amount remaining after time [tex]\( t \)[/tex],
- [tex]\( N_0 \)[/tex] is the initial amount,
- [tex]\( T \)[/tex] is the half-life period,
- [tex]\( t \)[/tex] is the time elapsed.
2. Given Data:
- Half-life [tex]\( T = 4500 \times 10^6 \)[/tex] years (which we need to convert to hours for consistency).
- Target remaining fraction [tex]\( N(t) / N_0 = 0.15 \)[/tex] or 15%.
3. Conversion:
- 1 year = 365.25 days (taking into account leap years),
- 1 day = 24 hours,
- Therefore, [tex]\( 4500 \times 10^6 \)[/tex] years = [tex]\( 4500 \times 10^6 \times 365.25 \times 24 \)[/tex] hours = 39,447,000,000,000 hours.
4. Calculating Time, [tex]\( t \)[/tex]:
We use the decay formula:
[tex]\[ 0.15 = \left( \frac{1}{2} \right)^{t/T} \][/tex]
Taking natural logarithms on both sides:
[tex]\[ \ln(0.15) = \frac{t}{T} \ln(0.5) \][/tex]
Solving for [tex]\( t \)[/tex]:
[tex]\[ t = \frac{\ln(0.15)}{\ln(0.5)} \times T \][/tex]
[tex]\[ t \approx 107,965,081,793,074.33 \text{ hours} \][/tex]
### Part 2: Substance with 2-Hour Half-life
5. Given Data:
- Half-life [tex]\( T = 2 \)[/tex] hours,
- Total time [tex]\( t = 6 \)[/tex] hours.
6. Calculating Fraction Remaining:
Using the half-life formula:
[tex]\[ \text{Fraction remaining} = \left( \frac{1}{2} \right)^{t/T} \][/tex]
Substituting the given values:
[tex]\[ \text{Fraction remaining} = \left( \frac{1}{2} \right)^{6/2} = \left( \frac{1}{2} \right)^{3} = \frac{1}{8} = 0.125 \][/tex]
### Experimental Demonstration
Objective: Demonstrate the concept of radioactive decay using a substance with a known half-life.
Materials Needed:
- A Geiger-Müller counter for measuring radiation,
- A sample of a radioactive substance with a short half-life (e.g., Barium-137m),
- Stopwatch,
- Graph paper and markers.
Procedure:
1. Setup: Place the Geiger-Müller counter near the radioactive source.
2. Measurement:
- Turn on the counter and start the stopwatch simultaneously.
- Record the count rate (number of decays per unit time) at regular intervals (e.g., every 30 seconds).
3. Data Collection: Continue measuring until you have sufficient data points showing a decrease in the count rate over time.
4. Analysis:
- Plot the count rate against time on graph paper.
- Identify the time interval in which the count rate halves (this is the half-life of the substance).
5. Conclusion: The plotted graph should show an exponential decay curve, demonstrating the concept of half-life.
Diagram:
```
+-------------------+
| Setup |
+-------------------+
| |
| [Counter] |
| | |
| | |
| [Radioactive] |
| [Sample] |
| |
| [Stopwatch]|
| |
+-------------------+
```
In summary:
- For Uranium-238, it takes approximately 107,965,081,793,074.33 hours to decay to 15% of its original amount.
- A substance with a 2-hour half-life will have 12.5% of its original amount remaining after 6 hours.
- The described experiment and setup visually and practically demonstrate the concept of radioactive decay.
