Answer :
Certainly! Let's break down the problem step by step.
### Initial Information:
- Initial Amount of Radium-226: 5.4 kg
- Half-Life of Radium-226: 173 years
- Elapsed Time: 100 years
### Step 1: Determine the Remaining Amount After 100 Years
Radium-226 undergoes exponential decay. The formula to find the remaining amount after a specific time period is:
[tex]\[ N(t) = N_0 \times \left(\frac{1}{2}\right)^{\frac{t}{\text{half-life}}} \][/tex]
Where:
- [tex]\( N(t) \)[/tex] is the remaining amount after time [tex]\( t \)[/tex].
- [tex]\( N_0 \)[/tex] is the initial amount.
- [tex]\( t \)[/tex] is the elapsed time.
- The half-life is the time it takes for half of the material to decay.
Substitute the given values into the formula:
[tex]\[ N(100) = 5.4 \times \left(\frac{1}{2}\right)^{\frac{100}{173}} \][/tex]
After performing the calculation, the remaining amount of Radium-226 after 100 years is approximately 3.62 kg.
### Step 2: Determine When All Radium-226 Will Be Gone
In reality, Radium-226 will never completely disappear mathematically due to exponential decay. However, we can calculate when it becomes negligible. A common approach is to consider it effectively "gone" when it's less than a tiny fraction, such as 0.001% of its initial amount.
For practical purposes, assume it is "gone" when less than:
[tex]\[ 5.4 \times 0.00001 = 5.4 \times 0.00001 = 0.000054 \text{ kg} \][/tex]
Using a method of calculation (not shown here), it is estimated that Radium-226 will be practically "gone" in approximately 2873 years.
### Step 3: Calculate the Annual Percentage Decrease
The formula to find the percentage decrease per year involves using the decay factor:
The percentage decrease per year can be calculated as:
[tex]\[ \text{Percentage Decrease Per Year} = \left(1 - \left(\frac{1}{2}\right)^{\frac{1}{173}}\right) \times 100\% \][/tex]
After calculation, the annual percentage decrease is approximately 0.40%.
In summary:
- Approximately 3.62 kg of Radium-226 will remain after 100 years.
- It will take about 2873 years for the Radium-226 to be considered negligible.
- Radium-226 decreases by approximately 0.40% each year.
### Initial Information:
- Initial Amount of Radium-226: 5.4 kg
- Half-Life of Radium-226: 173 years
- Elapsed Time: 100 years
### Step 1: Determine the Remaining Amount After 100 Years
Radium-226 undergoes exponential decay. The formula to find the remaining amount after a specific time period is:
[tex]\[ N(t) = N_0 \times \left(\frac{1}{2}\right)^{\frac{t}{\text{half-life}}} \][/tex]
Where:
- [tex]\( N(t) \)[/tex] is the remaining amount after time [tex]\( t \)[/tex].
- [tex]\( N_0 \)[/tex] is the initial amount.
- [tex]\( t \)[/tex] is the elapsed time.
- The half-life is the time it takes for half of the material to decay.
Substitute the given values into the formula:
[tex]\[ N(100) = 5.4 \times \left(\frac{1}{2}\right)^{\frac{100}{173}} \][/tex]
After performing the calculation, the remaining amount of Radium-226 after 100 years is approximately 3.62 kg.
### Step 2: Determine When All Radium-226 Will Be Gone
In reality, Radium-226 will never completely disappear mathematically due to exponential decay. However, we can calculate when it becomes negligible. A common approach is to consider it effectively "gone" when it's less than a tiny fraction, such as 0.001% of its initial amount.
For practical purposes, assume it is "gone" when less than:
[tex]\[ 5.4 \times 0.00001 = 5.4 \times 0.00001 = 0.000054 \text{ kg} \][/tex]
Using a method of calculation (not shown here), it is estimated that Radium-226 will be practically "gone" in approximately 2873 years.
### Step 3: Calculate the Annual Percentage Decrease
The formula to find the percentage decrease per year involves using the decay factor:
The percentage decrease per year can be calculated as:
[tex]\[ \text{Percentage Decrease Per Year} = \left(1 - \left(\frac{1}{2}\right)^{\frac{1}{173}}\right) \times 100\% \][/tex]
After calculation, the annual percentage decrease is approximately 0.40%.
In summary:
- Approximately 3.62 kg of Radium-226 will remain after 100 years.
- It will take about 2873 years for the Radium-226 to be considered negligible.
- Radium-226 decreases by approximately 0.40% each year.
