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------------------------------------------------ Radium-226 has a half-life of 173 years.

1. If you start with 5.4 kg of Radium-226, how much will remain after 100 years?

2. When will all the Radium-226 be gone?

3. By what percentage is the Radium-226 decreasing per year?

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.