The half-life of Radium-226 is 1590 years. If a sample contains 100 mg, how many mg will remain after 2000 years? mg Give your answer accurate to at least 2 decimal places.

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The half-life of Radium-226 is 1590 years. If a sample contains 100 mg, how many mg will remain after 2000 years?

### Radioactive Decay Calculation Tutorial

**Problem Statement:**

The half-life of Radium-226 is 1590 years. If a sample contains 100 mg, how many mg will remain after 2000 years?

[Interactable Text Box]
```plaintext
___ mg
```
*Give your answer accurate to at least 2 decimal places.*

### Explanation and Solution:

The concept of half-life is used to describe the time it takes for a substance to reduce to half its initial amount due to radioactive decay. Here, we use the decay formula:

\[ N(t) = N_0 \left( \frac{1}{2} \right)^{\frac{t}{T_{1/2}}} \]

Where:
- \( N(t) \) = amount remaining after time \( t \)
- \( N_0 \) = initial amount
- \( t \) = elapsed time
- \( T_{1/2} \) = half-life of the substance

Plugging in the values given:
- \( N_0 = 100 \) mg
- \( t = 2000 \) years
- \( T_{1/2} = 1590 \) years

First, compute the exponent:
\[ \frac{t}{T_{1/2}} = \frac{2000}{1590} \approx 1.26 \]

Now apply to the formula:
\[ N(2000) = 100 \left( \frac{1}{2} \right)^{1.26} \approx 100 \times 0.4156 \approx 41.56 \text{ mg} \]

So, approximately 41.56 mg of Radium-226 will remain after 2000 years.
Transcribed Image Text:### Radioactive Decay Calculation Tutorial **Problem Statement:** The half-life of Radium-226 is 1590 years. If a sample contains 100 mg, how many mg will remain after 2000 years? [Interactable Text Box] ```plaintext ___ mg ``` *Give your answer accurate to at least 2 decimal places.* ### Explanation and Solution: The concept of half-life is used to describe the time it takes for a substance to reduce to half its initial amount due to radioactive decay. Here, we use the decay formula: \[ N(t) = N_0 \left( \frac{1}{2} \right)^{\frac{t}{T_{1/2}}} \] Where: - \( N(t) \) = amount remaining after time \( t \) - \( N_0 \) = initial amount - \( t \) = elapsed time - \( T_{1/2} \) = half-life of the substance Plugging in the values given: - \( N_0 = 100 \) mg - \( t = 2000 \) years - \( T_{1/2} = 1590 \) years First, compute the exponent: \[ \frac{t}{T_{1/2}} = \frac{2000}{1590} \approx 1.26 \] Now apply to the formula: \[ N(2000) = 100 \left( \frac{1}{2} \right)^{1.26} \approx 100 \times 0.4156 \approx 41.56 \text{ mg} \] So, approximately 41.56 mg of Radium-226 will remain after 2000 years.
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