To determine the half-life of radium-226, we can use the relationship between the half-life and the time it takes for a sample to decay to a certain fraction of its original amount.
Given that the sample of radium-226 decays to \( \frac{1}{4} \) of its original amount after 3200 years, we can express this decay in terms of half-lives:
\[ \frac{1}{4} = \left( \frac{1}{2} \right)^n \]
where \( n \) is the number of half-lives. Since \( \frac{1}{4} \) can be expressed as \( \left( \frac{1}{2} \right)^2 \), we have:
\[ \frac{1}{4} = \left( \frac{1}{2} \right)^2 \]
This means \( n = 2 \). Since 2 half-lives result in a decay to \( \frac{1}{4} \) of the original amount, we can find the half-life (\( t_{1/2} \)) as follows:
\[ 2 \cdot t_{1/2} = 3200 \text{ years} \]
Dividing both sides by 2 gives:
\[ t_{1/2} = \frac{3200 \text{ years}}{2} = 1600 \text{ years} \]
Thus, the half-life of radium-226 is \( \boxed{1600} \) years.
1 answer