- To determine how many half-lives it takes for a radioactive substance to decay to one-eighth of its original amount, we can express the remaining quantity after \( n \) half-lives as:
\[ \frac{1}{2^n} \]
We want to find \( n \) such that:
\[ \frac{1}{2^n} = \frac{1}{8} \]
Since \( \frac{1}{8} = \frac{1}{2^3} \), we can set:
\[ 2^n = 2^3 \]
This implies that:
\[ n = 3 \]
So the answer is d) three.
- For the 100-gram sample of radioactive element X, with a half-life of 5 days, we need to determine how much remains after 10 days. Since 10 days corresponds to 2 half-lives (because \( 10 \div 5 = 2 \)), we can calculate the remaining quantity after each half-life:
- After the first half-life (5 days), the amount remaining is:
\[ \frac{100 \text{ g}}{2} = 50 \text{ g} \]
- After the second half-life (another 5 days), the amount remaining is:
\[ \frac{50 \text{ g}}{2} = 25 \text{ g} \]
So after 10 days, \( 25 \) grams of element X remain. Thus, the answer is d) 25 g.
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