Pregunta
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The half-life of a radioactive isotope is the time it takes for a quantity of the isotope to be reduced to half its initial mass. 5 tarting with 210 grams of a radioactive isotope, how much will be left after 5 half-lives? Use the calculator provided and round your answer to the nearest gram.

Ask by Olson Mejia. in the United States
Mar 11,2025

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After 5 half-lives, 7 grams of the radioactive isotope will remain.

Solución

Let \( m_0 = 210 \) grams be the initial mass. After each half-life, the mass is reduced to half, so after \( n \) half-lives the mass \( m \) is given by \[ m = m_0 \times \left(\frac{1}{2}\right)^n \] For \( n = 5 \) half-lives, we have \[ m = 210 \times \left(\frac{1}{2}\right)^5 = 210 \times \frac{1}{32} = \frac{210}{32} \approx 6.5625 \text{ grams} \] Rounding to the nearest gram, the mass remaining is \[ \boxed{7 \text{ grams}} \]

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Bonus Knowledge

After one half-life, you would have 105 grams left. After two half-lives, it would be 52.5 grams, and after three half-lives, you’d be down to 26.25 grams. Continuing this process, after four half-lives, you’d have about 13.13 grams remaining, and finally, after five half-lives, you would be left with approximately 6.56 grams. Rounding to the nearest gram, you would have 7 grams left after five half-lives. Radioactive decay isn’t just a scientific phenomenon; it paves the way for various real-world applications! For instance, carbon-14 dating, crucial in archaeology, helps determine the age of organic materials. By measuring the remaining carbon-14 in a fossil or ancient artifact, scientists can estimate when it was alive or created. It’s like unlocking history with a touch of science magic!

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