The half-life of the radioisotope ¹⁵⁸Eu is 0.77 h.How much time is required for a 160.0-g sample of ¹⁵⁸Eu to decay to 1.59 g?

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To find the time required for a 160.0-g sample to decay to 1.59 g, we first determine the number of half-lives that have passed. The formula for the amount of substance remaining after a certain number of half-lives is:$$ \text{Remaining mass} = \text{Initial mass} \times \left(\frac{1}{2}\right)^{\text{number of half-lives}} $$Rearranging to solve for the number of half-lives, we get:$$ \text{number of half-lives} = \frac{\log(\text{Remaining mass} / \text{Initial mass})}{\log(1/2)} $$Plugging in the given values:$$ \text{number of half-lives} = \frac{\log(1.59 / 160.0)}{\log(1/2)} \approx 6.33 $$Since each half-life is 0.77 hours, the total time is:$$ \text{Total time} = \text{number of half-lives} \times \text{half-life duration} \approx 6.33 \times 0.77 \approx 4.87 $$Rounding to the nearest option gives us 5.1 hours.

The Half-Life of the Radioisotope 158Eu Is 0

The Half-Life of the Radioisotope ¹⁵⁸Eu Is 0