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7 August, 08:42

A 3.5 gram sample of a radioactive element was formed in a 1960 explosion of an atomic bomb at Johnson Island in the Pacific test site. The half-life of the radioactive element is 28 years. How much of the original sample will remain in the year 2030? Show your work.

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  1. 7 August, 09:11
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    The answer is 0.62g.

    Solution:

    From year 1960 to year 2030, it has been

    2030-1960 = 70 years

    The half-life of the radioactive element is 28 years, then the sample will go through

    70 years * (1 half-life/28 years) = 2.5 half-lives

    Starting with a 3.5 gram sample, we will have

    3.5 * (1/2) after one half-life passes

    3.5 * (1/2) * (1/2) = 3.5 * (1/4) after two half-lives pass

    3.5 * (1/4) * (1/2) = 3.5 * (1/8) after three half-lives pass and so on

    Therefore, we can write the remaining amount of the sample after the number n of half-lives have passed as

    mass of sample = initial mass of sample/2^n

    The mass of the remaining sample for n = 2.5half-lives can be now calculated as

    mass of sample = 3.5 grams / 2^2.5 = 0.62 g
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