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.

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