Question

The half-life of a radioactive nucleus is 5 years. The fraction of the original sample that would decay in 15 years is:

• A. \(\frac{1}{8}\) of initial value
• B. \(\frac{7}{8}\) of initial value
• C. \(\frac{1}{4}\) of initial value
• D. \(\frac{3}{4}\) of initial value

Answer 1

The Correct Option is A

To solve the problem of determining the fraction of the original radioactive nucleus sample that would decay in 15 years, we need to use the concept of half-life.

The half-life of a radioactive substance is the time required for half of the radioactive nuclei in a sample to decay. According to the problem, the half-life is 5 years.

1. Firstly, calculate the number of half-lives in 15 years:

\(n = \frac{15}{5} = 3\)

1. The radioactive substance will have gone through 3 half-lives in 15 years.
2. After one half-life (5 years), the amount of substance remaining is half of the original:

\(A_1 = \frac{1}{2} \times A_0\)

1. After two half-lives (10 years), the amount left is:

\(A_2 = \frac{1}{2} \times A_1 = \left(\frac{1}{2}\right)^2 \times A_0 = \frac{1}{4} \times A_0\)

1. After three half-lives (15 years), the amount left is:

\(A_3 = \frac{1}{2} \times A_2 = \left(\frac{1}{2}\right)^3 \times A_0 = \frac{1}{8} \times A_0\)

1. This means that the fraction of the original sample remaining is \(\frac{1}{8}\).
2. Thus, the fraction that has decayed is:

\(1 - \frac{1}{8} = \frac{7}{8}\)

Hence, the fraction of the original sample that would decay in 15 years is \(\frac{7}{8}\) of the initial value.

The correct answer, therefore, is \(\frac{7}{8}\) of initial value.

Note: There seems to be a discrepancy between the computed answer here and the one provided initially. The correct decayed fraction should indeed be \(\frac{7}{8}\).