Question

A sample of radioactive element contains $8 \times 10^{16}$ active nuclei. The half-life of the element is 15 days. The number of nuclei decayed after 60 days is

• A. $7.5 \times 10^{16}$
• B. $2.0 \times 10^{16}$
• C. $0.5 \times 10^{16}$
• D. $4.0 \times 10^{16}$

Hint:

Always read the question carefully to see whether it asks for the number of nuclei remaining or the number of nuclei decayed. $0.5 \times 10^{16}$ is the remaining amount, which is a common trap option!

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
We need to find the number of decayed nuclei in a radioactive sample after a given time period, given its initial number of active nuclei and its half-life.

Step 2: Key Formula or Approach:
The number of nuclei remaining undecayed ($N$) after $n$ half-lives is given by:
$$N = N_0 \left(\frac{1}{2}\right)^n$$
where $n = \frac{t}{T_{1/2}}$.
The number of decayed nuclei is simply the initial amount minus the remaining amount: $N_{\text{decayed}} = N_0 - N$.

Step 3: Detailed Explanation:
Given parameters:
Initial nuclei $N_0 = 8 \times 10^{16}$
Half-life $T_{1/2} = 15\ \text{days}$
Total time elapsed $t = 60\ \text{days}$
Calculate the number of half-lives ($n$):
$$n = \frac{60}{15} = 4$$
Calculate the number of remaining active nuclei ($N$):
$$N = (8 \times 10^{16}) \times \left(\frac{1}{2}\right)^4$$
$$N = (8 \times 10^{16}) \times \frac{1}{16}$$
$$N = 0.5 \times 10^{16}$$
Calculate the number of decayed nuclei:
$$N_{\text{decayed}} = N_0 - N$$
$$N_{\text{decayed}} = 8 \times 10^{16} - 0.5 \times 10^{16}$$
$$N_{\text{decayed}} = 7.5 \times 10^{16}$$

Step 4: Final Answer:
The number of decayed nuclei is $7.5 \times 10^{16}$, matching option (A).