Step 1 : Understanding the Question:
The topic of this question is Nuclear Physics, specifically Radioactivity and the Law of Radioactive Decay. Radioactive substances are unstable and decay over time. The "half-life" is a fixed time interval during which exactly half of the remaining radioactive atoms in a sample will decay. The question asks us to calculate what portion (fraction) of the initial sample remains active after a specific duration that spans multiple half-lives.
Step 2 : Key Formulas and approach:
The approach involves calculating the number of half-life cycles that have passed:
1. Number of half-lives ($n$): $n = \frac{\text{Total Time}}{\text{Half-life}}$
2. Remaining Fraction formula: $\frac{N}{N_0} = (\frac{1}{2})^n$
Where $N$ is the amount remaining and $N_0$ is the initial amount.
Step 3 : Detailed Explanation:
First, we identify the given values: Half-life ($T_{1/2}$) = $10$ days, and Total time ($t$) = $30$ days.
Step 1: Calculate the number of half-life periods that occur within 30 days.
$n = \frac{30}{10} = 3$ half-lives.
Step 2: Understand the decay process stage by stage.
After the 1st half-life (10 days): The sample becomes $1/2$ of its original size.
After the 2nd half-life (20 days): Half of the $1/2$ remains, which is $1/4$ ($1/2 \times 1/2$).
After the 3rd half-life (30 days): Half of the $1/4$ remains, which is $1/8$ ($1/4 \times 1/2$).
Step 3: Use the formula to confirm: Fraction = $(\frac{1}{2})^3 = \frac{1}{8}$.
Note that radioactivity is a probabilistic process, so while we cannot predict when a single atom will decay, we can accurately predict the behavior of a large sample like this.
Step 4 : Final Answer:
After 30 days (which equals 3 half-lives), the remaining fraction of the radioactive substance is $1/8$. Thus, the correct option is (D).