Question

A radioactive substance has a half-life of 5 days. If its initial mass is 80 g, the mass remaining after 15 days is:

• A. 40 g
• B. 20 g
• C. 10 g
• D. 5 g

Hint:

For small integers of half-lives (like $1, 2, 3$, or $4$), the step-by-step halving method is usually faster and less prone to calculation errors than using the formula.
Simply divide the mass by $2$ successively for each half-life: $80 \rightarrow 40 \rightarrow 20 \rightarrow 10$.

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
This question is from "Modern Physics," specifically focusing on Radioactive Decay (or "Chemical Kinetics" in physical chemistry).
We are asked to find the remaining mass of a radioactive sample after a given amount of time, using the initial mass and the half-life.

Step 2: Key Formula or Approach:
The mass of a radioactive substance remaining after $n$ half-lives is calculated using the formula:
\[ N = N_0 \left(\frac{1}{2}\right)^n \]
where $N_0$ is the initial mass, and $n$ is the number of half-lives that have elapsed:
\[ n = \frac{t}{T_{1/2}} \]

Step 3: Detailed Explanation:

• We are given the following values:
Initial mass ($N_0$) = $80\text{ g}$
Half-life ($T_{1/2}$) = $5\text{ days}$
Total elapsed time ($t$) = $15\text{ days}$

• First, let us calculate the number of half-lives ($n$) that have passed:
\[ n = \frac{15\text{ days}}{5\text{ days}} = 3 \]

• This means the substance undergoes decay through exactly 3 half-lives.

• Substituting the values into our decay formula:
\[ N = 80 \times \left(\frac{1}{2}\right)^3 \]
\[ N = 80 \times \frac{1}{8} = 10\text{ g} \]

• Alternatively, we can trace this step-by-step:
- Initially: $80\text{ g}$
- After 1st half-life (5 days): $\frac{80}{2} = 40\text{ g}$ remaining.
- After 2nd half-life (10 days): $\frac{40}{2} = 20\text{ g}$ remaining.
- After 3rd half-life (15 days): $\frac{20}{2} = 10\text{ g}$ remaining.



Step 4: Final Answer:
The mass of the radioactive substance remaining after 15 days is $10\text{ g}$, which corresponds to option (C).