Question

Bismuth has a half-life period of $5$ days. A sample originally has a mass of $1000\text{ mg}$, then the mass of Bismuth after $30$ days is

• A. $16.625\text{ mg}$
• B. $13.625\text{ mg}$
• C. $14.625\text{ mg}$
• D. $15.625\text{ mg}$

Hint:

When calculations involve simple integers, you can quickly write out the stepwise division by 2: $$1000 \rightarrow 500 \rightarrow 250 \rightarrow 125 \rightarrow 62.5 \rightarrow 31.25 \rightarrow 15.625$$ Counting exactly 6 steps yields the solution effortlessly without managing large exponents.

Answer 1

The Correct Option is D

Step 1: Understanding the Question:
The problem presents a radioactive decay process for Bismuth with a known initial mass and half-life. We need to determine the remaining residual mass after a total elapsed time of $30$ days.

Step 2: Key Formula or Approach:
The remaining mass $N$ after $n$ half-lives can be computed using the standard decay relation: $$N = N_0 \left(\frac{1}{2}\right)^n$$ where $n = \frac{\text{Total Time}}{\text{Half-life Period}}$ and $N_0$ is the initial mass.

Step 3: Detailed Explanation:
Given values: Initial mass, $N_0 = 1000\text{ mg}$ Half-life, $T_{1/2} = 5\text{ days}$ Total time, $t = 30\text{ days}$ First, find the total number of half-life cycles completed ($n$): $$n = \frac{30}{5} = 6$$ Now, substitute these parameters into our exponential decay formula: $$N = 1000 \left(\frac{1}{2}\right)^6$$ $$N = \frac{1000}{64}$$ $$N = 15.625\text{ mg}$$

Step 4: Final Answer:
The remaining mass of Bismuth after $30$ days is $15.625\text{ mg}$, which corresponds to option (D).