Question

Activity of a radioactive sample decreases to $\left(\dfrac{1}{4}\right)^{th}$ of its original value in 4 days. Then in 16 days its activity will become $x$ times the original value. The value of $x$ is

• A. $\dfrac{1}{16}$
• B. $\dfrac{1}{32}$
• C. $\dfrac{1}{256}$
• D. $\dfrac{1}{128}$

Hint:

Activity falling to a quarter takes two half-lives.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept
Activity halves every half-life. A fall to \(\left(\frac14\right)\) of the original value takes two half-lives.

Step 2: Half-life
\[ 2T_{1/2}=4\ \text{days}\Rightarrow T_{1/2}=2\ \text{days} \]

Step 3: Number of half-lives in 16 days
\[ n=\frac{16}{2}=8 \]

Step 4: Activity
\[ \frac AA_0=\left(\frac12\right)^8=\frac1{256} \]

Step 5: Check the options
\(\frac1{16}\) would be after 4 half-lives (8 days). The value \(\frac1{256}\) is option (C). Another way: after 4 days it is \(\frac14\), after 8 days \(\frac1{16}\), after 12 days \(\frac1{64}\), after 16 days \(\frac1{256}\).

Final Answer:
Activity quarters every 4 days, so in 16 days it becomes 1/256 of the original, option (C). \[ \boxed{\frac{1}{256}} \]