Question

A human body required the 0.01 M activity of radioactive substance after 24 h. Half-life of radioactive substance is 6 h. Then, injection of maximum activity of radioactive substance that can be injected will be

• A. 0.08
• B. 0.04
• C. 0.16
• D. 0.32

Answer 1

The Correct Option is C

Remaining activity = 0.01 M after 24 h
Remaining activity
$\, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, $=Initial activity $\times\Bigg(\frac{1}{2}\Bigg)^n$
Used half-life time(n)=$\frac{Total time}{t_{1/2}}=\frac{24}{6}=4$
So $\, \, \, \, \, \, \, $0.01 = Initial activity $\times\Bigg(\frac{1}{2}\Bigg)^4$
Initial activity = 0.01 $\times$ 16 = 0.16

Answer 2

To determine the maximum activity of the radioactive substance that can be injected into the human body to achieve an activity of 0.01 M after 24 hours, we need to use the concept of radioactive decay and the half-life formula.
Given:
- Required activity after 24 hours: \( A_{24} = 0.01 \) M
- Half-life of the radioactive substance: \( t_{1/2} = 6 \) hours
 Formula for Radioactive Decay:
The activity of a radioactive substance after a certain time period can be calculated using the formula:
\[A_t = A_0 \left(\frac{1}{2}\right)^{\frac{t}{t_{1/2}}}\]
where:
- \( A_t \) is the activity after time \( t \)
- \( A_0 \) is the initial activity
- \( t \) is the time elapsed
- \( t_{1/2} \) is the half-life of the substance
Steps to Determine the Maximum Initial Activity (\( A_0 \)):
1. Determine the decay factor:
  \[\left(\frac{1}{2}\right)^{\frac{t}{t_{1/2}}} = \left(\frac{1}{2}\right)^{\frac{24}{6}} = \left(\frac{1}{2}\right)^4 = \frac{1}{16}\]
2. Use the decay formula to find \( A_0 \):
  \[  A_{24} = A_0 \left(\frac{1}{16}\right)  \]
  \[  0.01 = A_0 \times \frac{1}{16}  \]
3. Solve for \( A_0 \):
  \[A_0 = 0.01 \times 16 = 0.16 \text{ M}\]
Conclusion:
The maximum initial activity of the radioactive substance that can be injected to achieve an activity of 0.01 M after 24 hours is 0.16 M. so the correct Answer is Option (C)