Question

A sample of radioactive element contains \( 8 \times 10^{16} \) active nuclei. The half-life of the element is 15 days. The number of nuclei decayed after 60 days is

• A. \( 0.5 \times 10^{16} \)
• B. \( 2 \times 10^{16} \)
• C. \( 7.5 \times 10^{16} \)
• D. \( 4 \times 10^{16} \)

Hint:

Radioactive decay is exponential. To find the number of nuclei decayed, calculate the remaining nuclei after each half-life and subtract from the initial value.

Answer 1

The Correct Option is C

Step 1: Formula for radioactive decay.
The number of nuclei decayed over time can be calculated using the formula for half-life decay: \[ N(t) = N_0 \left( \frac{1}{2} \right)^{\frac{t}{T_{1/2}}} \] where \( N_0 \) is the initial number of nuclei, \( t \) is the time elapsed, and \( T_{1/2} \) is the half-life of the element. The number of nuclei decayed is given by: \[ \text{Nuclei decayed} = N_0 - N(t) \] Step 2: Applying the given values.
Given that \( N_0 = 8 \times 10^{16} \), \( T_{1/2} = 15 \) days, and \( t = 60 \) days, we first find the number of remaining nuclei after 60 days: \[ N(60) = 8 \times 10^{16} \left( \frac{1}{2} \right)^{\frac{60}{15}} = 8 \times 10^{16} \left( \frac{1}{2} \right)^4 = 8 \times 10^{16} \times \frac{1}{16} = 0.5 \times 10^{16} \] Thus, the number of nuclei decayed is: \[ \text{Nuclei decayed} = 8 \times 10^{16} - 0.5 \times 10^{16} = 7.5 \times 10^{16} \] Step 3: Conclusion.
Thus, the number of nuclei decayed after 60 days is \( 7.5 \times 10^{16} \), corresponding to option (C).