Question

There are 10\(^{10}\) radioactive nuclei in a given radioactive element. Its half-life time is 1 minute. How many nuclei will remain after 30 seconds ? (\(\sqrt{2}\) = 1.414)

• A. 10\(^5\)
• B. 2 \(\times\) 10\(^{10}\)
• C. 7 \(\times\) 10\(^9\)
• D. 4 \(\times\) 10\(^{10}\)

Hint:

For radioactive decay problems, always check if the elapsed time is a simple multiple or fraction of the half-life. If \(t = n \cdot T_{1/2}\), the number of nuclei remaining is \(N_0 / 2^n\). In this case, \(n = 1/2\), so the answer is \(N_0 / 2^{1/2} = N_0 / \sqrt{2}\).

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
We are given the initial number of radioactive nuclei, the half-life of the element, and a specific time. We need to calculate the number of nuclei that have not yet decayed after this time.
Step 2: Key Formula or Approach:
The law of radioactive decay gives the number of undecayed nuclei \(N\) at time \(t\) as: \[ N = N_0 \left(\frac{1}{2}\right)^{t/T_{1/2}} \] where: - \(N_0\) is the initial number of nuclei. - \(T_{1/2}\) is the half-life. - \(t\) is the elapsed time.
Step 3: Detailed Explanation:
First, let's list the given values and ensure they are in consistent units.
Initial number of nuclei, \(N_0 = 10^{10}\).
Half-life, \(T_{1/2} = 1\) minute = 60 seconds.
Time, \(t = 30\) seconds.
Now, let's calculate the exponent \(n = t/T_{1/2}\): \[ n = \frac{30 \text{ s}}{60 \text{ s}} = \frac{1}{2} \] This means that the elapsed time is equal to half of one half-life.
Now, substitute these values into the decay formula: \[ N = N_0 \left(\frac{1}{2}\right)^n = 10^{10} \left(\frac{1}{2}\right)^{1/2} \] \[ N = \frac{10^{10}}{\sqrt{2}} \] We are given the value \(\sqrt{2} = 1.414\). \[ N = \frac{10^{10}}{1.414} \] To calculate this, we can approximate \(1/1.414 \approx 0.707\). \[ N \approx 0.707 \times 10^{10} \] \[ N \approx 7.07 \times 10^9 \] Step 4: Final Answer:
The number of nuclei remaining after 30 seconds is approximately \(7.07 \times 10^9\). This matches option (C), \(7 \times 10^9\).