There are 10¹⁰ radioactive nuclei in a given radioactive element, its half-life time is 1 minute. How many nuclei will remain after 30 seconds? ( √2 = 1.414 )

Solution

Physics sir · 3 steps

30 seconds is half a half-life, so multiply by (1/2)^(1/2) = 1/√2, not by 1/2.

1. Decay law. After time t, the number left is N = N₀ (1/2)^(t/T_(1/2)). The exponent counts how many half-lives have passed, and it need not be a whole number.
2. Count half-lives. Here t = 30 s and T_(1/2) = 60 s, so t/T_(1/2) = 1/2.

   N = 10¹⁰ × (1/2)^(1/2) = 10¹⁰/√2 = 10¹⁰/1.414 ≈ 7.07 × 10⁹

3. So. About 7 × 10⁹ nuclei remain, which is option (b).

The idea to keep. Half a half-life does not remove half of what half-life removes. Decay is exponential, so after a fraction f of a half-life the remaining share is (1/2)ᶠ.

Answer from the NTA final answer key. Solution by PrepWiser.