The nuclear activity of a radioactive element becomes ( 1/8 )ᵗʰ of its initial value in 30 years. The half-life of radioactive element is _____________ years.

Solution

Physics sir · 3 steps

One eighth is one half three times over, so 30 years must hold exactly three half-lives.

1. Decay law. Activity falls by half every half-life. After n half-lives it is A = A₀ (1/2)ⁿ.

   A/A₀ = 1/8 = (1/2)³ ⇒ n = 3

2. Divide the time. Three half-lives take 30 years, so one half-life is T_(1/2) = 30/3 = 10 years.
3. So. The half-life of the element is 10 years.

The common slip: 3.75. Dividing 30 years by 8, the fraction itself, gives 3.75 years. Divide by the number of half-lives, which is 3.

The idea to keep. Write the remaining fraction as a power of 1/2. The power is the number of half-lives, and the total time divided by it gives T_(1/2).

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