Question

One mole of a parent nuclide is undergoing radioactive decay. The fraction of atoms of this nuclide that has decayed after three half-lives is.................

Hint:

The fraction of atoms decayed in radioactive decay can be calculated using the formula \(1 - \left(\frac{1}{2}\right)^n\), where \(n\) is the number of half-lives.

Answer 1

Correct Answer: 0.875

Step 1: What one half life does.
After each half life, half of the leftover parent atoms remain. So the fraction still present keeps halving step by step.

Step 2: Track the surviving fraction.
Start with the whole amount. After one half life $1/2$ is left, after two $1/4$ is left, and after three \[ \left(\tfrac{1}{2}\right)^3 = \tfrac{1}{8} \] is left.

Step 3: Turn survivors into decayed.
The part that has decayed is just what is no longer there. So decayed fraction is \[ 1 - \tfrac{1}{8}. \]

Step 4: Do the subtraction.
Working it out, \[ 1 - 0.125 = 0.875. \] So most of the sample has decayed.

Step 5: State the answer.
The decayed fraction after three half lives is 0.875.
\[ \boxed{0.875} \]