Concept:Radioactive decay follows a geometric progression where the remaining amount halves after each half-life period. The formula for the remaining mass is based on the number of half-lives elapsed.
Formula:$$ N = N_0 \left(\frac{1}{2}\right)^n $$
Where \(n\) is the number of half-lives, defined as \(n = \frac{t}{T_{1/2}}\).
Solution:- Given: Initial mass \(N_0 = 64\text{ g}\), Final mass \(N = 2\text{ g}\), Total time \(t = 20\text{ days}\).
- Substitute into the formula: $$ 2 = 64 \left(\frac{1}{2}\right)^n $$
- Solve for \(n\): $$ \frac{2}{64} = \frac{1}{32} = \left(\frac{1}{2}\right)^n \implies n = 5 $$
- Calculate the half-life: $$ T_{1/2} = \frac{t}{n} = \frac{20}{5} = 4\text{ days} $$
Why other options are incorrect:Option A assumes 4 half-lives passed. Option B miscalculates the power of 2. Option D is a math error in division.
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