The half-life of a radio-active element is 9 days, what fraction of atoms has decayed in 36 days?
This chemistry question involves key chemical concepts and calculations. The detailed solution below walks through each step, from identifying the reaction type to computing the final answer.
ScanSolve AI Answer
Step 1: Calculate the number of half-lives.
The number of half-lives ($n$) is the total time divided by the half-life.
$$n = \frac{\text{Total time}}{\text{Half-life}}$$
$$n = \frac{36 \text{ days}}{9 \text{ days}}$$
$$n = 4$$
Step 2: Calculate the fraction of atoms remaining.
After $n$ half-lives, the fraction of atoms remaining is given by $(\frac{1}{2})^n$.
$$\text{Fraction remaining} = \left(\frac{1}{2}\right)^n$$
$$\text{Fraction remaining} = \left(\frac{1}{2}\right)^4$$
$$\text{Fraction remaining} = \frac{1^4}{2^4}$$
$$\text{Fraction remaining} = \frac{1}{16}$$
Step 3: Calculate the fraction of atoms that has decayed.
The fraction decayed is $1 - \text{fraction remaining}$.
$$\text{Fraction decayed} = 1 - \frac{1}{16}$$
$$\text{Fraction decayed} = \frac{16}{16} - \frac{1}{16}$$
$$\text{Fraction decayed} = \frac{15}{16}$$
The fraction of atoms that has decayed in 36 days is $\boxed{\frac{15}{16}}$.
