Question

If 87.5% of atoms of a radioactive element decay in 6 days, then the fraction of atoms of the element that decay in 8 days is

• A. \( \frac{1}{8} \)
• B. \( \frac{7}{8} \)
• C. \( \frac{1}{16} \)
• D. \( \frac{15}{16} \)

Hint:

Memorize the common fractions for half-life decay: - 1 half-life: 1/2 remaining (50% decayed) - 2 half-lives: 1/4 remaining (75% decayed) - 3 half-lives: 1/8 remaining (87.5% decayed) - 4 half-lives: 1/16 remaining (93.75% decayed) Recognizing 87.5% as 7/8 decayed (or 1/8 remaining) instantly tells you that 3 half-lives have passed.

Answer 1

The Correct Option is D

Step 1: Determine the half-life of the element.
If 87.5% of the atoms have decayed, the fraction of atoms remaining is \( 100% - 87.5% = 12.5% \).
\( 12.5% = \frac{12.5}{100} = \frac{1}{8} \).
The fraction of atoms remaining, \( N/N_0 \), after 'n' half-lives is given by \( \frac{N}{N_0} = \left(\frac{1}{2}\right)^n \).
So, \( \frac{1}{8} = \left(\frac{1}{2}\right)^3 = \left(\frac{1}{2}\right)^n \).
This means that 3 half-lives have passed.
We are given that this decay takes 6 days. So, 3 half-lives = 6 days.
Therefore, one half-life (\(T_{1/2}\)) is \( \frac{6}{3} = 2 \) days.
Step 2: Calculate the fraction remaining after 8 days.
The time elapsed is \( t = 8 \) days.
The number of half-lives in this period is \( n = \frac{t}{T_{1/2}} = \frac{8}{2} = 4 \).
The fraction of atoms remaining after 4 half-lives is:
\( \frac{N}{N_0} = \left(\frac{1}{2}\right)^4 = \frac{1}{16} \).
Step 3: Calculate the fraction that has decayed in 8 days.
The fraction decayed is \( 1 - \text{(fraction remaining)} \).
Fraction decayed = \( 1 - \frac{1}{16} = \frac{15}{16} \).