Question

A radioactive sample has a half-life of 10 days. The fraction of the initial nuclei decayed after 40 days is:

• A. \( 1/4 \)
• B. \( 3/4 \)
• C. \( 1/16 \)
• D. \( 15/16 \)

Hint:

Always read the final line of a radioactivity question with extreme caution. A very common exam trap is to choose option (C) \(1/16\), which is the fraction of active nuclei undecayed (remaining). Always remember: \(\text{Decayed Fraction} + \text{Remaining Fraction} = 1\).

Answer 1

The Correct Option is D

Concept: Radioactive decay follows first-order kinetics. The half-life (\(T_{1/2}\)) is the time required for a radioactive sample to decrease to half of its initial value. The remaining fraction of active nuclei (\(N/N_0\)) left after \(n\) half-lives is given by the formula: \[ \frac{N}{N_0} = \left(\frac{1}{2}\right)^n \] Where \(n\) is the total number of half-lives that have elapsed: \[ n = \frac{\text{Total elapsed time }(t)}{\text{Half-life }(T_{1/2})} \] Crucially, the question asks for the fraction of nuclei decayed, which is the total initial amount minus what remains: \[ \text{Fraction decayed} = 1 - \frac{N}{N_0} \]

Step 1: Calculating the number of half-lives ($n$) that have elapsed.
From the given information:
• Half-life of the sample, \(T_{1/2} = 10 \text{ days}\)
• Total decay time, \(t = 40 \text{ days}\) Finding the number of half-life cycles: \[ n = \frac{40}{10} = 4 \]

Step 2: Determining the remaining and decayed fractions.
Calculate the fraction of intact radioactive nuclei remaining after 4 half-lives: \[ \frac{N}{N_0} = \left(\frac{1}{2}\right)^4 = \frac{1}{16} \] Now, solve for the fraction of the initial nuclei that have decayed over this period: \[ \text{Fraction decayed} = 1 - \frac{1}{16} = \frac{16 - 1}{16} = \frac{15}{16} \]