Radioactive decay: How many days are required for a radioactive sample, with a half-life of 5.7 d and an initial activity of 1.07 × 10⁵ Bq, to decay to an activity of 100 Bq?

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Quizplus · Accepted Answer

The half-life formula is used to calculate the time required for a substance to decay to a certain level. The formula is $t = \frac{\ln(\frac{N_0}{N})}{\lambda}$, where $t$ is the time, $N_0$ is the initial quantity, $N$ is the final quantity, and $\lambda$ is the decay constant. The decay constant $\lambda$ can be found from the half-life $T_{1/2}$ using the formula $\lambda = \frac{\ln(2)}{T_{1/2}}$. However, for this problem, we can directly use the concept of half-lives without calculating the decay constant explicitly. Given: - Initial activity $N_0 = 1.07 \times 10^{15} Bq$- Final activity $N = 100 Bq$- Half-life $T_{1/2} = 5.7$ daysThe number of half-lives ($n$) required to reach from $N_0$ to $N$ can be calculated using the formula $N = N_0 \times (\frac{1}{2})^n$. Rearranging for $n$, we get $n = \log_2(\frac{N_0}{N})$.Plugging in the given values: $n = \log_2(\frac{1.07 \times 10^{15}}{100})$.This calculation gives the number of half-lives needed to decay from $1.07 \times 10^{15} Bq$ to $100 Bq$. Each half-life is 5.7 days, so the total time $t$ is $n \times 5.7$ days.Without needing to reach for a calculator, the principle here is that the activity decreases by half every 5.7 days. To go from $1.07 \times 10^{15} Bq$ to $100 Bq$, it takes several half-lives. The exact calculation would show that it takes about 10 half-lives, which is $10 \times 5.7 = 57$ days, making option A correct.

Radioactive Decay: How Many Days Are Required for a Radioactive