The half-life of ^ 226 Ra is 1602 y . Determine the activity of 0.1...

Molar mass of $\text{RaCl}_2 = 226 + 2 \times 35.5 = 297 \text{ g mol}^{-1}$

Number of moles of $\text{RaCl}_2 = \dfrac{0.1}{297} \text{ mol}$

Since each molecule of $\text{RaCl}_2$ contains one atom of $\text{Ra}$, the number of $\text{Ra}$ atoms is:

$N = \dfrac{0.1}{297} \times 6.022 \times 10^{23} \approx 2.028 \times 10^{20}$

The half-life of ^{226}\text{Ra}$in seconds is:<br><br>$T_{1/2} = 1602 \times 365 \times 24 \times 3600 \text{ s} \approx 5.05 \times 10^{10} \text{ s}$<br><br>The decay constant$\lambda$is given by:<br><br>$\lambda = \dfrac{0.693}{T_{1/2}} = \dfrac{0.693}{5.05 \times 10^{10}} \approx 1.37 \times 10^{-11} \text{ s}^{-1}$<br><br>The activity$A$of the sample is:<br><br>$A = \lambda N = 1.37 \times 10^{-11} \times 2.028 \times 10^{20} \approx 2.78 \times 10^9 \text{ disintegrations s}^{-1}$<br><br>Rounding to the nearest given option, the activity is$2.8 \times 10^9 \text{ disintegrations s}^{-1}.