91Ó°ÊÓ

The decay constant \( \lambda \) is related to the half-life \( t_{1/2} \) by the equation \( \lambda = \frac{\ln(2)}{t_{1/2}} \). The half-life of \(^2^3^8\text{U}\) is \( 4.47 \times 10^9 \) years. First, convert the half-life to seconds:\[ t_{1/2} = 4.47 \times 10^9 \times 3.156 \times 10^7 \text{ seconds/year} = 1.411 \times 10^{17} \text{ seconds} \] Then, calculate the decay constant:\[ \lambda = \frac{\ln(2)}{1.411 \times 10^{17}} \approx 4.92 \times 10^{-18} \text{ s}^{-1} \]

Activity \( A \) is defined by \( A = \lambda N \), where \( N \) is the number of radioactive atoms. A curie (1 Ci) is defined as \( 3.7 \times 10^{10} \text{ disintegrations per second}\). So, rearrange the formula to find \( N \):\[ N = \frac{A}{\lambda} = \frac{3.7 \times 10^{10}}{4.92 \times 10^{-18}} \approx 7.52 \times 10^{27} \text{ atoms} \] To find the mass, convert atoms to moles using Avogadro's number (\( 6.022 \times 10^{23} \text{ atoms/mole} \)):\[ \text{Moles of } ^2^3^8\text{U} = \frac{7.52 \times 10^{27}}{6.022 \times 10^{23}} \approx 12493.7 \text{ moles} \] Finally, multiply by the molar mass of uranium (238 g/mol):\[ \text{Mass of uranium} = 12493.7 \times 238 \approx 2.97 \times 10^{6} \text{ g} \]

First, calculate the number of moles in 10 g of \(^2^3^8\text{U}\):\[ \text{Moles of } ^2^3^8\text{U} = \frac{10}{238} \approx 0.042 \text{ moles} \] This corresponds to \( N \) atoms using Avogadro's number:\[ N = 0.042 \times 6.022 \times 10^{23} \approx 2.53 \times 10^{22} \text{ atoms} \] The activity \( A \) or the rate of alpha emission is given by \( A = \lambda N \):\[ A = (4.92 \times 10^{-18}) \times (2.53 \times 10^{22}) \approx 1.24 \times 10^5 \text{ alpha particles per second} \]