According to current regulations, the maximum permissible dose of strontium-90 in the body of an adult is $1 \mu \mathrm{Ci}\left(1 \times 10^{-6} \mathrm{Ci}\right)\(. Using the relationship rate \)=k N$, calculate the number of atoms of strontium-90 to which this dose corresponds. To what mass of strontium-90 does this correspond? The half-life for strontium-90 is $28.8 \mathrm{yr}$

We will use the formula for decay constant based on half-life, which is given by: \(k = \frac{ln(2)}{t_{1/2}}\), where \(t_{1/2}\) is the half-life of strontium-90. Substitute the given half-life value for strontium-90 (28.8 years) and calculate the decay constant k: \(k = \frac{ln(2)}{28.8}\) \(k = 0.0241 \, \mathrm{yr}^{-1}\)

We will use the relationship between activity (rate) and the number of radioactive atoms, given by: \(A = kN\), where A is the activity, k is the decay constant, and N is the number of strontium-90 atoms. The maximum permissible activity (rate) for an adult is given as 1 µCi = \(1 \times 10^{-6}\) Ci. We must convert this to decays per year. Knowing that 1 Ci = \(3.7 \times 10^{10} \, \mathrm{decays/s}\), we can convert to decays per year: \(\mathrm{1 \, µCi \times 10^{-6} \, Ci/µCi \times 3.7 \times 10^{10} \, decays/s/Ci \times (365 \times 24 \times 3600) \, s/yr}\) \(\mathrm{A = 3.7 \times 10^4 \, decays/yr}\) Now, we can solve for the number of strontium-90 atoms, N: \(N = \frac{A}{k}\) \(\mathrm{N = \frac{3.7 \times 10^4 \, decays/yr}{0.0241 \, yr^{-1}}}\) \(N = 1.54 \times 10^{6} \, \mathrm{atoms}\)

To calculate the mass of strontium-90, we will use the number of atoms found in step 2, the molar mass of strontium-90 (89.907 g/mol), and Avogadro's number (\(6.022 \times 10^{23}\) atoms/mol). First, find the number of moles of strontium-90: \(\mathrm{moles = \frac{1.54 \times 10^{6} \, atoms}{6.022 \times 10^{23} \, atoms/mol}}\) \(\mathrm{moles = 2.56 \times 10^{-18} \, mol}\) Now, calculate the mass of strontium-90 using the molar mass: \(\mathrm{mass = 2.56 \times 10^{-18} \, mol \times 89.907 \, g/mol}\) \(\mathrm{mass = 2.3 \times 10^{-16} \, g}\) The mass of strontium-90 corresponding to the maximum permissible dose of 1 µCi for an adult is approximately \(2.3 \times 10^{-16}\) grams.