The bromine- 82 nucleus has a half-life of \(1.0 \times 10^{3}\) min. If you wanted \(1.0 \mathrm{~g}{ }^{82} \mathrm{Br}\) and the delivery time was \(3.0\) days, what mass of NaBr should you order (assuming all of the \(\mathrm{Br}\) in the \(\mathrm{NaBr}\) was \({ }^{82} \mathrm{Br}\) )?

First, let's figure out the number of half-lives that will pass during the delivery time. We know the delivery takes 3 days. We need to convert this into minutes, to match the half-life unit. 3 days = \(3 × 24 × 60 = 4320\) minutes Now, we can calculate the number of half-lives: Number of half-lives = \(\frac{\text{Total Time}}{\text{Half-Life}}\) Number of half-lives = \(\frac{4320}{(1.0 \times 10^{3})}\) Number of half-lives ≈ 4.32

Now, we will use the number of half-lives to find the initial mass of bromine-82 required, given that we want 1.0 g of \(^{82}Br\). The formula we will use is: Initial Mass = \(\text{Final Mass} × 2^{\text{number of half-lives}}\) Initial Mass = \(1.0 × 2^{4.32}\) Initial Mass ≈ 21.13 g of \(^{82}Br\)

Finally, we need to calculate how much NaBr we should order. First, let's find the molar mass of NaBr by adding the molar mass of Na (22.99 g/mol) and Br (79.90 g/mol). Molar Mass of NaBr = 22.99 + 79.90 = 102.89 g/mol The ratio of Br in NaBr is equal to the molar mass of Br (79.90 g/mol) divided by the molar mass of NaBr (102.89 g/mol). Ratio of Br in NaBr = \(\frac{79.90}{102.89}\) Now, to find the mass of NaBr required, we will divide the mass of bromine-82 needed by the ratio of Br in NaBr. Mass of NaBr required = \(\frac{\text{Initial Mass of } ^{82}Br}{\text{Ratio of Br in NaBr}}\) Mass of NaBr required = \(\frac{21.13}{\frac{79.90}{102.89}}\) Mass of NaBr required ≈ 27.40 g To obtain 1.0 g of \(^{82}Br\) after a delivery time of 3 days, you should order approximately 27.40 g of NaBr.