Question

Yttrium- 87 has a rate constant of \(2.6 \times 10^{-6} \mathrm{~s}^{-1}\). What is the activity of a 5.00 -mg sample?

Short answer

Answer: The activity of the 5.00-mg sample of Yttrium-87 is approximately 7.83 × 10^{13} Bq.

Step-by-step solution

Find the atomic mass of Yttrium-87

We can find the atomic mass of Yttrium-87 from the periodic table. The atomic mass of Yttrium(Y) is approximately 87.

Calculate the number of moles of Yttrium-87

To calculate the number of moles of Yttrium-87, we will use the formula: moles of Y = (mass of Y)/(atomic mass of Y) moles of Y = (5.00 mg)/(87 g/mol) Since mass is given in milligrams, we need to divide by 1000 to convert mg to g, so: moles of Y = (5.00/1000 g)/(87 g/mol) moles of Y = 5.00 × 10^{-5} mol

Calculate the total number of Yttrium-87 nuclei

Now we need to calculate the total number of Yttrium-87 nuclei (N) using Avogadro's number, which is 6.022 x 10^{23}: N = moles of Y × Avogadro's number N = (5.00 × 10^{-5} mol)(6.022 × 10^{23} mol^{-1}) N = 3.01 × 10^{19} nuclei

Calculate the activity of the Yttrium-87 sample

Now, we can use the rate constant (k) and the total number of nuclei (N) to find the activity of the Yttrium-87 sample: A = k × N A = (2.6 × 10^{-6} s^{-1})(3.01 × 10^{19} nuclei) A = 7.83 × 10^{13} Bq (becquerels) So the activity of the 5.00-mg sample of Yttrium-87 is approximately 7.83 × 10^{13} Bq.