Question

Radioactive iodine, \(^{131} \mathrm{I},\) with a half-life of $8.0252 \mathrm{d},$ is used in some forms of medical diagnostics. (a) If the initial activity of a sample is \(64.5 \mathrm{mCi},\) what is the mass of $^{131} \mathrm{I}$ in the sample? (b) What will the activity be 4.5 d later?

Short answer

Answer: The mass of the radioactive iodine sample is 5.96 x 10^{-7} g, and its activity after 4.5 days is 40.5 mCi.

Step-by-step solution

Find the decay constant

The decay constant, λ, can be found using the half-life, T, formula: λ = ln(2) / T. Since the half-life of radioactive iodine is 8.0252 days, we can find the decay constant as follows: λ = ln(2) / 8.0252 = 0.08652 d^{-1}.

Calculate the number of radioactive iodine nuclei

The activity, A, of a radioactive sample is related to the number of radioactive nuclei, N, and the decay constant, λ, by the formula: A = λN. We can rearrange this equation to calculate N: N = A / λ. With A = 64.5 mCi and λ = 0.08652 d^{-1}, we now calculate N: N = (64.5 x 10^{-3} Ci) / 0.08652 d^{-1}. Note that 1 Ci = 3.7 x 10^{10} decays/s. So, N = (64.5 x 10^{-3} x 3.7 x 10^{10} decays/s) / 0.08652 d^{-1} = 2.746 x 10^{15} nuclei.

Find the mass of the iodine sample

The mass, m, of a radioactive sample can be found using the number of radioactive nuclei, N, and the molar mass, M, of the element: m = (N x M) / N_A, where N_A = 6.022 x 10^{23} nuclei/mol (Avogadro's number). For radioactive iodine, M = 131 g/mol. Using the values of N and M, we calculate the mass of the iodine sample: m = (2.746 x 10^{15} nuclei x 131 g/mol) / (6.022 x 10^{23} nuclei/mol) = 5.96 x 10^{-7} g.

Calculate the fraction of nuclei remaining after 4.5 days

The fraction of radioactive nuclei remaining after a certain time, t, is given by the equation: Fraction_remaining = e^{-λt}. In our case, t = 4.5 days and λ = 0.08652 d^{-1}, so we can calculate the fraction of radioactive nuclei remaining after 4.5 days: Fraction_remaining = e^{-(0.08652 d^{-1})(4.5 d)} = 0.628.

Find the activity after 4.5 days

The activity, A_t, of the iodine sample after 4.5 days can be calculated by multiplying the initial activity, A, by the fraction of radioactive nuclei remaining: A_t = A x Fraction_remaining. With A = 64.5 mCi and Fraction_remaining = 0.628, we calculate the activity after 4.5 days: A_t = (64.5 mCi) x 0.628 = 40.5 mCi. Thus, the mass of the radioactive iodine sample is 5.96 x 10^{-7} g, and its activity after 4.5 days is 40.5 mCi.