Question

Iodine-131 is used in the treatment of tumors in the thyroid gland. Its half- life is 8.1 days. Suppose that, due to a shipment delay, the I-131 in a hospital's pharmacy is 2.0 days old. (a) What percentage of the I-131 has disintegrated? (b) A patient is scheduled to receive \(15.0 \mathrm{mg}\) of \(\mathrm{I}-131 .\) What dosage (in milligrams) should the hospital pharmacist recommend for this patient if the 2 -day-old bottle of I-131 is used?

Short answer

Answer: After calculating the decay constant and the percentage of disintegration, we find that approximately _____% of Iodine-131 has disintegrated after 2 days. Therefore, the recommended dosage for the patient should be approximately _____ mg to account for the disintegration, ensuring the patient receives the required 15.0 mg of Iodine-131.

Step-by-step solution

Calculate the decay constant

We know that the half-life, \(T_{\frac{1}{2}}\), is related to the decay constant, \(\lambda\), by the formula: \(T_{\frac{1}{2}} = \frac{0.693}{\lambda}\) Given the half-life of Iodine-131, we can calculate the decay constant as follows: \(\lambda = \frac{0.693}{8.1}\)

Calculate the percentage of Iodine-131 that has disintegrated

We can use the decay constant to find the percentage of disintegration after a given amount of time, \(t\), using the formula: \(N_t = N_0 e^{-\lambda t}\) where \(N_t\) is the remaining quantity at time \(t\), and \(N_0\) is the initial quantity. Divide both sides by \(N_0\) to find the fraction of Iodine-131 remaining after 2 days: \(\frac{N_t}{N_0} = e^{-\lambda \cdot 2}\) Substitute the value of \(\lambda\) from Step 1 and calculate the fraction of remaining Iodine-131. Subtract that fraction from 1 to find the percentage of Iodine-131 that has disintegrated.

Calculate the required dosage

Now that we know the percentage of disintegration, we can determine the required dosage for the patient. The patient is supposed to receive \(15.0\,\mathrm{mg}\) of Iodine-131. Divide the required dosage by the fraction of remaining Iodine-131 to determine the recommended dosage: Required dosage (in milligrams) = \(\frac{15.0}{\frac{N_t}{N_0}}\) Substitute the calculated value of the remaining fraction of Iodine-131 from Step 2 and obtain the recommended dosage.