Iodine-131 is a convenient radioisotope to monitor thyroid activity in humans. It is a beta emitter with a half-life of 8.02 days. The thyroid is the only gland in the body that uses iodine. A person undergoing a test of thyroid activity drinks a solution of NaI, in which only a small fraction of the iodide is radioactive. (a) Why is NaI a good choice for the source of iodine? (b) If a Geiger counter is placed near the person's thyroid (which is near the neck) right after the sodium iodide solution is taken, what will the data look like as a function of time? (c) A normal thyroid will take up about \(12 \%\) of the ingested iodide in a few hours. How long will it take for the radioactive iodide taken up and held by the thyroid to decay to \(0.01 \%\) of the original amount?

NaI is a good choice for the source of iodine in thyroid tests because of the following reasons: 1. NaI is highly soluble in water, making it easy to prepare in a solution that can be ingested by a person. 2. Iodine is the active component that will be monitored in the thyroid, as it is the only gland in the body that uses iodine. 3. Sodium (Na) is chemically inert in this context and does not interfere with the iodine uptake by the thyroid gland nor does it cause any harmful side effects. Thus, NaI works well for ingesting the iodine and monitoring its uptake by the thyroid gland.

After ingestion of the sodium iodide solution, the radioactive iodine (I-131) will be taken up by the thyroid gland. Right after drinking the solution, the Geiger counter will detect a higher number of radioactive iodine decays. As time passes, the amount of radioactive iodine in the thyroid will decrease due to decay and the Geiger counter will detect progressively lower counts. The data will follow an exponential decay function, which is characteristic for radioactive decay processes. Mathematically, the number of radioactive iodine decays detected by the Geiger counter can be represented by the function: \[ N(t) = N_0 e^{-\lambda t} \] Where: - \(N(t)\) is the number of radioactive iodine decays at time \(t\), - \(N_0\) is the initial number of radioactive iodine decays, - \(\lambda\) is the decay constant, - \(t\) is the time since ingesting the sodium iodide solution.

To calculate the time it takes for the radioactive iodide taken up and held by the thyroid to decay to \(0.01 \%\) of the original amount, we'll use the radioactive decay formula: \[ N(t) = N_0 e^{-\lambda t} \] First, we need to find the decay constant \(\lambda\) using the half-life formula: \[ t_{1/2} = \frac{\ln{2}}{\lambda} \] Given the half-life (\(t_{1/2}\)) of Iodine-131 as 8.02 days, we can find the decay constant: \[ \lambda = \frac{\ln{2}}{8.02} \approx 0.0865 \, \text{day}^{-1} \] Now, we're given that the normal thyroid takes up about \(12\%\) of the ingested iodide. We're trying to find the time \(t\) required for the radioactive iodide to decay to \(0.01 \%\) of the original amount. Let's set up the equation: \[ 0.0001 N_0 = (0.12 N_0) e^{-0.0865 t} \] Divide both sides by \(0.12 N_0\): \[ \frac{0.0001}{0.12} = e^{-0.0865 t} \] Now, take the natural logarithm of both sides: \[ \ln{\frac{0.0001}{0.12}} = -0.0865 t \] Finally, solve for \(t\): \[ t = \frac{\ln{\frac{0.0001}{0.12}}}{-0.0865} \] \[ t \approx 51.36 \, \text{days} \] So, it will take approximately 51.36 days for the radioactive iodide taken up and held by the thyroid to decay to \(0.01 \%\) of the original amount.