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 Nal, in which only a small fraction of the iodide is radioactive. (a) Why is Nal 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?

Sodium iodide (Nal) is a good choice for the source of iodine because it is water-soluble, allowing easy absorption by the body. Moreover, as iodine is selective to the thyroid gland, it ensures that the radioactive element (iodine-131) is primarily absorbed by the thyroid, making it a highly specific and effective monitoring tool for thyroid activity.

The data obtained from the Geiger counter placed near the person's thyroid right after taking the sodium iodide solution will show an initial high count, indicating the presence of radioactive iodine. As time passes, the count will decrease exponentially, following a decay curve. This is because the half-life of iodine-131 is 8.02 days, and with each passing half-life, the amount of radioactive material present around the thyroid will reduce by half.

We are given that a normal thyroid will take up about 12% of the ingested iodide. We are supposed to find the time it takes for this radioactive iodide to decay to 0.01% of the original amount. The decay of radioactive isotopes can be described by the formula: \[ N_t = N_0 * (1/2)^{t/T} \] where: - \(N_t\) is the remaining amount of the substance at a given time \(t\) - \(N_0\) is the initial amount of the substance - \(T\) is the half-life of the substance Let's assume the initial radioactive iodide ingested was 100 units. Therefore, the thyroid would take up 12 units (12% of 100 units). To find the time it takes for this amount to decay to 0.01% of the original amount (which is 0.01 units), we can set up the equation: \[ 0.01 = 12 * (1/2)^{t/8.02} \] We can then solve for \(t\): 1. Divide both sides by 12: \[ 0.0008333 = (1/2)^{t/8.02} \] 2. Take the logarithm base 2 of both sides: \[ \log_2{0.0008333} = \frac{t}{8.02} \] 3. Multiply both sides by 8.02 to isolate \(t\): \[ t = 8.02 * \log_2{0.0008333} \] 4. Calculate the value of \(t\): \[ t \approx 53.69\; \text{days} \] Hence, it takes approximately 53.69 days for the radioactive iodide taken up and held by the thyroid to decay to 0.01% of the original amount.