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. (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?

Verified step by step guidance

1. Identify the initial and final percentages of the radioactive iodide. The initial uptake by the thyroid is 12% of the ingested iodide, and we want to find out when it decays to 0.01% of this initial amount.
2. Use the formula for radioactive decay: \( N = N_0 \times (0.5)^{t/t_{1/2}} \), where \( N \) is the final amount, \( N_0 \) is the initial amount, \( t \) is the time, and \( t_{1/2} \) is the half-life.
3. Substitute the known values into the decay formula. Here, \( N/N_0 = 0.01/12 \), and \( t_{1/2} = 8.02 \) days.
4. Solve the equation \( 0.01/12 = (0.5)^{t/8.02} \) for \( t \). This involves taking the logarithm of both sides to isolate \( t \).
5. Calculate \( t \) using the logarithmic form: \( t = t_{1/2} \times \frac{\log(0.01/12)}{\log(0.5)} \).

Key Concepts

Radioactive Decay

Radioactive decay is the process by which an unstable atomic nucleus loses energy by emitting radiation. This decay occurs at a predictable rate characterized by the half-life, which is the time required for half of the radioactive substance to decay. Understanding this concept is crucial for calculating how long it will take for a given amount of a radioactive isotope, like iodine-131, to reduce to a specific fraction of its original quantity.

Half-Life

Half-life is a fundamental concept in nuclear chemistry that describes the time it takes for half of a sample of a radioactive substance to decay. For iodine-131, the half-life is 8.02 days, meaning that after this period, only half of the original amount remains. This concept is essential for determining the time required for the radioactive iodide in the thyroid to decay to a specified percentage of its initial amount.

Exponential Decay

Exponential decay refers to the decrease of a quantity at a rate proportional to its current value, commonly observed in radioactive substances. The amount of radioactive material remaining can be modeled using the equation N(t) = N0 * (1/2)^(t/T), where N0 is the initial amount, T is the half-life, and t is the elapsed time. This mathematical relationship is vital for calculating how long it will take for the radioactive iodide in the thyroid to decay to 0.01% of its original amount.