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. b. 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 percentage of radioactive iodide taken up by the thyroid, which is 12% of the ingested amount.
2. Recognize that the problem involves radioactive decay, which follows first-order kinetics. Use the formula for first-order decay: \( N_t = N_0 e^{-kt} \), where \( N_t \) is the remaining amount, \( N_0 \) is the initial amount, \( k \) is the decay constant, and \( t \) is time.
3. Calculate the decay constant \( k \) using the half-life formula: \( k = \frac{\ln(2)}{\text{half-life}} \). Substitute the given half-life of 8.02 days into the formula.
4. Set up the equation to find the time \( t \) when the remaining radioactive iodide is 0.01% of the initial amount. Substitute \( N_t = 0.01\% \times N_0 \) into the decay equation.
5. Solve the equation \( 0.01\% \times N_0 = N_0 e^{-kt} \) for \( t \) by taking the natural logarithm of both sides and isolating \( t \).

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 takes 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 concept is vital for solving problems related to the decay of radioactive isotopes over time.