Problem 48 - Chemistry A Molecular Approach

Question

At 8: 00 A.M., a patient receives a 1.5 - \(\mu\) g dose of \(I\) - 131 to treat thyroid cancer. If the nuclide has a half-life of eight days, what mass of the nuclide remains in the patient at \(5: 00 \mathrm{P}\). M. the next day? (Assume no excretion of the nuclide from the body.)

Step-by-Step Solution

Answer

The mass of the nuclide remaining in the patient at 5:00 P.M. the next day is approximately 1.4315 \(\frac{33}{24}\) \mu g.

Step 1: Determine the Time Elapsed

Calculate the total time elapsed from the initial dose at 8:00 A.M. to 5:00 P.M. the next day. Since there are 24 hours in a day, adding the 9 hours (from 8:00 A.M. to 5:00 P.M.) to the first 24-hour day gives 33 hours total.

Step 2: Convert Time to Days

Convert the time elapsed from hours to days to match the half-life unit. There are 24 hours in one day, so divide 33 hours by 24 to get the elapsed time in days.

Step 3: Calculate the Number of Half-Lives

Divide the elapsed time in days by the half-life of the nuclide to find out how many half-lives have passed.

Step 4: Calculate Remaining Mass of Nuclide

Use the half-life decay formula to calculate the remaining mass of the nuclide. The formula is: \( N(t) = N_0 \times (1/2)^{t/T} \), where \(N(t)\) is the remaining quantity at time \(t\), \(N_0\) is the initial quantity, and \(T\) is the half-life period. In this case, we will use \(N_0 = 1.5 \mu g\) and the values calculated in the previous steps for \(t\) and \(T\).

Key Concepts

Half-Life

The term 'half-life' is a critical concept in nuclear chemistry, referring to the amount of time it takes for half of a quantity of a radioactive isotope, or 'radioisotope', to decay. It is a measure of the rate at which a radioactive substance undergoes decay, essentially telling us how 'fast' or 'slow' the process of radioactive decay is for a specific isotope.

Understanding half-life is crucial for various applications, including medical treatments like the use of I-131 in thyroid cancer therapy, dating fossils and archaeological artifacts, and managing nuclear waste. It's also a cornerstone in performing decay calculations since half-life directly influences the amount of a radioactive substance remaining after a certain period.

In our textbook exercise, the half-life of I-131 is eight days. By knowing this and the amount of time elapsed since the substance was introduced, we can calculate the remaining mass of the radioisotope in the patient's body.

Nuclear Chemistry

Nuclear chemistry explores the changes in the nucleus of atoms that transmute one element into another. The core focus includes understanding radioactive decay, nuclear fission and fusion, and the behaviour of unstable isotopes.

The study of nuclear chemistry is essential for various fields such as medical diagnosis and treatment, energy production, and even forensic science. For instance, the radioisotope I-131, used to treat thyroid cancer, falls within the purview of nuclear chemistry.

The principles of nuclear chemistry help us to predict and calculate changes in the nucleus, such as the decay of I-131 in the exercise. Through these calculations, clinicians can determine the appropriate dose for treatment and ensure the safe handling of radioactive materials.

Radioisotope

A radioisotope is an isotope of an element that has an unstable nucleus and hence, emits radiation during its decay to a stable form. Radioisotopes have myriad uses in science and industry, including medical imaging, cancer treatment, and as tracers in biochemical research.

The key to harnessing radioisotopes safely and effectively is understanding their half-lives and decay patterns, which are unique to each radioisotope. In our exercise, I-131 is the radioisotope in question, chosen for its appropriate radioactive properties to treat thyroid conditions. By knowing its half-life, healthcare providers can gauge how long the isotope will remain active in the body, helping to tailor individual treatment plans.

Exponential Decay

The concept of exponential decay is a mathematical model that describes the process of reducing a quantity by a constant percentage rate over a period of time. In the context of radioisotopes, exponential decay perfectly describes how the quantity of the substance decreases over time.

When we say the decay of a radioactive substance is exponential, we mean that the rate of decay is proportional to the remaining quantity of the substance. The formula for exponential decay in our exercise is: \( N(t) = N_0 \times (1/2)^{t/T} \), where \( N(t) \) is the remaining quantity at time \( t \), \( N_0 \) is the initial quantity, and \( T \) is the half-life period. This formula allows us to predict the amount of radioisotope left after a given period, such as the 33 hours since the I-131 was administered to the patient.