Problem 46 - Chemistry: Principles and Reactions

Question

Iodine-131 is used to treat tumors in the thyroid. Its first-order half-life is \(8.1\) days. If a patient is given a sample containing \(5.00 \mathrm{mg}\) of \(\mathrm{I}-131\), how long will it take for \(25 \%\) of the isotope to remain in her system?

Step-by-Step Solution

Answer

Answer: It takes approximately 10.63 days for only 25% of the initial amount of Iodine-131 isotope to remain.

Step 1: Find the decay constant (k)

We can find the decay constant using the half-life for Iodine-131. The relation between decay constant (k) and half-life (t_1/2) is as follows: \( t_{1/2} = \frac{ln(2)}{k} \) We are given half-life, so let's rearrange the formula to get k: \( k = \frac{ln(2)}{t_{1/2}} \) Now, plug in the half-life value for Iodine-131: \( k = \frac{ln(2)}{8.1\, \text{days}} \approx 0.0855\; \text{day}^{-1} \)

Step 2: Determine the remaining percentage of Iodine-131 after the decay

We are given that we want 25% of the initial amount to remain. First, let's express this remaining percentage as a fraction: \( \frac{A(t)}{A_0} = \frac{25}{100} = 0.25 \) Where A(t) is the amount of Iodine-131 after time t, and A0 is the initial amount of Iodine-131 (5mg).

Step 3: Use the integrated rate law to find the time (t)

The first-order integrated rate law equation is: \( ln\left(\frac{A(t)}{A_0}\right) = -kt \) Now, plug in the values for \(\frac{A(t)}{A_0}\) and \(k\): \( ln(0.25) = -0.0855\; t \) Calculate the value of t: \( t = \frac{ln(0.25)}{-0.0855} \approx 10.63\; \text{days} \)

Step 4: Final Answer

The time it takes for 25% of Iodine-131 to remain in the patient's system is approximately 10.63 days.

Key Concepts

Iodine-131

Iodine-131 is a radioactive isotope commonly used in the treatment of thyroid conditions, including hyperthyroidism and certain types of thyroid cancer. It emits beta particles, which help in destroying the hyperactive thyroid cells with precision. This targeted approach is beneficial because it minimizes damage to the surrounding healthy tissue. Iodine-131 has a dual-functionality as a diagnostic tool and a therapeutic agent, making it integral in medical practices associated with thyroid health. Externally, it emits gamma radiation, which can be detected in scans, confirming the location and activity of the thyroid tissues. Thus, it plays a crucial role in both monitoring and treating thyroid-related issues.

Half-life

The half-life of a radioactive substance is the time required for half of the original amount of the substance to decay. For Iodine-131, this half-life is 8.1 days. During this time, half of the iodine-131 atoms will have undergone radioactive decay. This concept is fundamental when determining how long a radioactive isotope remains active in the body or its environment.

• Knowing the half-life helps in scheduling repeat treatments and understanding the duration of radiation exposure.
• It aids in predicting the decay process in medical treatments.

This makes it easier to manage and safely implement radioactive materials in medical settings.

First-order reaction

Radioactive decay of Iodine-131 follows a first-order reaction kinetics. This means the rate of decay is proportional to the amount of the substance left. In other words, the larger the quantity of undecayed iodine-131, the faster it will decay. In first-order reactions such as this, the half-life is consistent and does not change with the amount of material present.

• This constant half-life is useful for predicting the amount of material left after a certain period.
• The formula \(ln\left(\frac{A(t)}{A_0}\right) = -kt\) describes the relationship between the remaining quantity of a substance, its initial amount, the decay constant, and the time elapsed.

This straightforward relationship makes calculations involving first-order reactions efficient and easy to comprehend.

Decay constant

The decay constant, often denoted as \(k\), is a measurement of how quickly a radioactive substance decays. It is vital in the calculations used to find the amount of substance remaining after a given period. For Iodine-131 with a half-life of 8.1 days, the decay constant \(k\) is approximately \(0.0855\; ext{day}^{-1}\). This was derived from the formula \( k = \frac{ln(2)}{t_{1/2}} \).

• The decay constant allows us to link the rate of decay with time and helps in predicting future concentrations of a substance in treatments.
• Its constant value for each distinct radioactive material, like Iodine-131, aids in ensuring precise dosages and timing in therapeutic applications.

Understanding the decay constant bridges the gap between theoretical calculations and practical biomedical applications, ensuring safe and effective patient care.