Question

Iodine-131 is used as a nuclear medicine to treat hyperthyroidism. The half- life of \({ }^{131} \mathrm{I}\) is 8.04 days. How long will it take for a \(500 \mathrm{mg}\) sample of \({ }^{131} \mathrm{I}\) to decay into \(1 \%\) of its original mass?

Short answer

It will take approximately \(53.65\) days for a \(500 \mathrm{mg}\) sample of Iodine-131 to decay into \(1 \%\) of its original mass.

Step-by-step solution

Determining the decay constant

Since we are given the half-life of Iodine-131, we can use the following relationship between the decay constant \(\lambda\) and the half-life \(t_{1/2}\): \[\lambda = \frac{\ln(2)}{t_{1/2}}\] Plugging in the given value of the half-life, which is 8.04 days, we get: \[\lambda = \frac{\ln(2)}{8.04}\] Calculate the decay constant: \[\lambda \approx 0.0863 \text{ per day}\]

Setting up the radioactive decay equation

Now that we have the decay constant, we can set up the radioactive decay equation with the given initial mass (\(500 \mathrm{mg}\)) and desired final mass (\(1 \% \text{ of the initial mass}\)): \[N(t) = N_0 e^{-\lambda t}\] Where \(N(t) = 500 \times 0.01 = 5 \mathrm{mg}\) and \(N_0 = 500 \mathrm{mg}\). Plug in these values into the equation: \[5 = 500 e^{-0.0863 t}\]

Solve for time

To solve for \(t\), we'll first need to divide by \(500\) on both sides of the equation: \[\frac{5}{500} = e^{-0.0863 t}\] Then take the natural logarithm of both sides: \[\ln{\frac{1}{100}} = -0.0863 t\] Now, divide by \(-0.0863\) to isolate \(t\): \[t = \frac{\ln{\frac{1}{100}}}{-0.0863}\] Calculate the time it takes to decay to \(1 \%\) of the initial mass: \[t \approx 53.65 \text{ days}\] It will take approximately \(53.65\) days for a \(500 \mathrm{mg}\) sample of Iodine-131 to decay into \(1 \%\) of its original mass.

Key concepts

Iodine-131

Iodine-131, often symbolized as \(^{131}\mathrm{I}\), plays a critical role in nuclear medicine. It is particularly utilized in the treatment of hyperthyroidism due to its radioactive properties. Iodine-131 emits beta particles and gamma rays, which can help reduce the overactive thyroid gland by targeting thyroid cells with precision. It is derived from nuclear fission and is used in both diagnostic and therapeutic applications in medicine.
It has the magical ability to concentrate in the thyroid gland, allowing selective treatment which makes it less harmful to other body tissues. This attribute makes it an invaluable tool in medicine. Understanding its decay and how it operates over time is crucial for its application in treatments.

Half-life

The concept of half-life is a cornerstone in radioactivity where it represents the time taken for half of the radioactive atoms in a sample to decay. For Iodine-131, its half-life is exactly 8.04 days. This implies that after 8.04 days, half of any amount of \(^{131}\mathrm{I}\) would have decayed. It is important because:

• It helps determine how long a radioactive sample remains active.
• Pivotal in calculating the time a sample is effective for medical purposes.
• Aids in predicting the decay pattern over time.

Half-life is an intuitive way to measure the rate of radioactive decay and helps medical professionals decide the proper dosage and treatment schedule for \(^{131}\mathrm{I}\) in patients.

Decay constant

The decay constant, symbolized as \(\lambda\), provides a numerical representation of the rate of radioactive decay, essentially demonstrating how quickly a sample will decay. For Iodine-131, the decay constant is calculated using its half-life with the equation:\[\lambda = \frac{\ln(2)}{t_{1/2}}\] This formula highlights the relationship between half-life and the decay rate, implying a direct calculation method where:

• \(\ln(2)\) is the natural logarithm of 2.
• \(t_{1/2}\) is the half-life, given as 8.04 days for Iodine-131.

In this context, the decay constant for Iodine-131 is approximately 0.0863 per day, indicating a gradual decay process, suitable for prolonged therapeutic use.

Radioactive decay equation

The radioactive decay equation expresses how a radioactive substance decreases over time. The general formula is given by:\[N(t) = N_0 e^{-\lambda t}\]where \(N(t)\) is the amount of substance remaining at time \(t\), \(N_0\) is the initial amount, and \(\lambda\) is the decay constant. For the problem regarding Iodine-131:- Initial mass \(N_0\) is 500 mg.- Desired mass \(N(t)\) is 1% of 500 mg, which equals 5 mg.By solving the equation, we understand the time it takes for a given mass of Iodine-131 to decay to a desired level. This computation is crucial in determining treatment duration in medical settings. Hence, understanding how the mass of a radioactive substance reduces with time, allows medical professionals to plan treatment procedures appropriately.