Question

The only major use of iodine in the human body is in the production of certain hormones by the thyroid gland, and iodine from the diet concentrates in this area of the body. Iodine-131 is used in the diagnosis and treatment of thyroid disease and has a half-life of 8 days. If a patient with thyroid disease consumes a sample of \(\mathrm{Na}^{131} \mathrm{I}\) containing \(10 \mu \mathrm{g}\) of \(^{131} \mathrm{I},\) how long will it take for the amount of \(^{131} \mathrm{I}\) to decrease to approximately \(1 / 1000\) of the original amount?

Short answer

It will take approximately \(26.57\) days for the amount of \(\mathrm{^{131}I}\) to decrease to approximately \(1 / 1000\) of the original amount.

Step-by-step solution

Identify the given information

We are given: - Initial amount of Iodine-131 \((\mathrm{^{131}I})\): \(10 \mu \mathrm{g}\) - Half-life of Iodine-131 \((\mathrm{^{131}I})\): 8 days - We want to find the time it takes for the amount of Iodine-131 to decrease to \(1/1000\) of the initial amount.

Use the radioactive decay formula

We can use the radioactive decay formula to find the time (t) it takes for the amount of Iodine-131 to decrease to \(1/1000\) of the original amount: \(N_t = N_0 \times \left(\frac{1}{2}\right)^{\frac{t}{t_{1/2}}}\) Where: \(N_t\): Final amount of Iodine-131 after time t \(N_0\): Initial amount of Iodine-131 (10 μg) \(t_{1/2}\): Half-life of Iodine-131 (8 days) \(t\): Time in days

Solve for T

We're given that \(N_t\) should be equal to \(1/1000\) of \(N_0\). Therefore: \(\frac{1}{1000} N_0 = N_0 \times \left(\frac{1}{2}\right)^{\frac{t}{8}}\) Now, we'll solve for t: \(\frac{1}{1000} = \left(\frac{1}{2}\right)^{\frac{t}{8}}\) Take the logarithm of both sides (base 2): \(\log_2 \left(\frac{1}{1000}\right) = \log_2 \left(\left(\frac{1}{2}\right)^{\frac{t}{8}}\right)\) Apply the power rule of logarithms: \(-10 = \frac{t}{8}\) Now, solve for t: \(t = -10 \times 8\) \(t = -80\) days

Address the negative value

We obtain a negative value for the time, which is impossible. This is because we're looking for the time taken for I-131 to decrease to 1/1000 of its initial value, which is normally smaller than the half-life. Therefore, we should use a positive exponent in our decay formula: \(\frac{N_t}{N_0} = \left(\frac{1}{2}\right)^{\frac{t}{t_{1/2}}}\) \(\frac{1}{1000} = \left(\frac{1}{2}\right)^{\frac{t}{8}}\) Repeat the previous steps: \(\log_2 \left(\frac{1}{1000}\right) = \log_2 \left(\left(\frac{1}{2}\right)^{\frac{t}{8}}\right)\) \(-10 \log_2(10) = \frac{t}{8}\) \(t = -8 \times 10 \log_2(10)\) \(t \approx 26.57\) days

Final answer

It will take approximately \(26.57\) days for the amount of \(\mathrm{^{131}I}\) to decrease to approximately \(1 / 1000\) of the original amount.

Key concepts

Iodine-131

Iodine-131, often abbreviated as \(^{131}\text{I}\), is a radioactive isotope of iodine. It has a significant role in medical applications, particularly concerning thyroid health. As a point of curiosity, iodine is a critical element utilized by the human body primarily for hormone production within the thyroid gland.
This isotope is utilized both in diagnostic procedures and treatments of various thyroid conditions, such as hyperthyroidism and thyroid cancer. Due to its radioactive nature, it helps healthcare providers target and visualize specific thyroid problems effectively.
Importantly, the radioactive properties of \(^{131}\text{I}\) allow it to emit radiation that can damage or destroy cells, which is why it's used as a targeted therapy in certain thyroid conditions. However, its use requires careful calculation and monitoring due to its radioactive nature and potential effects.

Half-life Calculation

The half-life of a substance is a crucial concept in understanding radioactive decay. Simply put, the half-life is the time required for half of the radioactive atoms in a sample to decay. For iodine-131, this period is 8 days. This means that every 8 days, half of the iodine-131 in a sample will have decayed, transforming into a non-radioactive or a different element.
To calculate the time it takes for a substance to decrease to a particular amount, we use the formula:\[N_t = N_0 \times \left(\frac{1}{2}\right)^{\frac{t}{t_{1/2}}}\]
In this formula:

• \(N_t\): final quantity of the substance after time \(t\)
  
• \(N_0\): initial quantity
  
• \(t_{1/2}\): half-life of the substance
  
• \(t\): time
  

This formula helps determine how long it will take for a substance to reduce to a specific fraction of its original amount. Understanding logarithmic functions simplifies solving this exponential decay formula.

Thyroid Treatment

Thyroid treatment through radioactive iodine, such as \(^{131}\text{I}\), is a common therapy for certain thyroid conditions. These treatments aim to reduce thyroid tissue in diseases like Graves' disease or to eliminate remaining cancerous thyroid cells post-surgery.
When a patient undergoes treatment with radioactive iodine, they swallow a prescribed dose. The iodine naturally travels to the thyroid gland, where it becomes concentrated. The radiation emitted by \(^{131}\text{I}\) works to gradually reduce the size and function of the overactive thyroid tissue.
This method allows for a targeted operation without invasive procedures. However, the radioactivity requires careful handling and monitoring, and doctors need to calculate the appropriate dosage accurately to ensure the undesired tissue is affected while minimizing exposure to healthy tissues.

Logarithmic Functions

Logarithmic functions are mathematical functions that help us understand exponential processes like radioactive decay more effectively. In our context, logarithms simplify solving equations involving exponential decay.
When dealing with the formula for radioactive decay:\[1/1000 = \left(1/2\right)^{\frac{t}{t_{1/2}}}\],
logarithms allow us to solve for the time \(t\).

• First, we express the equation in logarithmic form using a base that reflects our exponential base, such as base 2.
  
• Then, you apply the logarithmic property that allows you to bring the exponent down as a coefficient, allowing for straightforward calculation of \(t\).
  

This technique showcases the practical use of logarithms by converting seemingly complex exponential equations into more manageable forms, which is crucial when calculating how long radioactive substances like iodine-131 take to significantly diminish in quantity.