Problem 13 - Precalculus with Calculus Previews

Question

Iodine-131, used in nuclear medicine procedures, is radioactive and has a half-life of 8 days. Find the decay constant \(k\) for iodine-131. If the amount remaining of an initial sample after \(t\) days is given by the exponential model \(A(t)=A_{0} e^{k t},\) how long will it take for \(95 \%\) of the sample to decay?

Step-by-Step Solution

Answer

The decay constant is \( k \approx 0.086625 \), and it takes about 34.57 days for 95% of the iodine-131 to decay.

Step 1: Understanding Half-life and Decay Constant

The half-life of iodine-131 is 8 days. The decay constant \( k \) is related to the half-life \( T_{1/2} \) by the formula \( k = \frac{\ln(2)}{T_{1/2}} \). First, calculate the decay constant \( k \) for iodine-131 using this relationship.

Step 2: Calculating the Decay Constant

Plug \( T_{1/2} = 8 \) into the formula: \( k = \frac{\ln(2)}{8} \). Calculate \( k \) using \( \ln(2) \approx 0.693 \). So, \( k \approx \frac{0.693}{8} \approx 0.086625 \).

Step 3: Setting Up the Exponential Decay Equation

We need to find the time \( t \) when 95% of the sample has decayed, meaning 5% of the sample remains. Thus, \( A(t) = 0.05A_{0} \). Substitute this into the decay formula, \( 0.05A_{0} = A_{0} e^{kt} \). Simplifying gives \( 0.05 = e^{kt} \).

Step 4: Solving for Time

Take the natural logarithm of both sides: \( \ln(0.05) = kt \). Use the decay constant \( k \approx 0.086625 \), so \( \ln(0.05) \approx -2.9957 \). Thus, \( t = \frac{-2.9957}{0.086625} \approx 34.57 \). This means it takes approximately 34.57 days for 95% of the sample to decay.

Key Concepts

Understanding Half-life in Radioactive Decay

The half-life is a crucial concept in understanding radioactive decay. It is the time required for half of the radioactive nuclei in a sample to decay. During each half-life period, the number of remaining radioactive atoms reduces by half. This is a constant property of a substance and allows for predicting how quickly a substance will decay.

For instance, iodine-131 has a half-life of 8 days. This means if you start with 100 grams of iodine-131, after 8 days, only 50 grams would remain radioactive. After another 8 days (16 days total), only 25 grams will remain active. This pattern continues, which is why understanding half-life helps in calculating the remaining amount of a substance over time.

Decay Constant and Its Calculation

The decay constant, often denoted as \( k \), is a parameter that indicates the rate or speed at which a radioactive substance decays. It is mathematically linked to the half-life through the formula:

• \( k = \frac{\ln(2)}{T_{1/2}} \)

Here, \( \ln(2) \approx 0.693 \) and \( T_{1/2} \) is the half-life in days. For iodine-131, with a half-life of 8 days, the decay constant can be calculated as:

• \( k = \frac{0.693}{8} \)
• \( k \approx 0.086625 \)

The decay constant provides a clearer expression of the decay rate, making it easier to model and predict the decay behavior over time.

Basics of Radioactive Decay

Radioactive decay is the process by which an unstable atomic nucleus loses energy by radiation. Such decay results in the transformation of an element into a different element or a different isotope. This natural process is characterized by the spontaneous emission of particles and energy.

• Exponential Nature: The decay of radioactive substances is generally exponential.
• Mathematical Model: It’s commonly modeled by the exponential decay formula \( A(t) = A_{0} e^{kt} \), where \( A_{0} \) is the initial amount, \( k \) is the decay constant, and \( t \) is time.

This mathematical representation helps in calculating the remaining quantity of a substance after a specific time, predicting things like how long it will take for a certain proportion of the substance to decay.

Specific Example: Iodine-131

Iodine-131 is a radioactive isotope commonly used in nuclear medicine, particularly in diagnostic imaging and radiotherapy. It’s favored in medical applications due to its ability to emit both beta particles and gamma rays, providing therapeutic and imaging benefits.

• Decay Properties: Having a half-life of 8 days, iodine-131 requires careful handling in medical settings to ensure that it decays to a level that’s safe for patients and healthcare workers.
• Practical Application: The known half-life allows medical professionals to calculate how long the isotope will stay active in the body, aiding in planning treatment cycles and ensuring effective and safe dosage.

Understanding these characteristics of iodine-131, together with its exponential decay behavior, is vital for its controlled and beneficial application in medical science.