Problem 13 - Calculus with Concepts in Calculus

Question

Iodine 131 , which has been used for treating cancer of the thyroid gland, is also used to detect leaks in water pipes. It has a half-life of \(8.14\) days. Suppose a water company wishes to use 100 milligrams of iodine 131 to search for a leak and must take delivery 2 days before using it. How much iodine 131 should be purchased?

Step-by-Step Solution

Answer

The company should purchase approximately 119 mg of iodine 131.

Step 1: Understanding the half-life concept

A half-life is the time required for a quantity to reduce to half its initial value. For iodine 131, every 8.14 days, its amount reduces to half. This means, given some initial quantity, after 8.14 days, only half of that material remains active.

Step 2: Identify the time before usage

The company needs the iodine 131 two days before usage, meaning during this period, the substance will decay, reducing the amount of active iodine available by the time it's used.

Step 3: Calculate decay constant

The decay constant \( k \) is used in exponential decay calculations and can be found using the relation \( k = \frac{\ln(2)}{\text{half-life}} \). Calculate \( k \) for iodine 131: \( k = \frac{\ln(2)}{8.14} \approx 0.0852\, \text{days}^{-1}.\)

Step 4: Set up the decay formula

The decay formula is \( N(t) = N_0 e^{-kt} \), where \( N(t) \) is the amount after time \( t \), and \( N_0 \) is the initial amount. You need to reverse-engineer the amount to find \( N_0 \), knowing the remaining amount is 100 mg after 2 days.

Step 5: Plug values into the decay formula

Given \( N(t) = 100 \) mg and \( t = 2 \) days, calculate the initial amount \( N_0 \). Substitute the known values: \[ 100 = N_0 e^{-0.0852 \times 2}. \]

Step 6: Solve for initial amount

Rearrange the formula to solve for \( N_0 \): \[ N_0 = \frac{100}{e^{-0.1704}} \approx \frac{100}{0.8437} \approx 118.55. \] Thus, the company should purchase approximately 119 mg of iodine 131.

Key Concepts

Exponential Decay

In processes involving exponential decay, a quantity decreases at a rate proportional to its current value. This is characteristic of radioactive substances. The change is rapid at first and then slows down over time.
For example, if you start with a certain amount, after one half-life, only half of it remains. After another half-life, you're left with a quarter of the original amount. This keeps happening until what's left is almost negligible.
Exponential decay is described by the formula:

• \( N(t) = N_0 e^{-kt} \)

- \( N(t) \) is the amount at time \( t \).- \( N_0 \) is the initial amount.- \( k \) is the decay constant.
This formula helps calculate how much of a substance remains after a certain period. In the case of iodine 131, this allows for predicting how much should be purchased for effective use after accounting for decay.

Iodine 131

Iodine 131 is a radioactive isotope commonly used in medical and industrial applications due to its unique properties. Its radioactive nature makes it useful for tracing and detection.
This isotope is particularly popular in the treatment of thyroid cancer because it can be absorbed by the thyroid, where it delivers a targeted dose of radiation. Apart from medical uses, it's also used for detecting leaks in pipelines, thanks to its detectable radioactive signature.
Understanding iodine 131's behavior is crucial, especially its half-life, which is 8.14 days. This means every 8.14 days, only half of any given quantity will remain. For practical applications, this timing must be taken into account to ensure the substance is effective when used. Planning the purchase and usage around this decay is essential.

Decay Constant

The decay constant, denoted by \( k \), is a fundamental part of the exponential decay formula. It determines how quickly a substance like iodine 131 decays over time.
For radioactive substances, \( k \) can be calculated using the half-life with the formula:

• \( k = \frac{\ln(2)}{\text{half-life}} \)

This formula helps quantify the rate at which a substance decays. For iodine 131, with a half-life of 8.14 days, the decay constant is approximately 0.0852 days\(^{-1}\).
Knowing \( k \) is essential because it allows for precise calculations of how much substance remains after a given time. This information helps in planning and decision-making, ensuring the right amount of iodine 131 is available when needed. Properly understanding and using the decay constant is key to working with radioactive materials.