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Chapter 9: Problem 93

A \(0.4 \mathrm{mg}\) initial sample of radioactive iodine- \(123,\) used to treat thyroid cancer, decreases to \(0.1 \mathrm{mg}\) in 26.4 hours. What is the half-life of iodine- \(123 ?\)

Short Answer

Expert verified
The half-life of iodine-123 is approximately 13.23 hours.

Step by step solution

01

Understand the Problem

We're given an initial mass of radioactive iodine-123, which decays over time. The problem asks us to find the half-life, which is the time it takes for half of the substance to decay.
02

Use the Decay Formula

The decay of a radioactive substance can be modeled by the equation \( m(t) = m_0 \times e^{-kt} \), where \( m_0 \) is the initial mass, \( m(t) \) is the mass at time \( t \), and \( k \) is the decay constant. Let's identify what we have: \( m_0 = 0.4 \) mg, \( m(t) = 0.1 \) mg, and \( t = 26.4 \) hours.
03

Calculate the Decay Constant

Substitute the known values into the decay formula: \( 0.1 = 0.4 \times e^{-26.4k} \). First, divide both sides by 0.4 to isolate the exponential term: \( e^{-26.4k} = \frac{0.1}{0.4} = 0.25 \). Take the natural logarithm of both sides to solve for \( k \): \( -26.4k = \ln(0.25) \). Therefore, \( k = \frac{-\ln(0.25)}{26.4} \approx 0.0524 \text{ hour}^{-1} \).
04

Determine Half-Life Using Decay Constant

The half-life \( T_{1/2} \) is given by the formula \( T_{1/2} = \frac{\ln(2)}{k} \). Substitute the decay constant \( k \approx 0.0524 \) into this formula: \( T_{1/2} = \frac{\ln(2)}{0.0524} \approx 13.23 \text{ hours} \).
05

Verify and Conclude

Check each calculation to ensure that no mistakes were made. The half-life calculation appears consistent with the given decay from 0.4 mg to 0.1 mg over 26.4 hours.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Half-Life Calculation
Calculating the half-life of a radioactive substance is essential in understanding how long it takes for half of the material to undergo decay. The concept of half-life is widely applicable in fields like medicine, archaeology, and nuclear chemistry.

To determine the half-life, we use the formula:
  • The half-life ( \( T_{1/2} \) ) is the time it takes for half of the initial substance to decay.
  • The formula is \( T_{1/2} = \frac{\ln(2)}{k} \) , where \( k \) is the decay constant.
In the solved exercise, iodine-123's decay from 0.4 mg to 0.1 mg in 26.4 hours helped us calculate the half-life as approximately 13.23 hours.

This calculation process gives us a time frame for how a substance decreases, allowing proper time management for its use or replacement.
Decay Constant
The decay constant is a crucial component in radioactive decay calculations. It represents how rapidly a radioactive substance undergoes decay and is denoted by \( k \).

Here's how to calculate the decay constant:
  • Use the decay formula: \( m(t) = m_0 \times e^{-kt} \) , where \( m_0 \) is the initial mass and \( m(t) \) is the mass at time \( t \).
  • Re-arrange it to solve for \( k \).
  • Take the natural logarithm of both sides to find \( k \).
In the example given, by plugging in our known values, we found \( k \approx 0.0524 \text{ hour}^{-1} \).

A higher value of \( k \) indicates a faster decay, giving insights into the longevity and stability of the radioactive element.
Exponential Decay
Exponential decay describes how quantities decrease over time at a rate proportional to their current value. It is a common phenomenon observed in natural sciences and is defined by the exponential decay equation:

\[ m(t) = m_0 \times e^{-kt} \]

In this equation:
  • \( m(t) \) conveys the mass of the substance at time \( t \).
  • \( m_0 \) represents the initial mass or starting quantity of the substance.
  • The expression \( e^{-kt} \) represents the exponential decay factor determined by \( k \), the decay constant.
In the context of the example, the iodine-123 sample decreases according to this equation, allowing prediction of its quantity at any given time. The initial condition, coupled with the decay constant, demonstrates how quickly substances decay, crucial for real-world applications like assessing radioactive exposure.

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