Problem 99 - Chemistry: An Atoms-Focused Approach

Question

A 1.00 mg sample of \(^{192}\) Ir was inserted into the artery of a heart patient. After 30 days, 0.756 mg remained. What is the half-life of \(^{192} \mathrm{Ir} ?\)

Step-by-Step Solution

Answer

Answer: The half-life of Iridium-192 is approximately 76.36 days.

Step 1: Understand the radioactive decay formula

The formula for radioactive decay is given by: \(N(t) = N_0 e^{-\lambda t}\), where \(N(t)\) is the remaining mass of the isotope at time \(t\), \(N_0\) is the initial mass, \(\lambda\) is the decay constant, and \(t\) is the time.

Step 2: Write down the given information

We are given the initial mass \(N_0 = 1.00 \thinspace mg\), the remaining mass after 30 days \(N(t) = 0.756 \thinspace mg\), and the time \(t = 30 \thinspace days\).

Step 3: Rearrange the decay formula for the decay constant

Divide both sides of the radioactive decay formula by \(N_0\), then we have: \(\frac{N(t)}{N_0} = e^{-\lambda t}\) Now, we want to isolate \(\lambda\), so take the natural logarithm on both sides: \(ln \left(\frac{N(t)}{N_0}\right) = -\lambda t\) Finally, solve for the decay constant \(\lambda\) by dividing both sides by \(-t\): \(\lambda = \frac{ln\left(\frac{N(t)}{N_0}\right)}{-t}\)

Step 4: Calculate the decay constant

Using the values given in the problem, let's substitute them into the decay constant formula: \(\lambda = \frac{ln\left(\frac{0.756}{1.00}\right)}{-30}\) \(\lambda \approx 0.009088 \thinspace days^{-1}\)

Step 5: Relate the decay constant to the half-life

The half-life (\(T_{1/2}\)) is related to the decay constant by the formula: \(T_{1/2} = \frac{ln(2)}{\lambda}\)

Step 6: Calculate the half-life of Iridium-192

Now, substitute the decay constant found in Step 4 into the half-life formula: \(T_{1/2} = \frac{ln(2)}{0.009088}\) \(T_{1/2} \approx 76.36 \thinspace days\) The half-life of Iridium-192 is approximately 76.36 days.

Key Concepts

Half-Life Calculation

The half-life of a radioactive substance is the time it takes for half of the initial quantity of the substance to decay. This is a key concept in understanding how radioactive decay works.
To calculate the half-life, we often use the relation between the decay constant and half-life, which is expressed by the formula:

• \( T_{1/2} = \frac{\ln(2)}{\lambda} \)

Here, \( \lambda \) is the decay constant, which we will discuss further in the next section. What makes the half-life concept interesting is that it remains constant regardless of how much of the substance is present.
In the context of the problem mentioned, the half-life of Iridium-192 was calculated using the decay constant derived from the data given. Once the decay constant was found to be approximately \(0.009088 \thinspace days^{-1}\), it was plugged into the half-life formula to find that the half-life of Iridium-192 is about 76.36 days.
This calculation shows that even if we start with a different mass, the half-life for Iridium-192 will remain the same, 76.36 days, as long as the environmental conditions remain unchanged.

Decay Constant Formula

The decay constant, denoted as \( \lambda \), is a measure of the probability of decay per unit time for a radioactive substance. It plays a crucial role in the process of determining the rate of decay.
The decay constant can be calculated using the following formula:

• \( \lambda = \frac{\ln\left(\frac{N(t)}{N_0}\right)}{-t} \)

Where:

• \( N(t) \) is the remaining mass of the radioactive sample after time \( t \)


• \( N_0 \) is the initial mass of the substance


• \( t \) is the time elapsed

By applying this formula to our problem, with the values given:

• Initial mass, \( N_0 = 1.00 \) mg


• Remaining mass after 30 days, \( N(t) = 0.756 \) mg


• Time, \( t = 30 \) days

We find that \( \lambda \approx 0.009088 \thinspace days^{-1} \).
The decay constant gives us insight into how quickly or slowly a particular isotope decays. A larger decay constant indicates a faster rate of decay, while a smaller one means the isotope decays more slowly.

Iridium-192 Half-Life

Iridium-192 is a radioactive isotope commonly used in industrial radiography and medical applications for treatment. Understanding its half-life is fundamental to its safe and effective application.
The problem statement you were given allowed the calculation of Iridium-192's half-life using the decay constant. This was found to be approximately 76.36 days.
Here’s why knowing the half-life is important for Iridium-192:

• It helps industries and healthcare providers schedule the replacement of their radioactive sources.


• It provides insights into how long the isotope will remain active and effective for its intended purpose.


• A clear half-life calculation ensures safety and proper usage by anticipating how long the isotope will maintain its radioactive potency.

Because of its significant role in various professional fields, precise knowledge of Iridium-192's half-life helps in developing protocols and safety measures. It ensures that the isotope remains effective for its required use period while also protecting those who work with it from unnecessary radiation exposure.