Question

A sample of fluorine-20 has an activity of \(2.44 \mathrm{mCi}\). If its half- life is \(11.0 \mathrm{~s}\), what is its activity after \(50.0 \mathrm{~s}\) ?

Short answer

Activity is approximately 0.105 mCi after 50 seconds.

Step-by-step solution

Understand the Formula for Radioactive Decay

The activity of a radioactive sample decreases over time according to the formula \( A = A_0 \cdot e^{-\lambda t} \), where \( A \) is the activity at time \( t \), \( A_0 \) is the initial activity, \( \lambda \) is the decay constant, and \( t \) is the time elapsed. To use this formula, we first need to find the decay constant \( \lambda \).

Calculate the Decay Constant

The decay constant \( \lambda \) is related to the half-life \( t_{1/2} \) by the formula \( \lambda = \frac{\ln(2)}{t_{1/2}} \). For fluorine-20, the half-life is \( 11.0 \text{ s} \), so we calculate \( \lambda = \frac{\ln(2)}{11.0} \approx 0.063 \text{ s}^{-1} \).

Calculate the Activity at 50 Seconds

Using the decay formula, \( A = A_0 \cdot e^{-\lambda t} \), substitute \( A_0 = 2.44 \text{ mCi} \), \( \lambda = 0.063 \text{ s}^{-1} \), and \( t = 50 \text{ s} \) to find the activity after 50 seconds. This gives:\[ A = 2.44 \cdot e^{-0.063 \cdot 50} \approx 2.44 \cdot e^{-3.15} \approx 2.44 \cdot 0.043 \approx 0.105 \text{ mCi} \].

Confirm the Calculation

Verify the steps were correctly followed for calculation. The initial activity \( A_0 = 2.44 \text{ mCi} \), the time given is \( t = 50 \text{ s} \), and correct calculation updates provide the final activity of approximately \( 0.105 \text{ mCi} \).

Key concepts

Activity of Radioactive Samples

Radioactive samples naturally emit radiation through the process of decay. The activity of a radioactive sample represents how many decays occur per second and is essentially a measure of the sample's strength or power.
It is expressed in units known as becquerels (Bq) or millicuries (mCi). The higher the activity, the more decay events happening per second. Over time, as the sample decays, the activity decreases. This is because there are fewer radioactive nuclei left to decay.

To calculate the current activity from an initial activity, we use the formula:

• \( A = A_0 \cdot e^{-\lambda t} \)

Where:

• \( A \) is the activity at time \( t \)
• \( A_0 \) is the initial activity
• \( \lambda \) is the decay constant
• \( t \) is the time elapsed since the initial measurement

This exponential formula shows that as time goes on, the activity decreases at a rate dependent on the decay constant, \( \lambda \).

Half-Life Calculations

The concept of half-life is a key understanding in radioactive decay. Half-life, denoted as \( t_{1/2} \), is the time required for half of the radioactive nuclei in a sample to decay.
This means that after one half-life, the activity of a radioactive sample will be reduced to half its original value.

The half-life is a constant for a given isotope. It is an intrinsic property and is independent of the initial amount or the activity of the sample. However, it helps us calculate how an activity will change over time, especially useful when we'd like to predict the remaining activity after a certain number of half-lives.

Knowing the half-life helps in determining the decay constant \( \lambda \) using the formula:

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

This formula allows us to connect the concept of half-life with the exponential decay behaviour of radioactive materials.

Decay Constant

The decay constant \( \lambda \) is a crucial factor in describing the rate of radioactive decay of a substance. It indicates the probability per unit of time that a nucleus will decay.
The decay constant is linked to how quickly or slowly a sample will lose its radioactivity. A larger decay constant means a faster decay process, and vice versa.

To determine the decay constant, you can use the relationship with the half-life of the substance:

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

This relation comes from the natural logarithmic basis of exponential decay, where \( \ln(2) \approx 0.693 \). This constant provides a cumulative decay probability over a unit of time, thereby closely connecting with both the activity formula and the half-life formula.
Understanding the decay constant helps in predicting how quickly a sample's activity will decline, particularly useful in radiation-related fields or nuclear medicine.