Problem 33 - Physics

Question

A device used in radiation therapy for cancer contains \(0.50 \mathrm{~g}\) of cobalt \(\frac{60}{27} \mathrm{Co}\) \((59.933819 \mathrm{u}) .\) The half-life of \(\underset{27}{60} \mathrm{C} \circ\) is \(5.27 \mathrm{yr}\). Determine the activity of the radioactive material.

Step-by-Step Solution

Answer

The activity of the radioactive cobalt sample is calculated using decay constant and number of nuclei to find the total radioactive activity.

Step 1: Understanding the Activity Formula

Activity (A) of a radioactive sample is given by the formula: \[ A = \lambda N \]where \( \lambda \) is the decay constant and \( N \) is the number of radioactive nuclei.

Step 2: Calculate the Decay Constant

The decay constant \( \lambda \) can be calculated using the formula: \[ \lambda = \frac{\ln(2)}{T_{1/2}} \] where \( T_{1/2} = 5.27 \text{ years} = 5.27 \times 365 \times 24 \times 3600 \text{ seconds} \) for this problem. Compute \( \lambda \).

Step 3: Convert Mass to Number of Nuclei

The number of nuclei \( N \) can be calculated from the mass of radioactive material using: \[ N = \frac{m}{M} \times N_A \]where \( m = 0.50 \text{ grams} \), \( M = 59.933819 \text{ g/mol} \), and \( N_A = 6.022 \times 10^{23} \text{ mol}^{-1} \) is Avogadro's number. Compute \( N \).

Step 4: Calculate Radioactive Activity

Substitute \( \lambda \) and \( N \) into the formula for activity: \[ A = \lambda N \]This will give the activity \( A \), typically in becquerels (Bq). Compute \( A \) to find the activity of the sample.

Key Concepts

Decay Constant

In the world of radioactive decay, the decay constant \( \lambda \) plays a crucial role. It is a constant that characterizes the rate at which a radioactive substance undergoes decay. You can think of it as a clock ticking against the radioactive material, dictating the speed of the decay process.

The decay constant is derived from the equation \( \lambda = \frac{\ln(2)}{T_{1/2}} \), where \( T_{1/2} \) represents the half-life of the substance. Here, \( \ln(2) \) is the natural logarithm of 2, a mathematical constant approximately equal to 0.693. By calculating \( \lambda \), we can predict how quickly a radioactive isotope will lose half of its quantity, which is vital in many scientific applications.

• A higher decay constant means a faster rate of decay.
• It provides essential data for calculating radioactive activity, which is crucial for understanding a substance's potency.

Half-Life

Half-life, denoted as \( T_{1/2} \), is a core concept in understanding radioactive decay. It is defined as the time taken for half of the radioactive nuclei in a sample to decay. This parameter helps in predicting how long a radioactive material can be effectively used or remain dangerous.

The relationship between half-life and decay constant is given by \( T_{1/2} = \frac{\ln(2)}{\lambda} \). Knowing either \( T_{1/2} \) or \( \lambda \) allows us to determine the other, providing insights into both the speed of decay and the longevity of a radioactive sample.

• A longer half-life means the substance remains active over a more extended period.
• The half-life is independent of the initial amount of the substance.

This is especially significant in medical and nuclear applications, such as radiation therapy in cancer treatment, where precise timing and safety are paramount.

Avogadro's Number

Avogadro's number \( N_A \) is one of the fundamental constants in chemistry and physics, defined as \( 6.022 \times 10^{23} \text{ mol}^{-1} \). It is the number of constituent particles, usually atoms or molecules, in one mole of a substance.

In the context of calculating the number of radioactive nuclei \( N \) in a sample when given its mass and molar mass, Avogadro's number is essential. The formula used is \( N = \frac{m}{M} \times N_A \), where \( m \) is the sample's mass and \( M \) is its molar mass.

• Avogadro's number bridges the gap between the atomic scale and the macroscopic scale.
• It enables the conversion of a substance's mass to the number of atoms, facilitating understanding of chemical reactions at the molecular level.

For radioactive materials, knowing how many nuclei are present is crucial for predicting how the sample's activity changes over time.

Radioactive Activity

Radioactive activity refers to the rate at which a sample of radioactive material decays, measured in decays per second. The unit of activity is the becquerel (Bq), where 1 Bq equals one decay per second.

The activity \( A \) is computed using the formula \( A = \lambda N \), combining the decay constant \( \lambda \) and the number of radioactive nuclei \( N \). Understanding and calculating radioactive activity is vital in radiation therapy, nuclear power, and safety protocols as it gives a measurable indicator of how much radiation a substance emits.

• A higher activity indicates a greater number of decays per second.
• Activity helps in determining the "strength" or "potency" of a radioactive source.

For applications like medical treatments, precise control over radioactive activity ensures effectiveness and safety, making it a cornerstone concept in the usage of radioactive materials.