Problem 20 - University Physics with Modern Physics

Question

BIO Radioactive isotopes used in cancer therapy have a "shelf-life," like pharmaceuticals used in chemotherapy. Just after it has been manufactured in a nuclear reactor, the activity of a sample of \(^{60} \mathrm{Co}\) is 5000 \(\mathrm{Ci} .\) When its activity falls below \(3500 \mathrm{Ci},\) it is considered too weak a source to use in treatment. You work in the radiology department of a large hospital. One of these \(^{60} \mathrm{Co}\) sources in your inventory was manufactured on October \(6,2004\) . It is now April \(6,2007\) . Is the source still usable? The half-life of \(^{60} \mathrm{Co}\) is 5.271 years.

Step-by-Step Solution

Answer

Yes, the source is still usable with approximately 3822 Ci of activity.

Step 1: Calculate the Time Elapsed

First, we need to calculate the time elapsed since the manufacture of the radioactive source. The source was manufactured on October 6, 2004, and we need to check its status on April 6, 2007. From October 6, 2004, to October 6, 2005 is 1 year, to October 6, 2006 is 2 years, and to October 6, 2007, would be 3 years. Since it is only April 6, 2007, we must calculate how many months into 2007 this is, which is 6 months. Thus, a total of 2 years and 6 months have elapsed, which is 2.5 years.

Step 2: Convert Time Elapsed to Fraction of Half-Life

With the half-life of the isotope being 5.271 years, we divide the elapsed time by the half-life to find out how many half-lives have passed. This is calculated as \( \frac{2.5}{5.271} \), which gives approximately 0.4745 half-lives.

Step 3: Calculate the Remaining Activity

Using the formula for radioactive decay \( A = A_0 \left(\frac{1}{2}\right)^{n} \), where \( A_0\) is the initial activity, \( A\) is the activity after time \( t\), and \( n\) is the number of half-lives that have passed, we substitute our values: \( A = 5000 \left(\frac{1}{2}\right)^{0.4745}\). Solve this to find \( A ≈ 3822 \, \text{Ci}\).

Step 4: Determine if the Source is Still Usable

Compare the current activity level to the threshold of usability (3500 Ci). Since the calculated activity is approximately 3822 Ci, which is greater than 3500 Ci, this means that the source is still usable.

Key Concepts

Cobalt-60

Cobalt-60 is a radioactive isotope that is commonly used in medical treatments, industrial applications, and scientific research. It is a synthetic element, which means it is not found in nature but is produced artificially in a nuclear reactor. The process involves bombarding cobalt-59, a naturally occurring stable isotope, with neutrons to create cobalt-60.
Cobalt-60 is particularly important in the field of medicine because of its radioactive properties. It emits gamma rays, which can penetrate deeply into tissues. This makes it a valuable tool in radiation therapy for cancer treatment. The gamma rays emitted by cobalt-60 can target and destroy cancer cells while causing minimal damage to the surrounding healthy tissue. This targeted approach helps in reducing side effects compared to more extensive radiation therapies.
As a radioactive source, it is important to handle cobalt-60 with care, ensuring that proper safety measures are in place to protect both healthcare workers and patients. Additionally, as cobalt-60 decays, its effectiveness for certain applications, such as in cancer therapy, decreases over time, necessitating regular monitoring of its activity levels.

Half-Life Calculation

The concept of half-life is crucial in understanding the decay process of radioactive isotopes like cobalt-60. The half-life is the time required for half of the radioactive atoms in a sample to decay. This concept helps predict how long a radioactive substance will remain active and at what rate its activity decreases over time.
Cobalt-60 has a half-life of approximately 5.271 years. That means every 5.271 years, half of the cobalt-60 atoms in a sample will have decayed. Calculating the remaining activity of a radioactive source involves using this half-life to determine how much of the original sample is still active after a certain period.
In practice, we use the formula: \[ A = A_0 \left(\frac{1}{2}\right)^n \]where:

• \( A \) is the remaining activity.
• \( A_0 \) is the initial activity.
• \( n \) is the number of half-lives that have passed.

This formula allows us to determine the current activity level of a radioactive sample after a certain number of years based on its known half-life. For example, if a cobalt-60 source initially had an activity of 5000 Ci and we calculate that 0.4745 half-lives have passed, the activity can be recalculated to assess if it is still suitable for use.

Cancer Therapy Isotopes

Radioactive isotopes, like cobalt-60, play a crucial role in modern cancer therapy. These isotopes are used to deliver targeted radiation therapy, known as brachytherapy, which involves placing a radioactive source close to or inside a tumor.
Brachytherapy aims to provide a high dose of radiation to cancer cells while minimizing exposure to surrounding healthy tissue. Isotopes used in such therapies are chosen for their ability to deliver focused radiation over a short distance, efficiently impacting the tumors.

• Cobalt-60 is favored due to its powerful gamma emissions, which allow for effective penetration of cancerous tissues.
• Other isotopes, such as iodine-131 and cesium-137, are also used depending on the specific treatment needs and tumor type.
• Each isotope has different properties and half-lives, making them suitable for various cancer treatments and durations.

Radioactive isotopes must be carefully monitored and stored due to their decaying nature, which affects their strength and usability over time. Understanding the half-life and proper application of isotopes is essential for their effective and safe use in medical treatments.