Problem 25 - Fundamentals of Physics

Question

Cancer cells are more vulnerable to \(\mathrm{x}\) and gamma radiation than are healthy cells. In the past, the standard source for radiation therapy was radioactive \({ }^{60} \mathrm{Co},\) which decays, with a half-life of \(5.27 \mathrm{y},\) into an excited nuclear state of \({ }^{60} \mathrm{Ni}\). That nickel isotope then immediately emits two gamma-ray photons, each with an approximate energy of \(1.2 \mathrm{MeV}\). How many radioactive \({ }^{60} \mathrm{Co}\) nuclei are present in a \(6000 \mathrm{Ci}\) source of the type used in hospitals? (Energetic particles from linear accelerators are now used in radiation therapy.)

Step-by-Step Solution

Answer

Approximately \(5.31 \times 10^{22}\) nuclei of \(^{60}\mathrm{Co}\) are present.

Step 1: Understand the Activity Unit

In this step, we recognize the activity unit we are dealing with. The activity of a radioactive source is measured in Curies (Ci). Here, we have a source with an activity of 6000 Ci. Recall that 1 Ci is equal to \(3.7 \times 10^{10}\) disintegrations per second.

Step 2: Convert Ci to Disintegrations per Second

We will convert the activity from Curies to disintegrations per second, which is the SI unit Becquerels (Bq) for practical calculations.\[6000 \text{ Ci} \times 3.7 \times 10^{10} \frac{\text{disintegrations}}{\text{s}}/ \text{Ci} = 2.22 \times 10^{14} \text{ disintegrations/s}\]

Step 3: Use the Decay Constant Formula

The decay constant \(\lambda\) is related to the half-life \(T_{1/2}\) by the formula \(\lambda = \frac{\ln(2)}{T_{1/2}}\). Given the half-life of \(^{60}\mathrm{Co}\) is 5.27 years, we need to convert this to seconds to match the units of the decay constant:\[T_{1/2} = 5.27 \text{ years} \times 365 \text{ days/year} \times 24 \text{ hours/day} \times 3600 \text{ seconds/hour} \approx 1.66 \times 10^{8} \text{ seconds}\]Now, calculate \(\lambda\):\[\lambda = \frac{\ln(2)}{1.66 \times 10^{8} \text{ s}} \approx 4.18 \times 10^{-9} \text{s}^{-1}\]

Step 4: Calculate the Number of Nuclei

To find the number of radioactive nuclei \(N\), we use the relationship between activity \(A\), decay constant \(\lambda\), and the number of radioactive nuclei \(N\) which is \(A = \lambda N\). Rearrange to solve for \(N\):\[N = \frac{A}{\lambda} = \frac{2.22 \times 10^{14} \text{ disintegrations/s}}{4.18 \times 10^{-9} \text{s}^{-1}} \approx 5.31 \times 10^{22}\]

Step 5: Interpret the Result

We found that the number of radioactive \(^{60}\mathrm{Co}\) nuclei present in the 6000 Ci source is approximately \(5.31 \times 10^{22}\). This large number illustrates the high level of radioactivity in the source, hence its effectiveness in medical applications.

Key Concepts

Half-Life

The concept of half-life is essential in understanding how radioactive substances decay over time. Half-life is defined as the time it takes for half of the radioactive isotopes in a sample to decay. This means, after one half-life, only half of the original amount of isotopes remains.
Half-life is a constant characteristic of each radioactive isotope, and it's crucial because it helps us predict how a substance will act in the future.

• For example, the half-life of ^{60}Co is 5.27 years. This means that every 5.27 years, half of any given amount of ^{60}Co will have decayed into ^{60}Ni.
• This property is used to calculate how much of a substance remains active over a period and is vital in fields like medical radiation therapy.

Half-life measurements are used globally in various applications, from archaeological dating with carbon-14 to safety calculations in nuclear power plants. Understanding half-life is crucial for making informed decisions based on the decay rates of radioactive materials.

Radioactive Isotopes

Radioactive isotopes, also known as radioisotopes, are atoms with an unstable nucleus that emit radiation to achieve stability. These isotopes are found naturally but can also be artificially created in laboratories.
They are critically important in a wide array of fields due to their unique properties:

• In medical science, radioisotopes are used for diagnostic imaging and treatment, taking advantage of their radioactive properties. For instance, ^{60}Co is employed in radiation therapy for cancer treatment.
• Each radioactive isotope has a specific half-life and decay pattern, which must be considered to ensure safety and effectiveness in applications.
• Radioisotopes are also used in industries for material inspection and in scientific research to trace chemical and biological pathways.

The production, handling, and disposal of radioactive isotopes require strict regulations to protect human health and the environment. Understanding these isotopes' behavior through their half-lives and decay processes is necessary for their safe and effective use.

Radiation Therapy

Radiation therapy is a medical treatment that uses high-energy radiation to destroy cancer cells. It targets and damages the DNA of these cells, thereby inhibiting their ability to grow and divide.
This therapy is beneficial because:

• Cancer cells are more susceptible to damage from radiation than healthy cells, allowing for targeted treatment.
• In the past, isotopes like ^{60}Co were the primary radiation sources. ^{60}Co decays by releasing gamma radiation, effectively used to treat various cancers.
• Currently, linear accelerators, which can produce precise radiation beams, are also widely used.

Radiation therapy is often employed alongside surgery and chemotherapy. It is crucial for treatment planning to account for the isotope's half-life and energy levels to maximize effectiveness and minimize harm to healthy tissues.

Gamma Radiation

Gamma radiation is a form of electromagnetic radiation, similar to X-rays, but with much higher energy. It is emitted during radioactive decay when the nucleus of an atom transitions to a lower energy state, releasing photons in the process.
Gamma rays have several applications due to their ability to penetrate materials:

• In medicine, gamma radiation from isotopes like ^{60}Co is used for external beam radiation therapy to treat cancers.
• Its high energy allows it to penetrate deeply into tissues, effectively targeting tumors hidden within the body.
• Gamma radiation is also used for sterilizing medical equipment and food products by killing bacteria and other pathogens.

While beneficial, gamma radiation requires stringent safety measures due to its ability to cause tissue damage. Professionals handling gamma radiation must understand its properties to utilize it safely and effectively in various applications.