Question

Cobalt- 60 is a strong gamma emitter that has a half-life of 5.26 yr. The cobalt- 60 in a radiotherapy unit must be replaced when its radioactivity falls to \(75 \%\) of the original sample. If an original sample was purchased in June 2010 , when will it be necessary to replace the cobalt- \(60 ?\)

Short answer

The cobalt-60 sample will need to be replaced in February 2012.

Step-by-step solution

Understand the problem

In this problem, we are given the half-life of cobalt-60 (5.26 years) and are asked to determine when the radioactivity falls to 75% of its original amount.

Write the formula for radioactive decay

To solve this problem, we can use the half-life formula for radioactive decay: \[N_t = N_0 \cdot (1/2)^{\frac{t}{t_{1/2}}}\] Where \(N_t\) is the radioactivity at time t, \(N_0\) is the initial radioactivity, t is the time, and \(t_{1/2}\) is the half-life.

Set up the equation

We are given that the sample's radioactivity must be 75% of its original amount, so we can set up the equation: \[0.75N_0 = N_0 \cdot (1/2)^{\frac{t}{5.26}}\]

Solve for time t

Now, we can solve for the time t when the radioactivity falls to 75%: \[\frac{0.75N_0}{N_0} = (1/2)^{\frac{t}{5.26}}\] \[0.75 = (1/2)^{\frac{t}{5.26}}\] To solve for t, we can use the logarithm: \[t = 5.26 \cdot \log_{1/2}(0.75)\] Use a calculator to find the value of t: \[t \approx 1.66 \text{ years}\]

Find the replacement date

The cobalt-60 sample was purchased in June 2010, so we need to add the time until the radioactivity falls to 75% to this date: June 2010 + 1.66 years ≈ June 2010 + 1 year and 8 months ≈ February 2012 So, it will be necessary to replace the cobalt-60 sample in February 2012.

Key concepts

Half-Life

The concept of half-life is a key element in understanding radioactive decay. The half-life of a substance is the time taken for half of the radioactive atoms in a sample to decay. This measure helps predict how long a radioactive material will remain active, and it is expressed in units of time, such as years or seconds.
For Cobalt-60, the half-life is 5.26 years. This means that every 5.26 years, half of the original Cobalt-60 atoms will have decayed into another element. This decay process is random for each atom, but statistically, it follows this predictable pattern. Understanding half-life is crucial for applications such as radiotherapy, as it determines how often the radioactive source needs replacing. Since the activity decreases over time, once the radioactivity falls to a certain level, it becomes less effective for its intended purposes.

Cobalt-60

Cobalt-60 is a radioactive isotope of cobalt, known for emitting gamma rays. These high-energy rays are what make Cobalt-60 useful for certain applications, especially in medical treatments.
Key Characteristics of Cobalt-60:

• Gamma Emissions: These rays are useful in medicine and industry because they penetrate deep into materials, which makes them effective for imaging and treatment.
• Decay: Over time, Cobalt-60 turns into a non-radioactive element through beta decay, releasing gamma radiation in the process.
• Half-Life: At 5.26 years, this half-life period signifies when Cobalt-60 will have lost half its original radioactivity, compelling regular replacement in devices it powers.

Cobalt-60's application in medicine is primarily due to its gamma emission, which is beneficial for targeting cancer cells while sparing surrounding healthy tissues.

Radiotherapy

Radiotherapy uses ionizing radiation to treat cancer by damaging the DNA of cancer cells, which leads to cell death. Cobalt-60 is one of the radioactive sources used in this treatment.
How Radiotherapy Works:

• Targeting Cancer Cells: Cobalt-60 gamma rays are directed at cancerous tissues, which, because of their high energy, damage the DNA of these cells effectively.
• Precision: This treatment requires precise calculations to ensure the maximum dose affects the tumor, minimizing exposure to healthy tissues.
• Recalibration Bases on Half-Life: Due to Cobalt-60's half-life, treatment machines must be recalibrated or sources replaced regularly to maintain efficacy.

Radiotherapy is a well-established cancer treatment method that utilizes the destructive power of radioactive materials to kill cancer cells, with Cobalt-60 being a pivotal source in this process.