Problem 78 - Chemistry Matter and Change

Question

A worker stands near a machine that uses a cobalt-60 gamma source to sterilize medical equipment. The worker's dose 2.0 \(\mathrm{m}\) from the source is 0.85 \(\mathrm{mrem} / \mathrm{s} \cdot \mathrm{m}^{2}\) What is the worker's dose at a distance of 3.5 \(\mathrm{m}\) ?

Step-by-Step Solution

Answer

The worker's dose at a distance of 3.5 meters from the cobalt-60 gamma source is approximately \(0.2776 \, \mathrm{mrem/s}\).

Step 1: Understand the inverse square law formula

The inverse square law formula for radiation intensity can be expressed as follows: \[I_1 \times d_1^2 = I_2 \times d_2^2\] Here, \(I_1\) represents the initial radiation dose (intensity) at distance \(d_1\) from the source, and \(I_2\) represents the new radiation dose (intensity) at distance \(d_2\) from the source. We are given the worker's dose at a distance of 2.0 meters (\(I_1 = 0.85 \, \mathrm{mrem/s} \cdot \mathrm{m}^2\)) and need to find the worker's dose at a distance of 3.5 meters (\(I_2\)).

Step 2: Plug in the given values

We can plug in the given values into the inverse square law formula: \[(0.85 \, \mathrm{mrem/s} \cdot \mathrm{m}^2) \times (2.0 \, \mathrm{m})^2 = I_2 \times (3.5 \, \mathrm{m})^2\]

Step 3: Solve for the worker's dose at 3.5 meters

To solve for the worker's dose at a distance of 3.5 meters (\(I_2\)), we can first calculate the left-hand side of the equation: \[(0.85 \, \mathrm{mrem/s} \cdot \mathrm{m}^2) \times (2.0 \, \mathrm{m})^2 = (0.85 \, \mathrm{mrem/s} \cdot \mathrm{m}^2) \times 4.0 \, \mathrm{m}^2 = 3.40 \, \mathrm{mrem/s} \cdot \mathrm{m}^2\] Next, divide the result by the square of the new distance, 3.5 meters: \[I_2 = \frac{3.40 \, \mathrm{mrem/s} \cdot \mathrm{m}^2}{(3.5 \, \mathrm{m})^2} = \frac{3.40 \, \mathrm{mrem/s} \cdot \mathrm{m}^2}{12.25 \, \mathrm{m}^2} = 0.2776 \, \mathrm{mrem/s}\]

Step 4: Final Answer

The worker's dose at a distance of 3.5 meters from the cobalt-60 gamma source is approximately \(0.2776 \, \mathrm{mrem/s}\).

Key Concepts

Radiation Dosage

Radiation dosage is a crucial concept, especially when dealing with radioactive sources. It describes the amount of radiation energy absorbed by an object or person. The dosage is usually measured in units like millirem (mrem), which helps quantify the potential biological effect on living tissues.
Understanding radiation dosage is important for ensuring safe levels of exposure, especially for people working near radioactive sources such as medical equipment sterilization units. The dosage calculation takes into account distance from the source, time of exposure, and the nature of the radioactive material. This ensures workers and patients are not exposed to harmful levels.

Cobalt-60 Gamma Source

Cobalt-60 is a radioactive isotope commonly used in medical and industrial applications. It generates gamma radiation through the decay of cobalt-60, which helps in the sterilization of medical equipment, food preservation, and even in cancer treatments.
Gamma rays produced by cobalt-60 have high energy, which can penetrate materials easily, making them effective for both killing bacteria and treating tumors. However, these high energy levels also mean that safety measures must be put in place to protect people working around such sources. This requires understanding the intelligent use of distance and shielding materials.

Distance and Intensity Relationship

The inverse square law describes the relationship between distance from a radiation source and the intensity of radiation. It states that as the distance from the source increases, the intensity decreases proportionally to the square of the distance.
In mathematical terms, if you double the distance from the source, the radiation intensity becomes one-fourth. This law is essential in fields such as radiography, where ensuring the right amount of exposure is vital. Understanding this relationship helps in designing safety protocols, as increasing distance from a source significantly reduces exposure.

• Helps ensure safe working conditions.
• Guides placement and operation of radioactive sources.
• Assists in calculating safe exposure levels for workers.

Radiation Safety

Ensuring radiation safety involves several strategies and guidelines to protect individuals from excessive exposure. These include maintaining adequate distance, using shielding, minimizing exposure time, and using personal protective equipment (PPE).
Regular training and education are vital for workers who are consistently around radioactive materials. These safety practices are essential because long-term or high-dose exposure can lead to serious health problems.

• Implementing shielding, such as lead barriers, to reduce exposure.
• Designing workspaces to maximize distance from sources.
• Protocols for monitoring and measuring radiation levels.

With these measures in place, radiation can be safely used in healthcare, research, and industrial applications without posing a risk to human health.