Problem 77 - Chemistry Matter and Change

Question

Figure 24.34 shows the position of two workers near a radioactive gamma source. The worker at Position A is standing 2.5 \(\mathrm{m}\) from the source and receives an exposure of 0.98 mrem \(/ \mathrm{s} \cdot \mathrm{m}^{2} .\) The worker at Position \(\mathrm{B}\) receives an exposure of 0.50 \(\mathrm{mrem} / \mathrm{s} \cdot \mathrm{m}^{2} .\) What is the distance of the worker at Position \(\mathrm{B}\) from the source?

Step-by-Step Solution

Answer

The distance of the worker at Position B from the source is 3.5 meters.

Step 1: Understanding the Inverse Square Law

Gamma radiation decreases with the square of the distance from the radioactive source. This is called the Inverse Square Law, given by: \[ I_1 \times r_1^2 = I_2 \times r_2^2 \] where \(I_1\) and \(I_2\) are the intensities (exposure rates), and \(r_1\) and \(r_2\) are the distances from the source.

Step 2: Substitute Known Values

Substitute the known exposures and distance into the inverse square law equation:\[ 0.98 \times (2.5^2) = 0.50 \times r_2^2 \]

Step 3: Calculate \(r_2^2\)

Calculate the left side of the equation: \[ 0.98 \times 2.5^2 = 0.98 \times 6.25 = 6.125 \]Set the equation:\[ 6.125 = 0.50 \times r_2^2 \] and solve for \(r_2^2\):\[ r_2^2 = \frac{6.125}{0.50} = 12.25 \]

Step 4: Solve for \(r_2\)

To find \(r_2\), take the square root of \(12.25\):\[ r_2 = \sqrt{12.25} = 3.5 \]

Key Concepts

Gamma Radiation

Gamma radiation is a form of electromagnetic radiation, much like X-rays, but with a much higher energy. It's emitted during radioactive decay and is capable of penetrating most materials. This penetration ability makes gamma rays useful for medical imaging and treatment. However, it also poses risks when exposed to living tissue, as it can ionize atoms and molecules, potentially causing damage.

Unlike alpha and beta particles, gamma radiation doesn't have a mass or charge. Its high energy means it can travel through larger distances within materials and the air.

• High energy and deep penetration capability.
• Emitted during radioactive decay.
• No mass or charge.

Gamma radiation requires heavy shielding, like lead or concrete, to reduce exposure risks effectively.

Radioactive Sources

Radioactive sources are materials that emit radiation naturally or when induced to decay. These sources contain isotopes that have unstable nuclei, leading them to release energy in the form of radiation as they seek stability.

Radioactive sources are used in medicine, industry, and scientific research. It's crucial to manage these sources with care to prevent unintended radiation exposure. Depending on their type, different sources emit different kinds of radiation - alpha, beta, or gamma. Knowing the properties and behaviors of these radiations helps in setting up proper safety protocols.

• Contains isotopes with unstable nuclei.
• Emits alpha, beta, or gamma radiation.
• Used in various industries and healthcare.

Always handle radioactive sources with caution to ensure the safety of all individuals in the vicinity.

Radiation Exposure

Radiation exposure occurs when a person encounters radiation, such as gamma rays, from a radioactive source. The exposure level depends on the intensity of the radiation, the distance from the source, and the duration of exposure.

Exposure is often measured in units like rems or millirems, which represent the amount of radiation absorbed by the body. Protecting oneself from excessive exposure involves understanding the types of radiation and using appropriate shielding methods. Time, distance, and shielding are critical factors used to control exposure.

• Depends on intensity, distance, and time.
• Measured in rems or millirems.
• Controlled using time, distance, and shielding.

Minimizing exposure reduces the risk of radiation-induced damage to tissues and organs.

Distance Calculation

Calculating the distance from a radiation source involves understanding the Inverse Square Law. This principle states that the intensity of radiation decreases with the square of the distance from its source.

Using the formula \(I_1 \times r_1^2 = I_2 \times r_2^2\), where \(I_1\) and \(I_2\) are radiation intensities at distances \(r_1\) and \(r_2\) respectively, we can determine unknown distances by measuring the change in intensity. This principle is handy for ensuring safe distances are maintained from radioactive sources.

• Intensity decreases with distance squared.
• Inverse Square Law: \(I_1 \times r_1^2 = I_2 \times r_2^2\).
• Helps calculate safe distances from sources.

Applying these calculations helps in both understanding exposure risks and planning safe work environments.