Problem 28 - College Physics

Question

In an industrial accident a \(65-\mathrm{kg}\) person receives a lethal whole- body equivalent dose of 5.4 Sv from \(\mathrm{x}\) rays. (a) What is the equivalent dose in rem? (b) What is the absorbed dose in rad? (c) What is the total energy absorbed by the person's body? How does this amount of energy compare to the amount of energy required to raise the temperature of 65 \(\mathrm{kg}\) of water \(0.010^{\circ} \mathrm{C} ?\)

Step-by-Step Solution

Answer

Equivalent dose: 540 rem; absorbed dose: 540 rad; energy absorbed: 351 J, which is greater than the 27.209 J needed to heat the water by 0.010°C.

Step 1: Convert dose from Sv to rem

The equivalent dose in rem can be calculated since 1 Sv equals 100 rem. Multiply the dose in Sv by 100 to convert it to rem:\[5.4 \, \text{Sv} \times 100 = 540 \, \text{rem}.\]

Step 2: Find absorbed dose in rad from Sv

Given that 1 Gy (Gray) is equal to 1 Sv for x-rays and 1 Gy equals 100 rad, therefore, the absorbed dose in 5.4 Sv is:\[5.4 \, \text{Sv} \times 100 = 540 \, \text{rad}.\]

Step 3: Calculate total absorbed energy

The absorbed dose in joules can be calculated by using the relation: \[1 \, \text{rad} = 0.01 \, \text{J/kg}.\]The total energy absorbed by the body is:\[E = 65 \, \text{kg} \times 540 \, \text{rad} \times 0.01 \, \text{J/kg/ rad} = 351 \, \text{J}.\]

Step 4: Compare energy to heating water

To find the energy required to raise the temperature of water, use the formula:\[Q = m \, c \, \Delta T,\]where \(m = 65 \, \mathrm{kg} \), \(c = 4.186 \, \text{J/g}\cdot\text{C}\), and \(\Delta T = 0.010 \, ^{\circ} \text{C} \).Convert mass to grams and calculate \(Q\):\[Q = 65000 \, \text{g} \times 4.186 \, \text{J/g}\cdot\text{C} \times 0.010 \, ^{\circ} \text{C} = 27.209 \, \text{J}.\]The absorbed energy \(351 \, \text{J}\) is greater than \(27.209 \, \text{J}\) needed to heat the water.

Key Concepts

Sieverts

The Sievert (Sv) is a unit of measurement for radiation dosage, specifically the equivalent dose. It indicates the biological effects of ionizing radiation. It accounts for the type of radiation and its impact on human tissue. This is crucial because different types of radiation can cause varying levels of harm. The Sievert measures how much energy is absorbed by human tissue and the potential for biological damage.

When considering whole-body exposure, like in the context of medical imaging or radiation therapy, the Sievert provides a clear picture of risk. In the context of the original problem, a lethal dose of 5.4 Sv is considered extremely high for humans. It's vital to convert this dose for better understanding by using the rem unit, especially in contexts like the United States where rem can be a more common reference.

Rem

The rem, or Roentgen equivalent man, is another unit for measuring radiation dosage. It's primarily used in the United States, similar to the Sievert. However, its scale is different. Specifically, 1 Sv is equivalent to 100 rem, which means it's scaled slightly differently to make it easier to communicate larger or smaller doses. - When converting from Sieverts to rem, you simply multiply by 100. - In the original problem, the 5.4 Sv dose converts to 540 rem. This kind of conversion is essential in different fields, such as nuclear power and safety operations, where clarity and precision in communication are crucial.

Gray

Gray (Gy) is the SI unit used primarily for measuring absorbed radiation dose. In contrast to Sieverts and rem, the Gray doesn't consider biological effects. Instead, Gray is all about energy absorbed by the material—how much radiation energy per unit mass. One Gray is equivalent to 1 Joule of radiation energy absorbed per kilogram of mass. - When considering x-rays, an absorbed dose in Gray is equivalent to an equivalent dose in Sieverts. - 1 Gy is equal to 100 rad, the old units formerly used for absorbed dose. In our problem scenario, converting the Sieverts to Gray directly shows the absorbed dose of radiation, ensuring a clearer understanding of energy interactions with the body.

Energy Absorption

Energy absorption in the context of radiation refers to the amount of radiation energy taken up by the body's tissues. This is a critical measure since it directly represents the potential for biological harm. The original exercise involves calculating how much energy, in Joules, is absorbed when a particular radiation dose is given. - We use the conversion factor: 1 rad = 0.01 J/kg.- Given in the problem, with 540 rad absorbed dose for a 65 kg person, the equation is: \[ E = 65 \times 540 \times 0.01 = 351 \, \text{Joules} \]When comparing absorbed energy to heating water, even a seemingly small absorbed energy (351 J) can exceed the energy needed to slightly raise water temperature, illustrating the potential impact on human life. Understanding energy absorption helps us grasp how radiation doses affect biological tissues.