Problem 36 - University Physics with Modern Physics

Question

BIO In an industrial accident a 65-kg person receives a lethal whole-body equivalent dose of 5.4 \(\mathrm{Sv}\) 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 kg of water 0.010 \(\mathrm{C}^{\circ}\) ?

Step-by-Step Solution

Answer

(a) 540 rem (b) 540 rad (c) 351 J; the absorbed energy is much less than the energy to heat water.

Step 1: Convert 5.4 Sv to rem

The conversion from sieverts (Sv) to rem is straightforward. By definition, 1 Sv is equivalent to 100 rem. Therefore, to find the dose in rem, we multiply 5.4 Sv by 100: \[ ext{Dose in rem} = 5.4 ext{ Sv} \times 100 = 540 ext{ rem}\]

Step 2: Calculate absorbed dose in rad

To determine the absorbed dose in rad, use the relationship between Sv and Gy where 1 Sv = 100 rem and 1 Gy = 100 rad. Since 1 Sv = 1 Gy for x-rays, the absorbed dose in Gy is the same as the given dose in Sv (5.4 Gy). Convert this to rad by multiplying by 100:\[ ext{Dose in rad} = 5.4 ext{ Gy} \times 100 = 540 ext{ rad}\]

Step 3: Calculate total energy absorbed in Joules

The absorbed dose in Gy also tells us the energy absorption per kilogram. Since 1 Gy = 1 J/kg, the energy absorbed is 5.4 J/kg for the whole body. Multiply by the body mass (65 kg) to find the total energy:\[ ext{Total energy absorbed} = 5.4 ext{ J/kg} \times 65 ext{ kg} = 351 ext{ J}\]

Step 4: Compare with energy to heat water

The energy required to raise the temperature of 65 kg of water by 0.010 °C is calculated using the formula \[Q = mc\Delta T\]where \(m = 65 \text{ kg}\), \(c = 4,186 \text{ J/(kg}\cdot\text{°C)}\), and \(\Delta T = 0.01 \text{°C}\). Thus:\[Q = 65 \times 4,186 \times 0.010 = 2,721.9 \text{ J}\]This is much larger than the energy absorbed from the radiation (351 J).

Key Concepts

Sieverts and Rem Conversion

When working with radiation dose calculations, it's important to understand the units involved: the sievert (Sv) and the rem (rem). These units measure the same thing—radiation dose—but are used in different contexts and regions. The sievert is the SI unit used internationally, while the rem is used more commonly in the United States.

• The conversion between these units is simple: 1 sievert equals 100 rem.

To convert from sieverts to rem, you multiply by 100. For example, converting a dose of 5.4 Sv to rem involves the calculation: \[ ext{Dose in rem} = 5.4 imes 100 = 540 ext{ rem} \]This conversion factor is constant because both units quantify the same biological effect of ionizing radiation on the human body, but in different scales.

Absorbed Dose Calculation

The absorbed dose is another key concept in radiation physics, quantified in units of gray (Gy) and rad. Like sieverts and rem, these units measure the same concept but use different scales. Here, a crucial principle comes into play:

• 1 Sv is equivalent to 1 Gy for certain types of radiation like x-rays and gamma rays.
• 1 Gy is also equal to 100 rad.

So, if an individual receives an absorbed dose of 5.4 Sv from x-rays, this directly translates to 5.4 Gy. Because 1 Gy equals 100 rad, converting this dose to rads is straightforward: \[ ext{Dose in rad} = 5.4 imes 100 = 540 ext{ rad} \]Thus, the absorbed dose of radiation, whether in grays or rads, signifies the amount of energy deposited by the radiation in a specific mass of tissue.

Energy Absorption Calculation

Understanding the energy absorbed by the body during radiation exposure is fundamental for assessing the biological effects. The absorbed dose expressed in grays or rads shows how much energy per unit mass is absorbed.

• 1 Gy equals 1 Joule per kilogram (J/kg).

Here, since the absorbed dose is 5.4 Gy, the energy absorbed per kilogram of the person's body is 5.4 J/kg. Multiplying this by the person's mass (65 kg) gives the total energy absorbed:\[ ext{Total energy absorbed} = 5.4 imes 65 = 351 ext{ Joules} \]This total energy represents how much ionizing radiation energy is absorbed in the person's entire body, playing a critical role in evaluating potential radiation injuries.

Thermal Energy Comparison

To truly grasp the magnitude of radiation's impact, it helps to compare the absorbed energy to something more familiar, like heating water. The energy required to raise the temperature of a given mass of water is computed using the formula:\[ Q = mc\Delta T \]where:

• \(m\) is the mass of water (65 kg),
• \(c\) is the specific heat capacity of water (4,186 J/(kg°C)),
• \(\Delta T\) is the temperature change (0.01°C).

Performing this calculation gives:\[ Q = 65 imes 4,186 imes 0.010 = 2,721.9 ext{ Joules} \]Comparing this with the radiation absorbed energy of 351 Joules, it's clear that the energy needed to slightly warm water far exceeds the energy absorbed from the radiation. This puts into perspective just how small energy doses from radiation can have significant biological effects, unlike their fairly minimal impact in raising temperature.