Problem 68 - Fundamentals of Physics

Question

A \(75 \mathrm{~kg}\) person receives a whole-body radiation dose of \(2.4 \times 10^{-4} \mathrm{~Gy},\) delivered by alpha particles for which the \(\mathrm{RBE}\) factor is 12. Calculate (a) the absorbed energy in joules and the dose equivalent in (b) sieverts and (c) rem.

Step-by-Step Solution

Answer

Absorbed energy: \(1.8 \times 10^{-2} \ \mathrm{J}\), Dose equivalent: \(2.88 \times 10^{-3} \ \mathrm{Sv}\), \(0.288 \ \mathrm{rem}\).

Step 1: Calculate the Absorbed Dose in Joules

First, we need to find the absorbed energy in joules. The absorbed dose in grays (Gy) is given to be \(2.4 \times 10^{-4} \ \mathrm{Gy}\). The formula to find the absorbed energy \(E\) in joules is \(E = D \times m\), where \(D\) is the dose in grays and \(m\) is the mass in kilograms. Here, \(m = 75\mathrm{~kg}\) and \(D = 2.4 \times 10^{-4}\mathrm{~Gy}\). Therefore, \(E = 2.4 \times 10^{-4} \ \mathrm{Gy} \times 75 \mathrm{~kg} = 1.8 \times 10^{-2} \ \mathrm{J}\).

Step 2: Calculate the Dose Equivalent in Sieverts

The dose equivalent \(H\) in sieverts (Sv) is calculated by multiplying the absorbed dose \(D\) by the relative biological effectiveness (RBE) factor \(Q\). The formula is \(H = D \times Q\). Here, \(D = 2.4 \times 10^{-4} \mathrm{~Gy}\) and \(Q = 12\). Thus, \(H = 2.4 \times 10^{-4} \times 12 = 2.88 \times 10^{-3} \ \mathrm{Sv}\).

Step 3: Convert the Dose Equivalent from Sieverts to Rem

To convert the dose equivalent from sieverts to rem, we use the conversion factor \(1 \mathrm{~Sv} = 100 \mathrm{~rem}\). Therefore, \(H = 2.88 \times 10^{-3} \ \mathrm{Sv}\) can be converted to rem by multiplying by 100: \(H = 2.88 \times 10^{-3} \times 100 = 0.288 \ \mathrm{rem}\).

Key Concepts

absorbed energy

When discussing radiation dose calculation, the concept of absorbed energy is crucial. Absorbed energy refers to how much radiation energy is taken in by an object or person. In this context, it represents the amount of radiation energy deposited by particles such as alpha particles in the body of an individual. Understanding the absorbed energy is vital as it forms the basis for calculating other dosimetric quantities.

In the given problem, the absorbed dose is expressed in grays (Gy), where 1 gray equals 1 joule of energy absorbed per kilogram of the object or tissue. To find the total absorbed energy in joules, we use the formula:

• \( E = D \times m \)
• where \( E \) is the absorbed energy in joules, \( D \) is the dose in grays, and \( m \) is the object's mass in kilograms.

For example, a body weighing 75 kg receiving a dose of \( 2.4 \times 10^{-4} \, \mathrm{Gy} \) would absorb \( 1.8 \times 10^{-2} \, \mathrm{J} \). This value tells us how much energy has interacted with the body, which is essential for evaluating potential biological effects.

dose equivalent

The dose equivalent is a more comprehensive measurement than just absorbed dose. It accounts for the type of radiation and its biological impact on human tissue. While the absorbed dose indicates the energy deposited, the dose equivalent provides an insight into the possible health effects of that energy.

The dose equivalent \( H \) is measured in sieverts (Sv) and can be calculated by multiplying the absorbed dose \( D \) by the relative biological effectiveness (RBE) factor \( Q \). This gives us:

• \( H = D \times Q \)
• where \( D \) is the absorbed dose in grays and \( Q \) is the RBE.

The RBE factor allows us to understand how a specific type of radiation, like alpha particles, can be more or less harmful than others with the same absorbed dose. This measurement helps better assess potential health risks.

sieverts

The sievert (Sv) is a standard unit of measurement for the dose equivalent, reflecting the potential biological effects of radiation. It integrates both the amount of radiation absorbed and its effectiveness in causing harm to living tissue.

In the calculation of dose equivalent in the problem, sieverts were used to express the dose's biological impact by considering both the absorbed dose and the RBE factor. The calculated dose equivalent was \( 2.88 \times 10^{-3} \, \mathrm{Sv} \). This reflects the combined measure of radiation and its likely interactions within the human body.

Understanding sievert values helps in assessing health risks associated with radiation exposure, guiding protective measures or medical assessments for safety.

rem

A rem, short for roentgen equivalent man, is another unit used for measuring the health effects of ionizing radiation. It is an older unit but is still commonly used in certain contexts like in the United States.

To convert from sieverts to rem, we leverage the conversion factor:

• \( 1 \, \mathrm{Sv} = 100 \, \mathrm{rem} \)

In the exercise, the dose equivalent converted to rem from sieverts was \( 0.288 \, \mathrm{rem} \). By using this conversion, you translate a sievert value into rem, helping in communication across different standards used globally. This conversion maintains consistency and understanding in medical or occupational settings.

RBE factor

The Relative Biological Effectiveness (RBE) factor is vital for understanding different types of radiation's biological impact. It reflects how different types of radiation with the same absorbed dose can cause different levels of biological damage.

The RBE factor allows us to modify the absorbed dose to reflect potential damage better. It serves as a multiplier when calculating the dose equivalent. In the context of alpha particles, known to cause more biological harm than gamma or X-rays, the RBE factor was given as \( 12 \).

• For example, with the absorbed dose in grays \( D \) and the RBE factor \( Q \), the dose equivalent \( H \) is:
• \( H = D \times Q \)

This adjustment helps in making risk assessments more accurate by considering the type of radiation and its health implications.