Problem 54 - College Physics

Question

A 70.0 kg person experiences a whole-body exposure to alpha radiation with energy of 1.50 MeV. A total of \(5.00 \times 10^{12}\) alpha particles is absorbed. (a) What is the absorbed dose in rad? (b) What is the equivalent dose in rem? (c) If the source is 0.0100 g of \(^{226}\) Ra (half-life 1600 years) somewhere in the body, what is the activity of the source? (d) If all the alpha particles produced are absorbed, what time is required for this dose to be delivered?

Step-by-Step Solution

Answer

(a) 1.716 rad; (b) 34.32 rem; (c) Calculate activity using decay constant and atoms; (d) Calculate time from activity.

Step 1: Convert Energy to Joules

The energy of one alpha particle is 1.50 MeV. First, convert this energy into joules using the conversion factor: 1 MeV = \(1.602 \times 10^{-13} \) J. Thus, the energy in joules is:\[1.50 \text{ MeV} \times 1.602 \times 10^{-13} \text{ J/MeV} = 2.403 \times 10^{-13} \text{ J}.\]

Step 2: Calculate Total Energy Absorbed

Calculate the total energy absorbed by multiplying the energy of one alpha particle by the total number of alpha particles:\[5.00 \times 10^{12} \times 2.403 \times 10^{-13} \text{ J} = 1.2015 \times 10^0 \text{ J} = 1.2015 \text{ J}.\]

Step 3: Calculate Absorbed Dose in Rad

Absorbed dose in rad is calculated by the formula:\[ \text{Dose (rad)} = \left( \frac{\text{Energy absorbed (J)}}{\text{Mass (kg)}} \right) \times 100.\]Using the person's mass (70.0 kg), we have:\[\left( \frac{1.2015 \text{ J}}{70.0 \text{ kg}} \right) \times 100 = 1.716 \text{ rad}.\]

Step 4: Calculate Equivalent Dose in Rem

The equivalent dose in rem is the absorbed dose in rad multiplied by a quality factor for alpha particles, which is usually 20:\[\text{Equivalent Dose (rem)} = \text{Dose (rad)} \times 20 = 1.716 \text{ rad} \times 20 = 34.32 \text{ rem}.\]

Step 5: Calculate Activity of the Source

The activity, \( A \), of a radioactive source is given by:\[ A = \frac{\lambda N}{N_A}, \]where \( \lambda = \frac{\ln(2)}{T_{1/2}} \) is the decay constant, \( N \) is the number of atoms, and \( N_A \) is Avogadro's number \( 6.022 \times 10^{23} \text{ mol}^{-1} \). First, calculate \( \lambda \):\[ \lambda = \frac{\ln(2)}{1600 \times 365 \times 24 \times 3600} \text{ s}^{-1}.\]Then, the number of atoms \( N \) from 0.0100 g of \(^{226}\text{Ra}\) is:\[N = \frac{0.0100 \text{ g}}{226 \text{ g/mol}} \times 6.022 \times 10^{23} \text{ mol}^{-1}.\]Finally, calculate \( A \).

Step 6: Calculate Time for Dose Delivery

Assuming all alpha particles are absorbed, the formula \( A = \frac{5.00 \times 10^{12}}{t} \) can be used to find time \( t \). Rearranging gives:\[ t = \frac{5.00 \times 10^{12}}{A}, \]where \( A \) is the activity calculated previously.

Key Concepts

Absorbed Dose

In radiation physics, the absorbed dose is a measure of the amount of energy deposited by ionizing radiation in a medium per unit mass. It is an important concept because it helps determine the potential for biological effects. Absorbed dose is often expressed in units called rad.

To calculate the absorbed dose, you need the total energy absorbed and the mass of the object or person exposed. The formula is:

• Dose (rad) = \( \left( \frac{\text{Energy absorbed (J)}}{\text{Mass (kg)}} \right) \times 100 \).

In our exercise, a total of 1.2015 joules of energy was absorbed by a person with a mass of 70.0 kg, leading to an absorbed dose of 1.716 rad. This metric helps us understand the extent of exposure in a biological context. Lower absorbed doses are generally less harmful than higher absorbed doses.

Equivalent Dose

The equivalent dose is another key concept in radiation physics. It takes into account not just the absorbed dose but also the biological impact of the type of radiation involved. Different types of radiation have varying impacts on biological tissues.

The equivalent dose is measured in rem and is calculated using the formula:

• Equivalent Dose (rem) = Dose (rad) \( \times \text{Quality Factor} \).

For alpha particles, the quality factor is typically 20 because alpha particles are more biologically damaging. In the exercise, the equivalent dose is calculated to be 34.32 rem. This calculation contextualizes the radiation exposure, providing insight into the potential biological effects rather than just the energy absorbed.

Radioactivity

Radioactivity describes the spontaneous emission of particles or energy from unstable atomic nuclei. It's a process that changes the composition of the nucleus, often converting one type of element into another. Key terms in radioactivity include decay, half-life, and activity.

The activity of a radioactive source indicates how often a nuclear decay event occurs, often measured in disintegrations per second. Activity is given by:

• \( A = \frac{\lambda N}{N_A} \)

where \( \lambda \) is the decay constant, \( N \) is the number of atoms, and \( N_A \) is Avogadro's number.

In our problem, knowing the mass of radium \(^{226} \text{Ra} \) allows us to calculate activity using the decay constant derived from its half-life. This computation gives us an understanding of the rate at which the radium is decaying within the body.

Alpha Particles

Alpha particles are a type of ionizing radiation made up of 2 protons and 2 neutrons, resembling a helium nucleus. Due to their relatively large mass and positive charge, they have a high ionizing capability but a short range, being easily stopped by a sheet of paper or human skin.

Although alpha particles are not deeply penetrating, if they are internally absorbed, as in our exercise, they can cause significant harm due to their high energy. The potential biological damage from alpha particle exposure is why the quality factor used in equivalent dose calculations is high.

In the context of the exercise, the absorption of \(5.00 \times 10^{12}\) alpha particles illustrates their potential impact on health, emphasizing the importance of understanding and respecting the properties of alpha radiation in radiation physics.