Question

A \(65-\mathrm{kg}\) person is accidentally exposed for \(240 \mathrm{~s}\) to a \(15-\mathrm{m} \mathrm{Ci}\) source of beta radiation coming from a sample of \({ }^{90}\) Sr. (a) What is the activity of the radiation source in disintegrations per second? In becquerels? (b) Each beta particle has an energy of \(8.75 \times 10^{-14} \mathrm{~J}\), and \(7.5 \%\) of the radiation is absorbed by the person. Assuming that the absorbed radiation is spread over the person's entire body, calculate the absorbed dose in rads and in grays. (c) If the \(\mathrm{RBE}\) of the beta particles is \(1.0, \mathrm{what}\) is the effective dose in mrem and in sieverts? (d) How does the magnitude of this dose of radiation compare with that of a mammogram (300 mrem)?

Short answer

The activity of the radiation source is \(5.55 \times 10^{8} dps\) or \(5.55 \times 10^{8} Bq\). The absorbed dose is approximately 0.00435 rads or 0.0000435 grays. The effective dose equals 0.00435 mrem or 0.0000435 sieverts. The effective dose is about 0.0145 times the dose of a mammogram (300 mrem).

Step-by-step solution

Calculate Activity in disintegrations per second (dps) and Becquerels (Bq)

To find the activity of the source in disintegrations per second, we need to convert the given activity in millicuries (mCi) to disintegrations per second. One curie is defined as the activity of a radioactive source with \(3.7 \times 10^{10}\) disintegrations per second (dps). So, we have: Activity in dps = Activity in mCi × Conversion factor Activity in dps = \(15mCi * (3.7 \times 10^{10} dps/Ci)\) Activity in dps = \(15 * 3.7 \times 10^{7} dps\) Since 1 Becquerel (Bq) is equal to 1 disintegration per second, the activity in Bq will be equal to the activity calculated in dps.

Calculate the total energy absorbed by the person

Given that each beta particle has an energy of \(8.75 \times 10^{-14} J\), we can calculate the total energy emitted by the source during the 240-second exposure. Since the source emits a certain number of particles per second, we can multiply the activity (dps) by the energy per particle and the exposure time (s) to obtain the total energy: Total energy = Activity in dps × energy per particle × exposure time Total energy = \(15 * 3.7 \times 10^{7} dps * 8.75 \times 10^{-14} J * 240s\) The problem states that only 7.5% of the total energy is absorbed by the person. So, we will multiply the total energy by 0.075 to find the absorbed energy by the person. Energy absorbed = 0.075 × Total energy

Calculate absorbed dose in rads and grays

To find the absorbed dose, we need to divide the absorbed energy by the mass of the person: Absorbed dose = Energy absorbed / mass of the person Absorbed dose = (0.075 × Total energy) / (65 kg) Since 1 rad is equal to \(0.01 J/kg\), we can calculate the absorbed dose in rads by dividing the absorbed dose by \(0.01 J/kg\). Additionally, 1 gray (Gy) is equal to \(1 J/kg\), so the absorbed dose in grays can be found by dividing the absorbed dose by \(1 J/kg\).

Calculate the effective dose in mrem and sieverts

We are given the RBE (Relative Biological Effectiveness) of the beta particles is 1.0. To find the effective dose, we multiply the absorbed dose by the RBE: Effective dose = Absorbed dose × RBE To convert the effective dose to millirems (mrem) and sieverts (Sv), we need to use the conversion factors. There are 100 rads in 1 sievert and 100 rem in 1 sievert. Therefore, 1 rad is equal to 10 mrem, and 1 gray is equal to 1 sievert. Effective dose in mrem = Effective dose in rads × 10 mrem/rad Effective dose in sieverts = Effective dose in grays

Compare the radiation dose with a mammogram

Finally, we need to compare the effective dose calculated with the radiation dose of a mammogram (300 mrem). We can do this by calculating the ratio of the two doses: Comparison = Effective dose in mrem / Dose of a mammogram The Comparison value will tell us how many times higher (or lower) the effective dose received by the person is than the dose of a mammogram.

Key concepts

Beta Radiation

Beta radiation is a type of radiation emitted by certain unstable atoms as they decay to become stable. It is composed of high-energy, high-speed electrons or positrons known as beta particles. These particles have more penetrating power than alpha particles but are still less penetrating compared to gamma rays. When beta particles are emitted, they can travel through several millimeters of skin but typically will not penetrate deep into the body.

Understanding beta radiation is crucial because it helps in evaluating and managing exposure to such radiation sources. The activity of a radioactive sample, such as those emitting beta radiation, is measured in disintegrations per second (dps) or in becquerels (Bq), where 1 Bq equals 1 disintegration per second. The level of radioactivity determines how much beta radiation is emitted, impacting the potential dose a person might receive from exposure.

• Beta radiation involves high-speed electrons or positrons.
• It has moderate penetration power; more than alpha but less than gamma rays.
• Measured in disintegrations per second or Becquerels for activity levels.

Absorbed Dose Calculation

Calculating the absorbed dose is an essential part of understanding radiation exposure. In the context of radiation dosimetry, the absorbed dose tells us how much radiation energy is absorbed per unit mass of an organism. This is a crucial step in gauging the potential biological effects of radiation exposure.

The total energy emitted by a source can be determined by multiplying the number of disintegrations (activity) by the energy per emitted beta particle and the exposure time. In situations where only a percentage of the radiation is absorbed, you would adjust the total energy calculation accordingly. In this context, only 7.5% of the total energy was absorbed by the person.

The absorbed dose is calculated using the formula:
\[\text{Absorbed Dose} = \frac{\text{Energy Absorbed}}{\text{Mass of the Person}}\]
The dose can then be expressed in rads or grays, with 1 gray equating to 1 joule of radiation energy per kilogram of tissue and 1 rad equivalent to 0.01 joules per kilogram.

• Calculated by dividing absorbed energy by mass.
• Expressed in rads (0.01 J/kg) or grays (1 J/kg).
• Essential for evaluating potential biological effects.

Relative Biological Effectiveness

Relative Biological Effectiveness (RBE) is a factor used to measure the biological damage caused by different types of radiation. It's an important concept that helps transform the absorbed dose into an effective dose, which better reflects the potential harm to biological tissues.

Not all types of radiation have the same biological impact when compared on a per unit energy basis. The RBE gives us a way to compare different forms of radiation by accounting for these varying biological effects. In this scenario, beta particles have an RBE of 1.0, indicating they have the same biological effect per unit of absorbed energy as the reference radiation (generally gamma rays or X-rays).

To find the effective dose, you multiply the absorbed dose by the RBE, yielding values in millirems or sieverts. This allows us to make informed comparisons between different radiation types and exposure levels, guiding safety measures.

• Used to measure biological damages from different radiation types.
• RBE allows conversion from absorbed dose to effective dose.
• In this scenario, an RBE of 1.0 for beta particles implies equivalent damage as standard reference radiation.

Radiation Exposure Comparison

Radiation exposure is a crucial area of study in dosimetry, as understanding the different levels of exposure can aid in assessing potential health risks. When comparing radiation doses, it is essential to consider both absorbed and effective doses, converting them into common units for easier interpretation.

In this context, comparing a calculated effective dose from beta radiation to a familiar radiation source like a mammogram provides tangible insight into the potential hazard level. A mammogram typically delivers 300 millirems (mrem) of radiation, a benchmark used here for comparison to assess the magnitude of exposure.

Such comparisons help contextualize the dose absorbed by a person; whether it’s much higher or lower than typical diagnostic procedures. This understanding allows for better risk assessments and decision-making in limiting unnecessary exposure and ensuring safety. The ratio of the calculated effective dose to that of a mammogram gives a clear indication of how significant the exposure is.

• Compares different radiation exposures to assess health risks.
• Use of common benchmarks like a mammogram (300 mrem) enhances understanding.
• Helps evaluate the significance of an exposure in terms of public safety and medical contexts.