Problem 37 - University Physics with Modern Physics

Question

BIO A 67 -kg person accidentally ingests 0.35 Ci of tritium. (a) Assume that the tritium spreads uniformly throughout the body and that each decay leads on the average to the absorption of 5.0 \(\mathrm{keV}\) of energy from the electrons emitted in the decay. The half-life of tritium is \(12.3 \mathrm{y},\) and the RBE of the electrons is \(1.0 .\) Calculate the absorbed dose in rad and the equivalent dose in rem during one week. (b) The \(\beta^{-}\) decay of tritium releases more than 5.0 keV of energy. Why is the average energy absorbed less than the total energy released in the decay?

Step-by-Step Solution

Answer

(a) Absorbed dose: 9.36 rad; Equivalent dose: 9.36 rem. (b) Some energy is lost to neutrinos and radiation pathways.

Step 1: Convert Curie to Decays per Second

The activity in curie (Ci) needs to be converted to decays per second (Becquerel). We know: 1 Ci = \(3.7 \times 10^{10}\) decays per second.Therefore, 0.35 Ci is:\[0.35 \times 3.7 \times 10^{10} = 1.295 \times 10^{10}\, \text{decays/sec}\]

Step 2: Calculate Number of Decays in One Week

The total number of decays over a week (7 days) can be calculated by multiplying the decays per second by the number of seconds in a week:\[1 \text{ week} = 7 \times 24 \times 60 \times 60 = 604800 \text{ seconds}\]\[\text{Total decays in a week} = 1.295 \times 10^{10} \times 604800\] \[\approx 7.83216 \times 10^{15} \text{ decays}\]

Step 3: Calculate Total Energy Absorbed

Each decay results in the absorption of 5.0 keV of energy. Convert this to joules (1 eV = \(1.602 \times 10^{-19}\) joules):\[5.0 \text{ keV} = 5.0 \times 10^3 \times 1.602 \times 10^{-19} \approx 8.01 \times 10^{-16} \text{ J}\]Total energy absorbed over the week:\[\text{Total energy} = 7.83216 \times 10^{15} \times 8.01 \times 10^{-16} \approx 6.2737 \text{ J}\]

Step 4: Calculate Absorbed Dose

The absorbed dose in rad is given by:\[\text{Absorbed Dose (rad)} = \frac{\text{Energy absorbed (J)}}{\text{Mass (kg)}} \times 100 (\text{J/kg to rad})\]\[\text{Absorbed Dose} = \frac{6.2737}{67} \times 100 \approx 9.36 \text{ rad}\]

Step 5: Calculate Equivalent Dose

The equivalent dose is the absorbed dose multiplied by the Relative Biological Effectiveness (RBE), which is 1.0 in this case:\[\text{Equivalent Dose (rem)} = \text{Absorbed Dose (rad)} \times \text{RBE}\]\[\text{Equivalent Dose} = 9.36 \times 1.0 = 9.36 \text{ rem}\]

Step 6: Explain Energy Absorption vs Release

The reason the average energy absorbed per decay is less than the total energy released is due to two main factors: some of the energy is carried away by neutrinos and specifics of energy deposition pathways (like energy carried away by bremsstrahlung or escaping radiation) that aren't absorbed in the tissues.

Key Concepts

Tritium Decay

Tritium is a radioactive isotope of hydrogen. It is naturally occurring but also produced in nuclear reactions. Tritium decays through beta decay, which is the process by which an unstable nucleus releases energy to form a more stable state. In the case of tritium, the decay process involves the conversion of a neutron into a proton, emitting a beta particle (an electron) and an antineutrino.
This decay process has a half-life of approximately 12.3 years. The half-life is the time required for half of the tritium atoms in a sample to decay. Understanding half-life is crucial because it influences how long the emitted radiation remains dangerous.
In radiation calculations, tritium's activity is measured in becquerels, which indicate the number of decays per second. Knowing the activity allows you to calculate how much radiation is expected over a specific time period, such as a week.

Absorbed Dose

The absorbed dose measures the amount of energy absorbed by a mass, typically body tissue, due to radiation. The unit for absorbed dose is the rad, although the SI unit is the gray (Gy), where 1 Gy is equivalent to 100 rad. This calculation is critical because it gives an indication of the potential biological effect of the radiation on human tissue.
In simple terms, to find the absorbed dose, you calculate the total energy deposited in the tissue and divide it by the mass of the tissue. In our previous step solution, we calculated that a person absorbed 6.2737 joules of energy, resulting in the total absorbed dose. This was achieved by multiplying the energy per decay by the total number of decays in one week, then converting that energy into the absorbed dose in rad.

Equivalent Dose

While the absorbed dose tells us how much energy is absorbed, it doesn't account for the biological impact of the type of radiation. This is where the equivalent dose becomes important. The equivalent dose considers both the amount of energy absorbed and the biological effectiveness of the radiation, providing a more accurate measure of potential biological damage.
Equivalent dose is measured in rem (roentgen equivalent man) or sieverts (Sv) in the International System of Units. The calculation involves multiplying the absorbed dose by a factor called the Relative Biological Effectiveness (RBE).
In our example, the RBE was given as 1.0, meaning beta radiation from tritium and other radiation expected to interact similarly with biological tissues. Therefore, the equivalent dose calculation remains the same as the absorbed dose when RBE is unity.

Beta Decay

Beta decay is a type of radioactive decay in which a beta particle (an electron or positron) is emitted. In tritium's case, beta minus ( β⁻ ) decay occurs, where an electron is emitted from the nucleus.
During beta decay, a neutron changes into a proton causing an increase in the atomic number while keeping the atomic mass unchanged. For tritium, this means it transforms into the stable helium isotope, ^{3}He . This process is accompanied by the emission of an antineutrino, which carries away some energy but is nearly massless and difficult to detect.
Beta particles can penetrate body tissues, which is why the absorbed energy must be considered when assessing potential damage from radiation exposure. However, the emitted energy is often more than that absorbed by the body because of physical processes that carry some energy away.

Radiobiological Effectiveness

Radiobiological Effectiveness (RBE) is a measure of the relative effectiveness of different types of radiation in causing biological damage. It's crucial for understanding and comparing the impacts of various radiation sources, as not all radiation types have the same biological effects.
RBE is dimensionless and serves as a weighting factor in calculating the equivalent dose. The potential impact of radiation is influenced by the type and energy of the radiation, as well as the biological tissue it interacts with.
In the exercise system, the RBE value for tritium beta particles is given as 1.0, suggesting it has a standard level of biological effectiveness when compared to other types of radiation like alpha particles, which typically have an RBE greater than 1. Understanding RBE is essential for accurate radiation dose assessments and risk evaluations in medical and industrial settings.