Problem 69 - Fundamentals of Physics

Question

An \(85 \mathrm{~kg}\) worker at a breeder reactor plant accidentally ingests \(2.5 \mathrm{mg}\) of \({ }^{239} \mathrm{Pu}\) dust. This isotope has a half- life of \(24100 \mathrm{y}\), decaying by alpha decay. The energy of the emitted alpha particles is \(5.2 \mathrm{MeV},\) with an \(\mathrm{RBE}\) factor of \(13 .\) Assume that the plutonium resides in the worker's body for \(12 \mathrm{~h}\) (it is eliminated naturally by the digestive system rather than being absorbed by any of the internal organs) and that \(95 \%\) of the emitted alpha particles are stopped within the body. Calculate (a) the number of plutonium atoms ingested, (b) the number that decay during the \(12 \mathrm{~h},\) (c) the energy absorbed by the body, (d) the resulting physical dose in grays, and (e) the dose equivalent in sieverts.

Step-by-Step Solution

Answer

3.03 mSv

Step 1: Calculate the number of plutonium atoms ingested

We begin by converting the mass of the plutonium ingested to the number of atoms. The atomic mass of plutonium-239 is approximately 239 g/mol.First, convert milligrams to grams:\[ 2.5 \text{ mg} = 2.5 \times 10^{-3} \text{ g} \]Next, use Avogadro's number, \(6.022 \times 10^{23} \text{ atoms/mol} \), to find the number of atoms:\[ \frac{2.5 \times 10^{-3} \text{ g}}{239 \text{ g/mol}} \times 6.022 \times 10^{23} \text{ atoms/mol} = 6.3 \times 10^{18} \text{ atoms} \]

Step 2: Calculate the number of atoms that decay in 12 hours

Use the decay formula to find the number of decayed atoms. The decay constant \(\lambda\) is given by:\[ \lambda = \frac{\ln(2)}{t_{1/2}} \]For plutonium-239, \(t_{1/2} = 24100 \times 365 \times 24 \times 3600 \text{ s} \), and \( \ln(2) \approx 0.693 \). Therefore:\[ \lambda = \frac{0.693}{24100 \times 365 \times 24 \times 3600} \approx 9.1 \times 10^{-13} \text{ s}^{-1} \]The number of decayed atoms in 12 hours \((12 \times 3600 \text{ seconds})\) is:\[ N_d = N_0 \times (1 - e^{-\lambda t}) \approx 6.3 \times 10^{18} \times (1 - e^{-9.1 \times 10^{-13} \times 43200}) \approx 2.5 \times 10^{10} \text{ atoms} \]

Step 3: Calculate the energy absorbed by the body

Each decaying atom releases \(5.2 \text{ MeV} \). Convert this energy to joules:\[ 5.2 \text{ MeV} = 5.2 \times 1.602 \times 10^{-13} \text{ J} = 8.33 \times 10^{-13} \text{ J} \]The energy absorbed by the body is released by 95% of the decayed atoms:\[ E = 2.5 \times 10^{10} \times 8.33 \times 10^{-13} \times 0.95 \approx 1.98 \times 10^{-2} \text{ J} \]

Step 4: Calculate the physical dose in grays

The physical dose \(D\) in grays is the energy absorbed per unit mass of the body. The worker's mass is \(85 \text{ kg} \).\[ D = \frac{E}{\text{mass}} = \frac{1.98 \times 10^{-2} \text{ J}}{85 \text{ kg}} \approx 2.33 \times 10^{-4} \text{ Gy} \]

Step 5: Calculate the dose equivalent in sieverts

The dose equivalent \(H\) in sieverts considers the type and energy of radiation, using the relative biological effectiveness (RBE) factor. Here, \(\text{RBE} = 13\).\[ H = D \times \text{RBE} = 2.33 \times 10^{-4} \text{ Gy} \times 13 \approx 3.03 \times 10^{-3} \text{ Sv} \]

Key Concepts

Half-life in Radioactive Decay

Radioactive decay is a process where an unstable atomic nucleus emits particles or radiation to become more stable. A key concept in understanding this process is the *half-life*. Half-life is the time required for half of a sample of a radioactive substance to decay. Half-life varies widely among different elements and isotopes, from fractions of a second to millions of years. For plutonium-239, the isotope in the provided exercise, the half-life is 24,100 years. This means it takes 24,100 years for half of any given amount of plutonium-239 to transform into a different element or isotope. Key points about half-life include:

• It is a constant for a given isotope, describing the rate of decay.
• More stable isotopes like plutonium-239 have long half-lives, meaning they decay slowly.
• The remaining radioactive material continues to decay by half every half-life duration, never reaching zero.

Knowing the half-life helps scientists calculate the remaining quantity of material after a certain period and is crucial for understanding the ecological and health impacts of radioactive materials.

Alpha Decay

Alpha decay is a common type of radioactive decay where an atomic nucleus ejects an alpha particle. An alpha particle consists of two protons and two neutrons. This process decreases the original atom's atomic number by two and its mass number by four.In the exercise, plutonium-239 undergoes alpha decay. When an atom of \({}^{239}Pu\) decays, it emits an alpha particle and transforms into a different element, uranium-235:\[ {}^{239}Pu \rightarrow {}^{235}U + \alpha \]Some essential points about alpha decay are:

• Alpha particles are relatively heavy and positively charged.
• They have high energy but are stopped easily by materials like paper or skin.
• They can be harmful when ingested or inhaled because they can damage internal tissues directly due to their low penetration but high ionization power.

Energy Absorption from Alpha Particles

When radioactive materials decay, they release energy, which can be absorbed by surrounding materials, including human tissues. In the exercise scenario, the energy from alpha particles emitted by decaying plutonium-239 atoms is mainly absorbed by the body, except for a small fraction that escapes. This energy absorption can be calculated based on the number of decayed atoms and the energy released per decay. For instance, each decayed atom in the scenario releases 5.2 MeV of energy. The energy absorbed is a product of this energy and the number of decayed atoms, multiplied by the fraction absorbed by the human body. Key considerations include:

• The absorbed energy is responsible for potential damage to cells and tissues.
• Calculation of absorbed energy helps assess the initial dose of radiation.
• This assessment is crucial in radiation protection to estimate potential health effects and required safety measures.

Dose Equivalent and Its Importance

The dose equivalent is a measure in radiation protection that takes into account not just the absorbed dose but also the biological effect of the radiation type. It is measured in sieverts (Sv) and calculated by multiplying the absorbed dose in grays (Gy) by a quality factor known as the Relative Biological Effectiveness (RBE). In this scenario, the absorbed dose was calculated to be approximately 2.33 x 10^-4 Gy. Given an RBE factor of 13, the dose equivalent becomes 3.03 x 10^-3 Sv. Why is dose equivalent important?

• Different radiation types have varying impacts on biological tissues. Alpha particles, for example, have a higher RBE due to their high ionization potential but low penetration depth.
• Calculating the dose equivalent provides a more accurate understanding of the potential biological harm that radiation exposure could cause.
• This helps in formulating safety guidelines and emergency measures in environments dealing with radioactive materials.