Problem 76 - University Physics with Modern Physics

Question

In the 1986 disaster at the Chernobyl reactor in the Soviet Union (now Ukraine), about \(\frac{1}{8}\) of the \(^{137} \mathrm{Cs}\) present in the reactor was released. The isotope \(^{137} \mathrm{Cs}\) has a half-life for \(\beta\) decay of 30.07 \(\mathrm{y}\) and decays with the emission of a total of 1.17 \(\mathrm{MeV}\) of energy per decay. Of this, 0.51 \(\mathrm{MeV}\) goes to the emitted electron and the remaining 0.66 \(\mathrm{MeV}\) to a \(\gamma\) ray. The radioactive \(^{137} \mathrm{Cs}\) is absorbed by plants, which are eaten by livestock and humans. How many \(^{137} \mathrm{Cs}\) atoms would need to be present in each kilogram of body tissue if an equivalent dose for one week is 3.5 \(\mathrm{Sv}\) ? Assume that all of the energy from the decay is deposited in that 1.0 \(\mathrm{kg}\) of tissue and that the RBE of the electrons is 1.5.

Step-by-Step Solution

Answer

Approximately 1.23 × 10^{13} \(^{137} \mathrm{Cs}\) atoms are needed per kilogram of tissue.

Step 1: Understand Radiation Dose

The equivalent dose (H) is given by:\[ H = D \times ext{RBE} \]where \( D \) is the absorbed dose in Grays (Joules per kilogram) and RBE is the relative biological effectiveness; for electrons, RBE = 1.5.

Step 2: Calculate Absorbed Energy

The dose \( D \) in Grays is related to energy deposited as:\[ D = \frac{E}{ ext{mass}} \]where \( E \) is the energy in Joules. Convert the MeV to Joules for the given energy per decay (1.17 MeV/decay = 1.17 x 1.602 x 10^{-13} J/decay).

Step 3: Calculate Total Energy Deposited

We are given an equivalent dose \( H = 3.5 \; \mathrm{Sv} \) and need to solve for the number of decays \( N_d \) such that:\[ H = N_d \times 1.17 \times 1.602 \times 10^{-13} \times 1.5 \]Rearrange this to find \( N_d \).

Step 4: Calculate Number of Atoms

Since each decay corresponds to one atom decaying, \( N_d \) equals the number of \(^{137} \mathrm{Cs}\) atoms needed to deliver the desired dose. Substitute the known values into the equation to compute \( N_d \).

Step 5: Solve for Number of Atoms

Solving for \( N_d \):\[ N_d = \frac{3.5}{1.17 \times 1.602 \times 10^{-13} \times 1.5} \approx 1.23 \times 10^{13} \text{atoms/kg} \]

Key Concepts

Chernobyl disaster

The Chernobyl disaster was one of the most catastrophic nuclear events in history. It occurred on April 26, 1986, at the Chernobyl Nuclear Power Plant in the Soviet Union, which is now in Ukraine. This accident released a significant amount of radioactive materials into the environment, creating widespread concern and health hazards. During the incident, a substantial quantity of the 137-Cs (Cesium-137) isotope was released.

Cesium-137 is a radioactive isotope that was a major component of the fallout from the disaster. It played a significant role in the long-term environmental impacts, as it was spread over a large area. Being soluble in water, it easily entered the ecosystem, contaminating plants and animals, which, in turn, posed risks to humans through the food chain.

The Chernobyl disaster highlighted the dangers of nuclear power when safety systems are neglected or fail. It led to massive efforts to clean up and contained extensive international research into radiation safety and nuclear energy protocols.

Radioactive decay

Radioactive decay is the process by which an unstable atomic nucleus loses energy by emitting radiation. This is a natural process that helps elements like Cesium-137 transform into a more stable state. During decay, isotopes can emit particles and energy; in the case of Cesium-137, it undergoes beta decay.

Beta decay involves the transformation of a neutron into a proton, with the emission of an electron (beta particle) and an antineutrino. The emitted electron carries away a portion of the decay energy, and in this scenario, it accounts for 0.51 MeV.

Understanding radioactive decay is crucial for evaluating how long an isotope remains hazardous. For Cesium-137, with a half-life of approximately 30.07 years, we can calculate how much time will pass until its radioactivity diminishes by half. This long half-life means it remains hazardous for extended periods, requiring careful management and disposal.

137-Cs isotope

The 137-Cs isotope is a byproduct of nuclear fission and is notorious for its role in nuclear fallout due to its longevity and mobility in the environment. It has a half-life of 30.07 years, making it relatively stable and persistent in the environment after being released.

Cesium-137 decays with the release of 1.17 MeV of energy per decay. This energy is divided between 0.51 MeV to the emitted electron and 0.66 MeV to a gamma ray. These emissions pose health risks, since gamma rays can penetrate body tissues and cause damage at the cellular level.

Due to its chemical characteristics, Cesium-137 can easily become part of biological systems. Once it enters the food chain, it can accumulate in the body tissue of living organisms, posing internal radiation risks. This makes handling and monitoring of this isotope essential in the context of nuclear safety.

Energy deposition

Energy deposition refers to how the energy released from radioactive decay is absorbed in the surrounding matter. When Cesium-137 decays, it emits radiation that releases energy. This energy can be absorbed by body tissues, causing internal exposure to radiation.

In the exercise, energy deposition is crucial for calculating the radiation dosage. The equivalent dose, measured in sieverts, accounts for energy deposited per kilogram of tissue. This is determined by the absorbed dose and further adjusted by the Relative Biological Effectiveness (RBE).

Energy deposition calculations help us understand how much radiation different tissues in the body receive. For Cesium-137, all decay energy is assumed to be deposited into 1 kg of tissue. This simplification is critical for calculating potential biological effects and understanding safety limits.

Biological effectiveness

Biological effectiveness is a measure of the impact of different types of radiation on living tissue. Different radiations have varying capabilities to cause biological damage. Factors like particle energy and type influence how damaging the radiation might be.

This concept is quantified as the Relative Biological Effectiveness (RBE). RBE is used to compare the biological impact of radiation types: It's the factor by which an absorbed dose is multiplied to reflect its actual biological damage. For electrons, the RBE is often taken as 1.5, meaning electrons are 1.5 times more damaging biologically than other forms of radiation under specific conditions.

Understanding RBE is vital when calculating radiation doses and assessing the potential health risks from exposure. It allows us to adjust for the type and energy of radiation involved, providing a more accurate estimation of risk to living organisms.