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

Step by step solution

01

Calculate the Energy Released per Decay

The total energy released per decay is given as 1.17 MeV. Converting this to joules, we know 1 MeV is equivalent to \(1.602 \times 10^{-13}\) J. Therefore, the energy released per decay in joules is \(1.17 \times 1.602 \times 10^{-13}\) J.

02

Calculate the Energy in Joules from One Decay

Using our previous conversion, \(E_{decay} = 1.17 \times 1.602 \times 10^{-13} = 1.87434 \times 10^{-13}\) J per decay.

03

Convert the Dose to Energy Deposited

Given that 3.5 Sv is the equivalent dose, which is \(3.5 \, \text{J/kg}\) for 1 kg of tissue. For a week, the energy deposited is \(3.5 \, \text{J/sec} \times 7 \times 24 \times 3600 \, \text{seconds}\). This equals 2.11 \(\times 10^6\) J.

04

Determine the Number of Decays Required

The total number of decay events needed to deposit 2.11 \(\times 10^6\) J on 1 kg of tissue is calculated as \(\frac{2.11 \times 10^6}{1.87434 \times 10^{-13}}\). This equals approximately \(1.13 \times 10^{19}\) decays.

05

Calculate the Number of \(^{137}Cs\) Atoms Needed

Each decay assumes one \(^{137}Cs\) atom, so approximately \(1.13 \times 10^{19}\) \(^{137}Cs\) atoms need to be present in the 1 kg of tissue.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Radioactive Decay

Radioactive decay is a natural process where unstable atomic nuclei lose energy by emitting radiation. This process results in the transformation of an element into a different element or a different isotope. At the core of radioactive decay is the release of energy, often in the form of particles or electromagnetic waves. This decay can occur through different modes:

• Alpha Decay: The nucleus emits an alpha particle (two protons and two neutrons). This results in the element changing into a different one with a mass number reduced by four and atomic number by two.
• Beta Decay: A neutron in the nucleus is transformed into a proton and an electron, with the electron (beta particle) being emitted.
• Gamma Decay: Following alpha or beta decay, the nucleus releases excess energy by emitting a gamma ray, which is a high-energy photon.

Understanding radioactive decay is crucial because it explains how certain elements change over time and how they emit energy, which can be both useful and dangerous.

Cs-137

Cs-137, or Cesium-137, is a radioactive isotope formed as a byproduct of nuclear fission, commonly found in nuclear reactors and weapon tests. In the Chernobyl disaster, a significant portion of the released radiation was due to Cs-137.
Cs-137 is particularly concerning due to its relatively long half-life and its ability to spread in the environment. It is absorbed by plants and enters the food chain, ultimately affecting humans and animals. This isotope emits beta particles during its decay, contributing to radiation exposure. Addressing the presence of Cs-137 in the environment is crucial to limit its long-term ecological and health impacts.

Radiation Dose

Radiation dose is the measure of radiation absorbed by an object or person. It is crucial in assessing potential harm and health risks associated with exposure. Measurement units include grays (Gy) and sieverts (Sv), with sieverts considering the biological effects of different radiation types.
For instance, a dose of 3.5 Sv in one week significantly affects human health, and managing the deposition of such doses is essential. The dose can derive from absorbing the energy of radioactive isotopes like Cs-137. Understanding radiation doses helps ensure safety standards in environments where radioactive materials exist.

Half-Life

The concept of half-life is a cornerstone of understanding radioactive decay processes. Half-life refers to the time required for half of a given amount of a radioactive element to decay into another element or isotope.
For Cs-137, the half-life is around 30.07 years, meaning it takes this period for half of its atoms to undergo decay. This long half-life signifies that Cs-137 remains in the environment for extended periods, posing ongoing risks after nuclear events. Studying half-lives allows scientists to predict decay patterns and manage radioactive elements safely.

Beta Decay

Beta decay is a mode of radioactive decay where an unstable atom emits a beta particle, which is identical to an electron, to achieve a more stable state. This process involves a neutron converting into a proton with the release of a beta particle (an electron) and an antineutrino.
Cs-137 undergoes beta decay, transforming into a stable isotope of barium. During its decay, Cs-137 releases energy, including gamma rays alongside the beta particles, contributing to radiation exposure concerns post-nuclear incidents like Chernobyl. Recognizing beta decay's role helps in understanding the behavior and eventual stabilization of radioactive materials.