Question

Write down equations to describe the \(\beta^{-}\) -decay of the following atoms: a) \({ }^{60} \mathrm{Co}\) b) \({ }^{3} \mathrm{H}\) c) \({ }^{14} \mathrm{C}\)

Short answer

Question: Write down the equations for β− decay of Cobalt-60, Tritium, and Carbon-14. Answer: a) Cobalt-60: ${ }^{60}\mathrm{Co} \rightarrow { }^{60}\mathrm{Ni} + e^{-} + \bar{\nu_e}$ b) Tritium: ${ }^{3}\mathrm{H} \rightarrow { }^{3}\mathrm{He} + e^{-} + \bar{\nu_e}$ c) Carbon-14: ${ }^{14}\mathrm{C} \rightarrow { }^{14}\mathrm{N} + e^{-} + \bar{\nu_e}$

Step-by-step solution

Write the general equation for \(\beta^{-}\) decay

To begin with, we need to write a general equation for \(\beta^{-}\) decay. In this decay, a neutron (n) transforms into a proton (p), emitting an electron (e) and an electron-antineutrino (\(\bar{\nu}_e\)) in the process. This can be represented as: n \(\rightarrow\) p + e\(^{-}\) + \(\bar{\nu}_e\)

Write the provided isotopes in the form A ZX

In order to write the decay equations, we need first to express the given isotopes using the standard notation A ZX, where A is the mass number, Z is the atomic number, and X is the element symbol. a) \({ }^{60} \mathrm{Co}\) (Cobalt-60) b) \({ }^{3} \mathrm{H}\) (Tritium or Hydrogen-3) c) \({ }^{14} \mathrm{C}\) (Carbon-14)

Write the decay equations for each isotope

Now, we will write the decay equations for each isotope, applying the conservation of mass number and atomic number: a) \({ }^{60} \mathrm{Co}\) \(\rightarrow\) \({ }^{60} \mathrm{Ni}\) + e\(^{-}\) + \(\bar{\nu}_e\) b) \({ }^{3} \mathrm{H}\) \(\rightarrow\) \({ }^{3} \mathrm{He}\) + e\(^{-}\) + \(\bar{\nu}_e\) c) \({ }^{14} \mathrm{C}\) \(\rightarrow\) \({ }^{14} \mathrm{N}\) + e\(^{-}\) + \(\bar{\nu}_e\) These are the decay equations describing the \(\beta^{-}\) decay of Cobalt-60, Tritium, and Carbon-14.

Key concepts

Nuclear Physics

Nuclear physics is the branch of physics that deals with the constituents and interactions of atomic nuclei. It encompasses the study of nuclear reactions, radioactive decay, and the forces that hold the nucleus together. In nuclear physics, we explore processes like fusion, fission, and decay, which have profound implications for both natural phenomena and technological applications.

• Atomic Mass: The total number of protons and neutrons in the nucleus, symbolized as A.
• Atomic Number: The number of protons in the nucleus, which determines the element, symbolized as Z.

Understanding nuclear physics helps us grasp the behavior of radioactive elements and the mechanisms of nuclear decay. This fundamental knowledge supports fields like nuclear energy, medical imaging, and astrophysics.

Radioactive Decay

Radioactive decay is a process by which an unstable atomic nucleus loses energy by radiation. During this process, the nucleus transmits particles or energy, transforming into a different element or isotope. The decay is spontaneous and results from the nuclear instability of certain isotopes, known as radioisotopes.
Common types of radioactive decay include:

• Alpha decay (emission of an alpha particle).
• Beta decay (emission of a beta particle and an antineutrino).
• Gamma decay (emission of gamma radiation).

In beta decay, the emitted particles allow the nucleus to achieve a more stable configuration. This helps in the transformation of elements and contributes to the natural radioactive processes found in the environment, medical applications, and the dating of archaeological findings.

Electron Antineutrino

The electron antineutrino is an extremely light and neutral particle emitted during beta decay. It is produced alongside an electron when a neutron transforms into a proton. Electron antineutrinos belong to a fundamental class of particles known as neutrinos. They interact very weakly with matter, making them difficult to detect but crucial for conserving energy and lepton number in radioactive processes.
Key features of electron antineutrinos include:

• Very small mass.
• Almost no charge, making them weakly interacting.
• Essential in balanced nuclear equations.

Understanding electron antineutrinos is essential for explaining the energy balance in nuclear reactions and enhancing our grasp of the subatomic world.

Neutron to Proton Transformation

Neutron to proton transformation is a key process in beta decay, particularly \( \beta^{-} \) decay, where a neutron within an atomic nucleus converts into a proton. This transformation is accompanied by the emission of an electron (beta particle) and an electron antineutrino. The process maintains the conservation of mass and charge while altering the atomic identity of the element.
During this transformation:

• The mass number (A) remains the same, while the atomic number (Z) increases by one.
• The element's identity changes due to the increase in proton count.
• This process is a natural way for unstable isotopes to gain stability.

The neutron to proton transformation is fundamental in changing the atomic structure and plays a crucial role in the evolution of elements over time. Understanding this transformation spans knowledge of both simple radioactive transitions and complex stellar nucleosynthesis.