Question

Write equations for the \(\beta^{-}\) decay of the following isotopes: a) \(\frac{60}{27} \mathrm{Co}\) b) \({ }_{1}^{3} \mathrm{H}\) c) \({ }_{6}^{14} \mathrm{C}\)

Short answer

Question: Write the beta-minus decay equations for the following isotopes: a) Cobalt-60, b) Tritium-3, c) Carbon-14. Answer: a) Cobalt-60 beta-minus decay equation: $$ \frac{60}{27} \mathrm{Co} \rightarrow \frac{60}{28} \mathrm{Ni} + e^{-} + \bar{\nu}_e $$ b) Tritium-3 beta-minus decay equation: $$ { }_{1}^{3} \mathrm{H} \rightarrow { }_{2}^{3} \mathrm{He} + e^{-} + \bar{\nu}_e $$ c) Carbon-14 beta-minus decay equation: $$ { }_{6}^{14} \mathrm{C} \rightarrow { }_{7}^{14} \mathrm{N} + e^{-} + \bar{\nu}_e $$

Step-by-step solution

a) Cobalt-60 Beta-minus Decay

The isotope is given as \(\frac{60}{27} \mathrm{Co}\). In beta-minus decay, the atomic number (Z) increases by one, while the mass number (A) remains the same. The resulting isotope after decay can be written as \(\frac{60}{28}x\). Using the periodic table, we find element 28 is nickel (Ni). Therefore, the beta-minus decay equation is: $$ \frac{60}{27} \mathrm{Co} \rightarrow \frac{60}{28} \mathrm{Ni} + e^{-} + \bar{\nu}_e $$

b) Tritium-3 Beta-minus Decay

The isotope is given as \({ }_{1}^{3} \mathrm{H}\) (tritium). The atomic number will increase by one, while the mass number remains the same, resulting in the isotope \({ }_{2}^{3}x\) after decay. Using the periodic table, element 2 is helium. So, the beta-minus decay equation for tritium is: $$ { }_{1}^{3} \mathrm{H} \rightarrow { }_{2}^{3} \mathrm{He} + e^{-} + \bar{\nu}_e $$

c) Carbon-14 Beta-minus Decay

The isotope is given as \({ }_{6}^{14} \mathrm{C}\). For beta-minus decay, the atomic number will increase by one, and the mass number remains unchanged, resulting in the isotope \({ }_{7}^{14}x\). The element with an atomic number of 7 is nitrogen. So, the beta-minus decay equation for carbon-14 is: $$ { }_{6}^{14} \mathrm{C} \rightarrow { }_{7}^{14} \mathrm{N} + e^{-} + \bar{\nu}_e $$

Key concepts

Nuclear Physics

Nuclear physics is the field of physics that studies the constituents and interactions of atomic nuclei. The most well-known applications of nuclear physics include nuclear energy production and nuclear weapons technology, but the science goes far beyond these. It encompasses how nuclei are held together, the forces at play within a nucleus, and the processes that lead to changes in nuclear structure.

At the heart of nuclear physics is the understanding of nuclear reactions, which includes the fascinating process of radioactive decay. When considering the intricacies of subatomic particles like protons, neutrons, and electrons, nuclear physics also explains the behaviors of these tiny constituents, and how changes on the nuclear level can lead to tangible observances, such as energy releases or the transformation of one element into another, as is seen in beta-minus decay.

Radioactive Decay

Radioactive decay is the spontaneous transformation of one atomic nucleus into another—a change that often results in the emission of particles or radiation. This can occur in various forms, including alpha decay, beta decay (both beta-minus and beta-plus), and gamma decay.

In beta-minus decay, which is particularly relevant here, a neutron in the nucleus is transformed into a proton with the emission of an electron \(e^{-}\) and an antineutrino \(\bar{u}_e\). This changes the atomic number of the element by increasing it by one, while the mass number remains unchanged. Understanding the process behind beta-minus decay is crucial for grasping how unstable isotopes undergo change to reach a more stable state and how this change can help identify the age of archaeological specimens through radiocarbon dating with isotopes like \(^{14}\)C.

Isotopes

Isotopes are variants of a particular chemical element that have the same number of protons (and thus belong to the same element), but different numbers of neutrons. This difference in neutron number results in varying atomic masses for the isotopes of a single element. Isotopes can be stable or unstable, the latter being the focus of radioactive decay experiments and theories.

For instance, Cobalt-60, Tritium-3, and Carbon-14 are all unstable isotopes used in beta-minus decay equations to exemplify the process. What's key in these equations is to recognize how an isotope's instability leads to its transformation into a different element with the same mass number but a different atomic number. This intrinsic need for balance in nuclear forces drives the variety of isotopes to exist and their potential for scientific and technological applications—or, in the case of radioisotopes, to serve as tracers and treatment in medical applications.