Question

The argon isotope \({ }_{18}^{37} \mathrm{Ar}\) undergoes electron capture. (a) Write a nuclear reaction for this process. (b) Where on the band of stability would you expect to find \({ }_{18}^{37} \mathrm{Ar}\) ? (c) What is the daughter nucleus produced?

Short answer

(a) The nuclear reaction for electron capture is \({ }_{18}^{37} \mathrm{Ar} + \mathrm{e^-} \rightarrow {}_{17}^{37} \mathrm{Cl} + \nu_e\). (b) We would expect to find \({ }_{18}^{37} \mathrm{Ar}\) below the band of stability, as its N/Z ratio (\( \frac{19}{18} \approx 1.06\)) is slightly less than the ideal 1:1 ratio for light elements. (c) The daughter nucleus produced is \({ }_{17}^{37} \mathrm{Cl}\) (Chlorine-37).

Step-by-step solution

Nuclear Reaction for Electron Capture

In an electron capture process, a proton in the nucleus captures an electron from an inner electron shell and is converted into a neutron. To represent this process, we write the argon isotope and the electron on the left-hand side, and the products of the reaction on the right-hand side. \({ }_{18}^{37} \mathrm{Ar} + \mathrm{e^-} \rightarrow \text{daughter nucleus} + \nu_e\)

Position on the Band of Stability

The band of stability represents the stable isotopes of elements, and it is determined by the ratio of neutrons (N) to protons (Z). For light elements, this ratio is approximately 1:1, and for heavier elements, it is about 1.5:1. In the case of \({ }_{18}^{37} \mathrm{Ar}\), we have Z=18 (number of protons) and the number of neutrons N=37-18=19. The N/Z ratio for this isotope is: N/Z = \( \frac{19}{18} \approx 1.06 \) Comparing this ratio to the ideal 1:1 ratio for light elements, we can see that the number of neutrons is slightly less than the ideal ratio. Therefore, we would expect to find \({ }_{18}^{37} \mathrm{Ar}\) below the band of stability.

Identifying the Daughter Nucleus

In the electron capture process, a proton is converted into a neutron. Since the number of protons decreases by 1 and the number of neutrons increases by 1, the daughter nucleus will have one less proton and one more neutron compared to the original isotope. In this case, the atomic number will decrease from 18 to 17, and the mass number remains the same (37). The daughter nucleus is \({ }_{17}^{37} \mathrm{Cl}\) (Chlorine-37). So the complete nuclear reaction is: \({ }_{18}^{37} \mathrm{Ar} + \mathrm{e^-} \rightarrow {}_{17}^{37} \mathrm{Cl} + \nu_e\)

Key concepts

Nuclear Reaction

In nuclear physics, a nuclear reaction is a process that changes the composition, energy, or structure of an atomic nucleus. This change is often induced by the collision of a particle with the nucleus, which in the case of electron capture, involves an inner electron being absorbed by the nucleus. It is also essential to understand that during such reactions, the laws of conservation of mass-energy and charge are strictly obeyed.

For example, during the electron capture by an Argon-37 isotope, a proton within its nucleus captures an orbiting electron and transforms into a neutron, emitting a neutrino (ue) in the process. Here, the initial Argon-37 nucleus combines with the electron resulting in a new nucleus with one less proton, hence a different element, and a neutrino.

Band of Stability

The band of stability is a graphical representation showing the stable nuclei in the form of a plot of their neutron number (N) against their proton number (Z). Most stable nuclei fall within a region on the plot known as the 'band of stability'. This band shows a relationship between the number of protons and neutrons. Light elements tend to have a neutron-to-proton ratio (N/Z) close to 1:1, while heavier elements typically have a ratio closer to 1.5:1.

The closer an isotope lies within this range, the more stable it is likely to be. Isotopes that are far from the band of stability are usually radioactive and will undergo decay processes such as alpha decay, beta decay, or electron capture in order to move towards stability.

Isotopes

The term isotopes refers to different forms of an element that have the same number of protons but a different number of neutrons. Essentially, isotopes of an element share the same atomic number (Z) but have different mass numbers (A).

Characteristics of Isotopes

• They may have different physical properties, such as melting points and densities.
• Their chemical properties are very similar since they have the same electron configuration.
• Some isotopes are stable, while others are radioactive, meaning they decay over time.

For instance, Argon-37 is an isotope of the element Argon, characterized by having 18 protons and 19 neutrons. Because its neutron number varies from the most stable isotope of Argon (which has 22 neutrons), Argon-37 is radioactive and prone to decay, such as electron capture.

Neutron to Proton Ratio

The neutron to proton ratio (N/Z) is a crucial factor determining the stability of a nucleus. It refers to the number of neutrons (N) in the nucleus relative to the number of protons (Z). Isotopes with an N/Z ratio that falls within the band of stability are generally stable.

When the ratio is too high or too low, the isotope tends to be unstable or radioactive. In such cases, the nucleus may undergo various decay processes to achieve a more stable state. An isotope with a high N/Z ratio might undergo beta decay to decrease the number of neutrons, whereas one with a low N/Z might capture an electron to increase its neutron count.

In the case of Argon-37, the N/Z ratio is slightly greater than 1, which indicates that it has more neutrons than required for stability in lighter elements, leading it to undergo electron capture to reach a more balanced state.