Question

In each of the following pairs, which nuclide would you expect to be the more abundant in nature: (a) \({ }_{48}^{115} \mathrm{Cd}\) or \({ }_{4} 11 \mathrm{Cd}\), (b) \({ }_{13}^{30} \mathrm{Al}\) or \({ }_{13}^{2} \mathrm{Al}\), (c) palladium-106 or palladium113, (d) xenon-128 or cesium-128? Justify your choices.

Short answer

In summary, the more abundant nuclides in each pair are: (a) \({ }_{48}^{115} \mathrm{Cd}\), (b) \({ }_{13}^{27} \mathrm{Al}\), (c) palladium-106, and (d) xenon-128. The choices are justified by considering factors such as the neutron-to-proton ratios and even numbers of protons and neutrons, which provide greater stability in these nuclides.

Step-by-step solution

Comparison of \({ }_{48}^{115} \mathrm{Cd}\) and \({ }_{41}^{115} \mathrm{Cd}\)

In the first pair, we have two isotopes of Cadmium (symbol Cd). In \({ }_{48}^{115} \mathrm{Cd}\), there are 48 protons and 67 neutrons. In \({ }_{41}^{115} \mathrm{Cd}\), there are 41 protons and 74 neutrons. \({ }_{48}^{115} \mathrm{Cd}\) has a lower neutron-to-proton ratio (67/48) compared to \({ }_{41}^{115} \mathrm{Cd}\) (74/41). Nuclides with lower neutron-to-proton ratios that fall closer to the stability line tend to be more stable and therefore more abundant in nature. Hence, we expect \({ }_{48}^{115} \mathrm{Cd}\) to be more abundant than \({ }_{41}^{115} \mathrm{Cd}\).

Comparison of \({ }_{13}^{30} \mathrm{Al}\) and \({ }_{13}^{27} \mathrm{Al}\)

In the second pair, we have two isotopes of Aluminum (symbol Al). In \({ }_{13}^{30} \mathrm{Al}\), there are 13 protons and 17 neutrons. In \({ }_{13}^{27} \mathrm{Al}\), there are 13 protons and 14 neutrons. Both isotopes have relatively similar neutron-to-proton ratios (17/13 for \({ }_{13}^{30} \mathrm{Al}\) and 14/13 for \({ }_{13}^{27} \mathrm{Al}\)). However, \({ }_{13}^{27} \mathrm{Al}\) is more stable due to its even number of neutrons, which allows them to form more stable nuclear pairs. Therefore, we expect \({ }_{13}^{27} \mathrm{Al}\) to be more abundant than \({ }_{13}^{30} \mathrm{Al}\).

Comparison of Pd-106 and Pd-113

In the third pair, we have two isotopes of Palladium (symbol Pd). Pd-106 has 46 protons and 60 neutrons, while Pd-113 has 46 protons and 67 neutrons. Pd-106 is more stable due to its relatively lower neutron-to-proton ratio (60/46) and even numbers of both protons and neutrons, allowing them to form stable nuclear pairs. Thus, we expect palladium-106 to be more abundant than palladium-113.

Comparison of Xe-128 and Cs-128

In the last pair, we have an isotope of Xenon (symbol Xe) and an isotope of Cesium (symbol Cs). Xe-128 has 54 protons and 74 neutrons, while Cs-128 has 55 protons and 73 neutrons. Xe-128 is more stable due to its larger number of neutrons and hence a greater neutron-to-proton ratio (74/54) than Cs-128 (73/55). Additionally, Xe-128 has an even number of protons and neutrons, which increases nuclear stability through nuclear pairing. Therefore, we expect xenon-128 to be more abundant than cesium-128.

Key concepts

Isotope Stability

Understanding why certain isotopes are more abundant in nature than others requires a grasp of isotope stability. Isotope stability is influenced by a delicate balance between the forces that hold an atom's nucleus together. Stable isotopes have a balanced configuration that allows for a strong nuclear force, overcoming the repulsion between protons.

Nuclear stability is not strictly a binary concept, as stable isotopes do not simply exist or not. Instead, stability is often associated with longer half-lives and less decay activity. Nature favors isotopes with higher stability, resulting in greater natural abundance.

Several factors determine stability, such as the energy levels within the nucleus, the balance of nuclear forces, and the ratio of neutrons to protons. An understanding of nuclear shell theory, where protons and neutrons occupy nuclear energy levels much like electrons do in the electron shell, is also vital. Complete nuclear shells, just like electron shells, confer added stability.

Neutron-to-Proton Ratio

The neutron-to-proton ratio plays a pivotal role in isotope stability. Early in the periodic table, a 1:1 ratio is preferred for stability, but as we progress to heavier elements, the ratio becomes more skewed towards neutrons due to the need to counterbalance the increased proton-proton repulsion.

When the neutron-to-proton ratio strays too far from the range of stability, the isotope becomes unstable and is likely to undergo radioactive decay to reach a stable ratio. This is why isotopes with neutron-to-proton ratios close to the stability line are more likely to be found naturally.

For lighter elements, such as Carbon-12 with a 1:1 neutron-to-proton ratio, the isotope is considered very stable. However, for heavier elements like Uranium-238, a higher proportion of neutrons is necessary for stability, yielding a neutron-to-proton ratio greater than 1:1.

Nuclear Pairing

Nuclear pairing is another factor influencing nuclear stability. It refers to the tendency of protons and neutrons to form pairs within the nucleus. Even numbers of protons and/or neutrons generally produce a more stable arrangement due to pairing energy.

Imagine pairing as a stabilizing force where each pair of nucleons (protons or neutrons) lowers the overall energy of the nucleus, much like the stability in a molecule resulting from paired electrons. In cases where nuclides have both even numbers of protons and neutrons, they are doubly even and often exhibit even greater stability.

This preference for pairing is evident in the pattern of stable nuclides observable on the chart of nuclides—a graphic representation of the stable and unstable nuclides. Those with even numbers of protons and neutrons are more clustered on the line of stability, showcasing their prevalence in nature.