Question

One of the nuclides in each of the following pairs is radioactive. Predict which is radioactive and which is stable: (a) \({ }_{19}^{39} \mathrm{~K}\) and \({ }_{19}^{40} \mathrm{~K}\), (b) \({ }^{209} \mathrm{Bi}\) and \({ }^{208} \mathrm{Bi}\), (c) nickel-58 and nickel-65.

Short answer

The radioactive isotopes are \({ }_{19}^{40}\mathrm{K}\), \({ }^{208}\mathrm{Bi}\), and Nickel-65, as their proton and neutron numbers are farther from the magic numbers, making them less stable compared to their counterparts.

Step-by-step solution

Identify isotopes and their numbers of protons and neutrons

First, let's list the isotopes given in the exercise and identify their numbers of protons (Z) and neutrons (N): (a) \({ }_{19}^{39}\mathrm{K}\) (Z=19, N=20) and \({ }_{19}^{40}\mathrm{K}\) (Z=19, N=21) (b) \({ }^{209}\mathrm{Bi}\) (Z=83, N=126) and \({ }^{208}\mathrm{Bi}\) (Z=83, N=125) (c) Nickel-58 (Z=28, N=30) and Nickel-65 (Z=28, N=37)

Compare with magic numbers

Now, let's compare the numbers of protons and neutrons for each isotope to the magic numbers. The closer they are, the more stable the isotope. (a) \({ }_{19}^{39}\mathrm{K}\): Z = 19 (not close to any magic numbers), N = 20 (magic number) \({ }_{19}^{40}\mathrm{K}\): Z = 19 (not close to any magic numbers), N = 21 (not close to any magic numbers) (b) \({ }^{209}\mathrm{Bi}\): Z = 83 (not close to any magic numbers), N = 126 (magic number) \({ }^{208}\mathrm{Bi}\): Z = 83 (not close to any magic numbers), N = 125 (close to magic number 126) (c) Nickel-58: Z = 28 (magic number), N = 30 (not close to any magic numbers) Nickel-65: Z = 28 (magic number), N = 37 (not close to any magic numbers)

Determine stability

Based on our comparison with the magic numbers, we can now predict which isotopes are more stable, and which are likely to be radioactive: (a) \({ }_{19}^{39}\mathrm{K}\) is more stable (N = magic number), so \({ }_{19}^{40}\mathrm{K}\) is likely radioactive. (b) \({ }^{209}\mathrm{Bi}\) is more stable (N = magic number), so \({ }^{208}\mathrm{Bi}\) is likely radioactive. (c) Nickel-58 is more stable (Z = magic number), so Nickel-65 is likely radioactive. So the radioactive isotopes are \({ }_{19}^{40}\mathrm{K}\), \({ }^{208}\mathrm{Bi}\), and Nickel-65.

Key concepts

Radioactive Isotopes

Radioactive isotopes, also known as radioisotopes, are atoms that emit radiation as they decay into a more stable form. Unlike stable isotopes, radioactive isotopes have an imbalance in the number of protons and neutrons that makes them unstable. This radioactivity often stems from having too many protons, too many neutrons, or a mix of both, which disrupts the energy balance within the nucleus.
The stability of isotopes greatly depends on how their nucleus is composed and how well it can resist decay.

• Radioactive isotopes decay over time, emitting energy in the form of radiation, such as alpha, beta, or gamma rays.
• The process by which they decay is random but quantifiable in terms of half-life, the time it takes for half of the radioactive atoms in a sample to decay.
• Isotopes that are closer to having equal numbers of protons and neutrons tend to be more stable.

When assessing whether an isotope might be radioactive, scientists often compare the neutron-to-proton ratio and consult known magic numbers, which reflect particularly stable configurations.

Magic Numbers

Magic numbers are key ingredients for understanding nuclear stability. They refer to specific numbers of protons or neutrons in the nucleus that are arranged in a complete or closed shell within the atomic nucleus. These numbers lead to greater stability in the atom and thus make it less likely to be radioactive.
Magic numbers are based on a model similar to electron shells in atoms, but rather they apply to the nucleons in the nucleus, which are the protons and neutrons.
Understanding magic numbers can help predict nuclear stability:

• Common magic numbers for protons and neutrons include 2, 8, 20, 28, 50, 82, and 126.
• Magical configurations arise due to the full occupancy of nuclear energy levels, similar to how noble gases reflect full electron shells and increased stability.
• An isotope close to possessing these magic numbers will exhibit heightened stability relative to others not near magic numbers.

The insight given by magic numbers aids scientists in predicting which isotopes are more likely to be stable or radioactive, based on its nuclear configuration.

Protons and Neutrons

Protons and neutrons are the fundamental building blocks of atomic nuclei. Together, they are known as nucleons. The balance between these nucleons greatly determines the nuclear stability of isotopes.
Protons carry a positive charge, while neutrons are neutral. This distinction plays a significant role in the forces within the nucleus:

• Protons, being positively charged, repel each other due to the electromagnetic force.
• Neutrons contribute to the strong nuclear force, which helps to bind the nucleus together, offsetting the repulsion between protons.
• In balancing these forces, a stable nucleus commonly has nearly equal numbers of protons and neutrons, especially in lighter elements.

The relative numbers of protons and neutrons act like a delicate scale.
When this balance tips too far toward excess protons or neutrons, instability and radioactivity may occur. Observing this balance, along with magic numbers, allows scientists to deduce nuclear stability.