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. Explain.

Short answer

The radioactive and stable nuclides are: (a) \({ }_{19}^{39} \mathrm{~K}\) is radioactive, and \({ }_{19}^{40} \mathrm{~K}\) is stable. (b) \({ }^{209} \mathrm{Bi}\) is stable, and \({ }^{208} \mathrm{Bi}\) is radioactive. (c) Nickel-58 is stable, and nickel-65 is radioactive.

Step-by-step solution

Determine the number of protons and neutrons in each nuclide

First, let's find out the number of protons and neutrons in each nuclide. We can do this by looking at the atomic number (number of protons) and mass number (sum of protons and neutrons). (a) For \({ }_{19}^{39} \mathrm{~K}\), there are 19 protons and 20 neutrons (since 39 - 19 = 20). For \({ }_{19}^{40} \mathrm{~K}\), there are 19 protons and 21 neutrons (since 40 - 19 = 21). (b) For \({ }^{209} \mathrm{Bi}\), there are 83 protons and 126 neutrons (since 209 - 83 = 126). For \({ }^{208} \mathrm{Bi}\), there are 83 protons and 125 neutrons (since 208 - 83 = 125). (c) For nickel-58, there are 28 protons and 30 neutrons (since 58 - 28 = 30). For nickel-65, there are 28 protons and 37 neutrons (since 65 - 28 = 37).

Determine stability

Now, we will look at the even/odd number of protons and neutrons for each nuclide and any possible magic numbers and based on that, we will predict which nuclide is radioactive and which one is stable. (a) \({ }_{19}^{39} \mathrm{~K}\) has an odd number of protons and an even number of neutrons. In general, nuclides with an odd number of protons and even number of neutrons are less stable. \({ }_{19}^{40} \mathrm{~K}\) has an odd number of protons and an odd number of neutrons. Nuclides with odd numbers of both protons and neutrons tend to be more stable. So, we predict that \({ }_{19}^{39} \mathrm{~K}\) is radioactive, and \({ }_{19}^{40} \mathrm{~K}\) is stable. (b) \({ }^{209} \mathrm{Bi}\) has 126 neutrons, which is a magic number. This would indicate that \({ }^{209} \mathrm{Bi}\) is stable. \({ }^{208} \mathrm{Bi}\) has an odd number of protons and an odd number of neutrons, making it less stable. So, we predict that \({ }^{209} \mathrm{Bi}\) is stable, and \({ }^{208} \mathrm{Bi}\) is radioactive. (c) Nickel-58 has an even number of protons and an even number of neutrons, making it more stable. Nickel-65 has an even number of protons and an odd number of neutrons, making it less stable. Therefore, we predict that nickel-58 is stable and nickel-65 is radioactive. In conclusion: (a) \({ }_{19}^{39} \mathrm{~K}\) is radioactive, and \({ }_{19}^{40} \mathrm{~K}\) is stable. (b) \({ }^{209} \mathrm{Bi}\) is stable, and \({ }^{208} \mathrm{Bi}\) is radioactive. (c) Nickel-58 is stable, and nickel-65 is radioactive.

Key concepts

Nuclear Stability

Nuclear stability is a concept that describes whether an atom's nucleus remains intact or undergoes radioactive decay. Atoms are made of a core of protons and neutrons, tightly bound together in the nucleus. A stable nucleus does not change over time, whereas a radioactive (unstable) nucleus does.

Factors affecting nuclear stability include the ratio of neutrons to protons, referred to as the N/Z ratio (neutron number over proton number). When this ratio is not optimal, the nucleus may become unstable and decay to achieve a more stable state. Generally, a higher ratio is needed for larger atoms to counteract the repulsive forces between protons.

• If there are too many neutrons, beta decay tends to occur, converting a neutron into a proton.
• Left with too few neutrons, positron emission or electron capture might happen.

The more stable arrangement often corresponds to even numbers of protons and neutrons. This results in a more symmetrical and energetically favorable state that resists decay.

Protons and Neutrons

The nucleus of an atom is composed of two types of subatomic particles: protons and neutrons. Protons are positively charged particles, while neutrons have no charge. Together, they define the mass number of an atom, with the sum determining the isotope of an element.

Protons play a crucial role in defining the identity of an element. The atomic number, which is the number of protons, uniquely identifies an element. Conversely, neutrons serve to stabilize the nucleus by offsetting the repulsive forces between positively charged protons.

• Nuclei with odd numbers of protons and neutrons are often less stable than those with an even number of both.
• Stable nuclei often exhibit a neutron-to-proton ratio that allows for balance between nuclear forces.

Understanding how protons and neutrons interact underlies predicting the stability of isotopes. For instance, in the problem, nickel-58 with its even numbers of protons and neutrons is more stable compared to nickel-65, which features an imbalance.

Magic Numbers

Magic numbers are particular numbers of protons or neutrons that result in highly stable atomic nuclei. These numbers are akin to a filled shell in electron configurations, indicating a complete layer of nucleons, similar to a noble gas configuration for atoms.

Magic numbers are determined by the experimental observation of significantly increased stability at certain counts: 2, 8, 20, 28, 50, 82, and 126. Nuclei at these magic numbers are often resistant to radioactive decay.

• The presence of a magic number of neutrons, as seen in $^{209}$ Bi, with its 126 neutrons, greatly enhances its stability.
• Similarly, having a magic number of protons or neutrons often provides a double layer of stability.

Such nuclear configurations are so energetically advantageous that they resist transformation to other isotopic forms. Understanding magic numbers helps predict nuclear reactions and the synthesis of new elements. They are a key component in comprehending why certain isotopes resist decay while others do not.