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 \({ }_{19}^{40} \mathrm{~K},(\mathrm{~b})^{209} \mathrm{Bi}\) and \({ }^{208} \mathrm{Bi}\), (c) nickel-58 and nickel-65. Explain.

Short answer

The radioactive nuclides in each pair are: K-40 (potassium-40), Bi-208 (bismuth-208), and Ni-65 (nickel-65). The corresponding stable nuclides are K-39, Bi-209, and Ni-58. We used the neutron-to-proton ratio and the magic number rule to determine the stability of each nuclide in the pairs.

Step-by-step solution

Background Information: Neutron-to-Proton Ratio and Magic Numbers

The stability of a nuclide is determined by the ratio of neutrons to protons in its nucleus. Generally, stable nuclides have a neutron-to-proton ratio close to 1 for lighter elements (Z ≤ 20) and higher for heavier elements. Another concept that can help predict nuclear stability is the magic number rule. Magic numbers are specific numbers of protons or neutrons that form complete nuclear shells. Nuclides with a magic number of protons or neutrons are more stable. The magic numbers are 2, 8, 20, 28, 50, 82, and 126. Now, let's apply these concepts to predict the stability of each given nuclide.

(a) Analyzing K-39 and K-40

We are given two potassium (K) isotopes: 1. \({}_{19}^{39} \mathrm{~K}\) has 19 protons and 20 neutrons 2. \({}_{19}^{40} \mathrm{~K}\) has 19 protons and 21 neutrons For lighter elements (Z ≤ 20), stable isotopes have a neutron-to-proton ratio close to 1. In this case: 1. K-39: The neutron-to-proton ratio is 20/19 ≈ 1.05 2. K-40: The neutron-to-proton ratio is 21/19 ≈ 1.11 #### Prediction: K-39 is the stable nuclide, and K-40 is the radioactive one.

(b) Analyzing Bi-209 and Bi-208

We are given two bismuth (Bi) isotopes: 1. \(^{209} \mathrm{Bi}\) has 83 protons and 126 neutrons 2. \(^{208} \mathrm{Bi}\) has 83 protons and 125 neutrons In this case, we will use the magic number rule. Neutron numbers 126 and 82 are magic numbers, which imply shell closure and higher stability. #### Prediction: Bi-209 is the stable nuclide, and Bi-208 is the radioactive one, as Bi-209 has a magic number of neutrons.

(c) Analyzing Ni-58 and Ni-65

We are given two nickel (Ni) isotopes: 1. Nickel-58 has 28 protons and 30 neutrons 2. Nickel-65 has 28 protons and 37 neutrons For elements with an atomic number between 20 and 30, stable isotopes tend to have a neutron-to-proton ratio ranging from 1.25 to 1.45. In this case: 1. Ni-58: The neutron-to-proton ratio is 30/28 ≈ 1.07 2. Ni-65: The neutron-to-proton ratio is 37/28 ≈ 1.32 #### Prediction: Nickel-58 is the stable nuclide, and Nickel-65 is the radioactive one, as the neutron-to-proton ratio of Ni-65 is higher and falls within the stable range for elements with atomic numbers between 20 and 30.

Key concepts

Neutron-to-Proton Ratio

A crucial factor in determining the stability of a nucleus is the neutron-to-proton (n/p) ratio. Generally, lighter elements, with atomic numbers less than or equal to 20, tend to be more stable when their n/p ratio is close to 1. For instance, Potassium-39, with 19 protons and 20 neutrons, has an n/p ratio of approximately 1.05, which is close to 1, making it stable. However, Potassium-40, with an n/p ratio of about 1.11, is slightly off from 1, rendering it radioactive.

As the atomic number increases beyond 20, the stable n/p ratio shifts higher, reflecting the greater number of neutrons required to stabilize heavier nuclei. Isotopes like Nickel-58, with an n/p ratio of about 1.07, are stable, whereas Nickel-65, with an n/p ratio of 1.32, becomes radioactive because it exceeds the typical range for stable isotopes in this atomic number zone.

Radioactive Isotopes

Radioactive isotopes, or radioisotopes, are unstable atoms that decay over time, releasing radiation. This instability often arises from an imbalanced neutron-to-proton ratio within the nucleus. Unlike stable isotopes that remain unchanged, radioactive isotopes convert into different elements or isotopes during the decay process.

For example, in the given exercise, Potassium-40 and Nickel-65 are considered radioactive. These isotopes do not conform to their respective stability criteria, hence losing particles to reach a more stable state. Radioactive isotopes are critical in numerous applications, such as medical diagnostics, treatment, and carbon dating, exploiting their predictable decay rates and energy emission.

Magic Numbers

The concept of magic numbers relates to nuclear physics and helps explain why certain isotopes exhibit increased stability. Magic numbers are specific numbers of protons or neutrons that form complete nuclear energy levels, resulting in a particularly stable configuration. These numbers include 2, 8, 20, 28, 50, 82, and 126.

In the given problem, Bismuth-209 contains 126 neutrons, one of these magic numbers, providing a robust shell closure that enhances its stability compared to Bismuth-208, which has 125 neutrons. The presence of a magic number can significantly influence stability, making otherwise unstable combinations remain intact over longer periods, resisting radioactive decay.

Isotope Stability Analysis

Analyzing the stability of isotopes involves evaluating various factors, including neutron-to-proton ratios and magic number presence. To predict if an isotope is stable or radioactive, these criteria are pivotal. Each isotope may vary in neutron number, influencing its stability based on these benchmark principles.

In the task described, by applying n/p ratio analysis, one understands why Potassium-39 and Nickel-58 are stable, whereas Potassium-40 and Nickel-65 are not. Furthermore, the magic number concept clarifies why Bismuth-209 is stable compared to Bismuth-208.

• Analyzing whether isotopes have magic numbers can quickly suggest stability.
• Comparing actual n/p ratios to expected ranges for stability in different atomic regions provides clarity.

Extensive assessment combining these strategies enables more accurate predictions about an isotope's stability or tendency toward radioactivity.