Question

In each pair of isotopes shown, indicate which one you would expect to be radioactive: (a) \({ }_{10}^{20} \mathrm{Ne}\) or \({ }_{10}^{17} \mathrm{Ne}\) (b) \({ }_{20}^{40} \mathrm{Ca}\) or \({ }_{20}^{45} \mathrm{Ca}\) (c) \({ }_{42}^{95} \mathrm{Mo}\) or \({ }_{43}^{92} \mathrm{Tc}\) (d) \({ }_{80}^{195} \mathrm{Hg}\) or \({ }^{196} \mathrm{Hg},\) (e) \({ }_{83}^{209} \mathrm{Bi}\) or \({ }_{96}^{242} \mathrm{Cm} .\)

Short answer

(a) \\({ }_{10}^{17} \\mathrm{Ne}\\) is radioactive; (b) \\({ }_{20}^{45} \\mathrm{Ca}\\) is radioactive; (c) \\({ }_{43}^{92} \\mathrm{Tc}\\) is radioactive; (d) \\({ }^{196} \\mathrm{Hg}\\) could be radioactive; (e) \\({ }_{96}^{242} \\mathrm{Cm}\\) is radioactive.

Step-by-step solution

Understanding Isotopes and Radioactivity

Isotopes are atoms of the same element that have different numbers of neutrons. Therefore, they have different mass numbers. Radioactivity is often linked to an imbalance in the neutron-to-proton ratio and elements with a high atomic number tend to have radioactive isotopes.

Analyze Neon Isotopes

For (a), we compare \({ }_{10}^{20} \mathrm{Ne}\) and \({ }_{10}^{17} \mathrm{Ne}\). The stable isotopes of Neon typically have a stable neutron-to-proton ratio. The isotope \({ }_{10}^{20} \mathrm{Ne}\) is likely stable because its neutron count supports a stable atom, whereas \({ }_{10}^{17} \mathrm{Ne}\) has fewer neutrons than protons, likely causing instability and making it radioactive.

Analyze Calcium Isotopes

In (b), we compare \({ }_{20}^{40} \mathrm{Ca}\) and \({ }_{20}^{45} \mathrm{Ca}\). \({ }_{20}^{40} \mathrm{Ca}\) is a stable isotope, while \({ }_{20}^{45} \mathrm{Ca}\) has an excess of neutrons, making it likely radioactive.

Analyze Molybdenum and Technetium Isotopes

For (c), we compare \({ }_{42}^{95} \mathrm{Mo}\) and \({ }_{43}^{92} \mathrm{Tc}\). Technetium generally has no stable isotopes, so \({ }_{43}^{92} \mathrm{Tc}\) is definitely radioactive, while \({ }_{42}^{95} \mathrm{Mo}\) is likely stable.

Analyze Mercury Isotopes

In (d), \({ }_{80}^{195} \mathrm{Hg}\) and \({ }^{196} \mathrm{Hg}\), \({ }_{80}^{195} \mathrm{Hg}\) is considered stable, whereas \({ }^{196} \mathrm{Hg}\), which does not specify the atomic number, could potentially lack sufficient neutrons or protons, but \({ }^{195} \mathrm{Hg}\) is generally more stable.

Analyze Bismuth and Curium Isotopes

For (e), we compare \({ }_{83}^{209} \mathrm{Bi}\) and \({ }_{96}^{242} \mathrm{Cm}\). \({ }_{83}^{209} \mathrm{Bi}\) is essentially the last stable isotope, whereas \({ }_{96}^{242} \mathrm{Cm}\) is highly radioactive due to its high atomic number, typical for transuranium elements.

Key concepts

Neutron-to-Proton Ratio

The neutron-to-proton ratio is a crucial concept in determining the stability of an isotope. When this ratio is either too high or too low, the isotope tend to be unstable and thus radioactive. This is because a stable nucleus requires a specific balance between neutrons and protons. Neutrons provide an attractive force that helps to stabilize protons within the nucleus, which are positive and repel each other.
When isotopes have too few neutrons, like \({ }_{10}^{17} \text{Ne}\), they can't effectively counter the repulsion between the positively charged protons, leading to instability. Conversely, having too many neutrons can also lead to imbalance as it increases the overall energy and size of the nucleus. Thus, isotopes tend to decay if they deviate from the optimal neutron-to-proton ratio.
The magic numbers, where isotopes with a specific number of nucleons are particularly stable, play into this, where particularly the numbers 2, 8, 20, 28, 50, 82, and 126 suggest stability. Generally, light elements have a neutron-to-proton ratio close to 1:1, while heavier elements shift to a higher ratio to maintain stability.

Stable Isotopes

Stable isotopes are isotopes that do not undergo radioactive decay. They persist over time without changing their nuclear structure. These isotopes have a balanced neutron-to-proton ratio, making them energetically favorable and immune to spontaneous transformations. Such isotopes are essential in various scientific fields, including geology and biochemistry.
Take for example, \({ }_{20}^{40} \text{Ca}\) in the exercise. It is considered stable because it adheres to the typical ratio for stability. Another example is \({ }_{10}^{20} \text{Ne}\), a neon isotope that is stable due to its optimal neutron-to-proton balance. Stable isotopes do not emit radiation and can be found naturally in earth-based resources.
Understanding the behavior of stable isotopes aids in radiometric dating methods and tracing biochemical pathways as these isotopes are integrated into living organisms without altering their nuclear composition.

Transuranium Elements

Transuranium elements are those with atomic numbers greater than that of uranium (92). Given their high atomic numbers, these elements usually exhibit radioactivity. Their nuclei tend to be unstable due to the significant number of protons that cause high repulsion forces within the nucleus.
In the provided exercise, \({ }_{96}^{242} \text{Cm}\) is a transuranium element. These elements are typically synthesized in laboratories through nuclear reactions rather than being naturally occurring. Most transuranium elements are short-lived due to their high instability and tendency to undergo rapid decay.
They play a critical role in scientific research and practical applications such as in nuclear reactors and, albeit limited, in medicine. Understanding their properties is crucial for advancing nuclear technology and studying atomic-level phenomena.

Neon Isotopes

Neon isotopes, such as \(^{20}\text{Ne}\) and \(^{17}\text{Ne}\), demonstrate interesting properties that relate closely to their neutron-to-proton ratios. \(^{20}\text{Ne}\) is a stable isotope with a suitable balance between neutrons and protons. Hence, it is one of the naturally occurring stable forms of neon.In contrast, \(^{17}\text{Ne}\) has a different story due to its insufficient neutrons compared to protons, making it a radioactive isotope. Such imbalance can lead this isotope to decay over time as it strives to reach a more stable state through radioactive decay processes. Neon isotopes are useful in various scientific applications, such as the study of isotopic abundances in terrestrial and extraterrestrial materials,helping researchers understand the formation and evolution of the solar system.