Question

For each pair of isotopes listed, predict which one is less stable: (a) \({ }_{3}^{6} \mathrm{Li}\) or \({ }_{3}^{9} \mathrm{Li}\) (b) \({ }_{11}^{23} \mathrm{Na}\) or \({ }_{11}^{25} \mathrm{Na}\) (c) \({ }_{20}^{48} \mathrm{Ca}\) or \({ }_{21}^{48} \mathrm{Sc}\)

Short answer

The less stable isotopes are { }_{3}^{9} m{Li}, { }_{11}^{25} m{Na}, and { }_{21}^{48} m{Sc}.

Step-by-step solution

Understand stability factors

Isotopes' stability is often determined by their neutron-proton ratio and how close this ratio is to the optimal value for a given element. Stable nuclei typically have a neutron-proton ratio around 1:1 for lighter elements and gradually increase for heavier elements.

Analyze isotope (a) { }_{3}^{6} m{Li} vs { }_{3}^{9} m{Li}

For { }_{3}^{6} m{Li}, the neutron-proton ratio is 1 (3 neutrons / 3 protons). For { }_{3}^{9} m{Li}, the ratio is 2 (6 neutrons / 3 protons). { }_{3}^{6} m{Li} is closer to the stable 1:1 ratio, making { }_{3}^{9} m{Li} the less stable isotope.

Analyze isotope (b) { }_{11}^{23} m{Na} vs { }_{11}^{25} m{Na}

In { }_{11}^{23} m{Na}, the neutron-proton ratio is approximately 1.09 (12 neutrons / 11 protons). For { }_{11}^{25} m{Na}, the ratio is about 1.27 (14 neutrons / 11 protons). { }_{11}^{23} m{Na} is closer to the ideal ratio, making { }_{11}^{25} m{Na} less stable.

Analyze isotope (c) { }_{20}^{48} m{Ca} vs { }_{21}^{48} m{Sc}

{ }_{20}^{48} m{Ca} has a neutron-proton ratio of 1.4 (28 neutrons / 20 protons), which is quite stable for its atomic number. On the other hand, { }_{21}^{48} m{Sc} has a neutron-proton ratio of 1.29 (27 neutrons / 21 protons). { }_{20}^{48} m{Ca} is stable due to being a magic number element, making { }_{21}^{48} m{Sc} less stable due to its proton excess.

Key concepts

Neutron-Proton Ratio

The stability of isotopes is heavily influenced by their neutron-proton ratio. This ratio is a basic measure of the balance between the number of neutrons and protons in the nucleus of an atom. For lighter elements, a 1:1 ratio often indicates a higher stability.

As elements become heavier, especially those with atomic numbers beyond calcium, a slightly greater number of neutrons than protons is preferred. This accounts for the increased repulsion that protons experience in larger nuclei.

Understanding this ratio helps in predicting isotope stability: isoptopes with ratios closer to what's typical for their atomic number are generally more stable.

Stable Nuclei

Nuclei's stability depends largely on how close their composition matches the ideal neutron-proton balance. Stable nuclei are those that do not easily undergo radioactive decay. They have a balance that prevents them from falling apart.

This balance is crucial as too few or too many neutrons compared to protons can make a nucleus unstable. In some cases, this lack of stability can result in isotopes seeking stability through decay, where they may emit particles to adjust their composition towards a stable state.

Nuclei closer to the balance have a lower tendency to disintegrate, thus categorizing them as stable.

Magic Numbers

Magic numbers in nuclear physics are specific numbers of protons or neutrons that create a complete and stable nuclear shell, similar to how noble gases have complete electron shells. When a nucleus has a magic number of protons or neutrons, it is notably more stable.

Magic numbers are important for predicting stability because nuclei with these counts tend to have a lower likelihood of radioactive decay, providing additional insight into why some isotopes are more stable than others.

Understanding magic numbers aids in explaining why certain numbers of nucleons (protons and neutrons) lead to extra stability in nuclei, making them resistant to changes.

Less Stable Isotopes

Isotopes that don't align with ideal neutron-proton ratios or lack magic numbers are often less stable. Their imbalance and susceptibility to decay categorize them as less stable.

Less stable isotopes are likely to experience transformations, meaning they might decay by emitting particles or radiation to achieve a more stable configuration.

Knowing the factors that contribute to an isotope's instability helps predict which isotopes are more likely to be found in nature in a stable form and which are more transient.