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

Less stable isotopes: (a) \({ }_{3}^{9} \mathrm{Li}\), (b) \({ }_{11}^{25} \mathrm{Na}\), (c) \({ }_{21}^{48} \mathrm{Sc}\)

Step-by-step solution

Compare neutron-proton ratio for (a) and (b)

For (a) \({ }_{3}^{6} \mathrm{Li}\) and \({ }_{3}^{9} \mathrm{Li}\), calculate neutron-proton ratios by subtracting the atomic number (subscript) from the isotopic number (superscript). Thus, for \({ }_{3}^{6} \mathrm{Li}\) the ratio is \( \frac{6 - 3}{3} = 1.0 \) and for \({ }_{3}^{9} \mathrm{Li}\) it is \( \frac{9 - 3}{3} = 2.0 \). Similar computation can be done for (b) \({ }_{11}^{23} \mathrm{Na}\) and \({ }_{11}^{25} \mathrm{Na}\) giving ratios 1.091 and 1.273 respectively.

Predict stability for (a) and (b)

For both (a) and (b), the isotope with the higher neutron-proton ratio is less stable. Therefore, \({ }_{3}^{9} \mathrm{Li}\) is less stable than \({ }_{3}^{6} \mathrm{Li}\) and \({ }_{11}^{25} \mathrm{Na}\) is less stable than \({ }_{11}^{23} \mathrm{Na}\).

Compare proton number and neutron-proton ratio for (c)

For (c) \({ }_{20}^{48} \mathrm{Ca}\) and \({ }_{21}^{48} \mathrm{Sc}\), the isotopes have the same number of nucleons but different atomic numbers. Calculate neutron-proton ratios as in Step 1. For \({ }_{20}^{48} \mathrm{Ca}\) the ratio is 1.4 and for \({ }_{21}^{48} \mathrm{Sc}\) it is 1.286.

Predict stability for (c)

For isotopes with the same number of nucleons, the number of protons determines stability. Isotopes with more protons are generally less stable because of increased repulsion among them. Therefore, \({ }_{21}^{48} \mathrm{Sc}\) is less stable than \({ }_{20}^{48} \mathrm{Ca}\), despite its lower neutron-proton ratio.

Key concepts

Neutron-Proton Ratio

Understanding the neutron-proton ratio is crucial for predicting the stability of isotopes. The nucleus of an atom is composed of protons, which are positively charged, and neutrons, which have no charge.

The stability of an isotope depends largely on the balance between these particles. An ideal neutron-proton ratio allows the strong nuclear force, which holds the nucleus together, to counteract the electrostatic repulsion between protons, which tries to push them apart.

When this balance is disturbed by having too few or too many neutrons compared to protons, the isotope becomes more prone to radioactive decay as it seeks a more stable state. For light elements (up to calcium, with atomic number 20), the ideal ratio is close to 1:1. As the atomic number increases, however, more neutrons are needed to help maintain nuclear stability, leading to stable isotopes typically having neutron-to-proton ratios greater than 1:1.

Nuclear Chemistry

Nuclear chemistry is the subfield of chemistry dealing with radioactive substances and nuclear processes. At its core, it explores the changes in the composition of atomic nuclei and the energy these transformations can produce.

The stability of atomic nuclei is a fundamental aspect of nuclear chemistry. It involves the study of how isotopes decay, the types of radioactive emissions they can produce, and the half-life of radioactive materials. Nuclear chemists work with isotopes to understand their properties, how they interact, and how they can be applied in various fields such as medicine, energy, and industrial processes.

It's important for students to grasp the concepts of neutron-proton ratios and atomic numbers to understand nuclear reactions, which include fission, fusion, and radioactive decay, all pivotal in nuclear chemistry.

Atomic Number

The atomic number is a fundamental concept in chemistry and physics. It represents the number of protons found within an atom's nucleus and is denoted by the letter 'Z'.

Each element in the periodic table is defined by its atomic number, making it unique. It determines the chemical properties of the element and its position in the periodic table. Since protons are positively charged, the atomic number also gives the amount of positive charge in the nucleus.

When exploring isotopes, atoms of the same element with differing numbers of neutrons, the atomic number remains constant; thus, isotopes have the same number of protons but can vary widely in their stability due to differing neutron numbers. This variance in neutron number can lead to different neutron-proton ratios, which as mentioned previously, is significant in determining nuclear stability.