Question

Which of the following is least likely to be stable? (a) \(\mathrm{Ca}^{40}\) (b) \(\mathrm{Al}^{30}\) (c) \(\mathrm{Sn}^{119}\) (d) \(\mathrm{Mn}^{55}\)

Short answer

Aluminum-30 (b) is least likely to be stable.

Step-by-step solution

Understanding Nuclear Stability

Stable nuclei are generally those that have an appropriate balance of protons and neutrons. Isotopes further from the band of stability, especially with an imbalance of protons to neutrons, tend to be less stable. This commonly involves confirming that isotopes follow known patterns of stability such as the neutron-proton ratio, magic numbers, and the presence of even numbers of nucleons.

Analyzing Given Isotopes

Examine each given isotope for known stability patterns. For instance, nuclei with even numbers of protons and neutrons are usually more stable. Elements with atomic numbers greater than 82 are typically unstable. Also, stable nuclei have neutron to proton ratios within certain ranges for their atomic number.

Making Comparisons

Compare the isotopes to determine which is least likely to be stable. (a) \( \text{Ca}^{40} \) - Calcium-40 has 20 protons and 20 neutrons, an even-even nucleus, which is typically stable. (b) \( \text{Al}^{30} \) - Aluminum-30 has 13 protons and 17 neutrons, an odd-even nucleus, and is away from the band of stability. (c) \( \text{Sn}^{119} \) - Tin-119 has 50 protons (a magic number) and 69 neutrons, suggesting stability. (d) \( \text{Mn}^{55} \) - Manganese-55 has 25 protons and 30 neutrons, an odd-odd nucleus, but is relatively stable.

Identifying the Least Stable Isotope

Upon comparison, \( \text{Al}^{30} \) (Option b) is least likely to be stable due to having an odd number of protons, a relatively high neutron to proton ratio compared to the atomic number, and not fitting well within the band of stability.

Key concepts

Neutron-to-Proton Ratio

The neutron-to-proton ratio is a critical concept in understanding nuclear stability. Atoms are composed of protons and neutrons in the nucleus, with protons possessing a positive charge while neutrons are neutral. However, having just the right ratio of these subatomic particles is key to the stability of an atom's nucleus.

For lighter elements (those with atomic numbers up to approximately 20), the stable ratio of neutrons to protons is about 1:1. As we move to heavier elements, the neutron-to-proton ratio gradually increases. This is because the repulsive forces between the positively charged protons grow with the increase in the number of protons; hence, more neutrons are necessary to buffer and stabilize the nucleus.

If the neutron-to-proton ratio is too low or too high, the nucleus will become unstable, leading to radioactivity in the form of alpha decay, beta decay, or other nuclear reactions. An unstable nucleus will continue to decay until it reaches a stable state, which could take fractions of a second to millions of years, depending on the isotope.

Band of Stability

The band of stability is a graphical representation that helps us visualize which nuclei are likely to be stable and which are not. Within a chart plotting the number of protons versus the number of neutrons, the stable isotopes plot within a particular region shaped like a belt, known as the 'band of stability.'

Nuclei that lie within this band have a balanced neutron-to-proton ratio and do not undergo radioactive decay under normal circumstances. For light elements, the ratio is close to 1:1, but as mentioned earlier, the ratio becomes higher as the elements get heavier because neutrons are needed to overcome the electrostatic repulsion between protons.

Isotopes outside this band tend to be radioactive and will seek stability through various modes of radioactive decay. For instance, isotopes above the band have too many neutrons and tend to decay by beta emission, while those below the band are neutron-deficient and may undergo positron emission or electron capture to achieve balance.

Magic Numbers in Nuclear Physics

In nuclear physics, the term 'magic numbers' refers to the number of nucleons (either protons or neutrons) in a nucleus that are arranged in complete shells. According to the shell model of the nucleus, nucleons exist in different energy levels or shells, similar to the arrangement of electrons in an atom.

The concept mirrors the noble gases in the periodic table—the 'magic' configuration lends extraordinary stability to the nucleus. The most widely recognized magic numbers for protons and neutrons are 2, 8, 20, 28, 50, 82, and 126. When a nucleus has a magic number of protons and/or neutrons, it is far more stable against nuclear reactions and radioactive decay.

For instance, in the exercise solution, tin-119 (with 50 protons) is deemed more stable despite having more neutrons than protons. This is because 50 is a magic number, granting it a stability that outweighs the deviation from the typical neutron-to-proton ratio. Interestingly, nuclei with a magic number of both protons and neutrons (called 'double magic') are especially stable, such as helium-4, oxygen-16, and calcium-40.