Actinium-225 decays by successive emission of three \(\alpha\) particles. (a) Write the nuclear equations for the three decay processes. (b) Compare the neutron-to-proton ratio of the final daughter product with that of actinium-225. Which is closer to the band of stability?

Understanding the alpha decay series of Actinium-225 helps illustrate one of the pathways by which unstable nuclei achieve stability. Alpha decay is a process by which a nucleus emits an alpha particle—a helium nucleus, composed of two protons and two neutrons—leading to a new element with an atomic number decreased by two and a mass number decreased by four.

Actinium-225 undergoes a series of alpha decays, progressively shedding alpha particles until a more stable nucleus is formed. During the first step of this decay series, Actinium-225 emits an alpha particle to become Francium-221. Subsequently, Francium-221 goes through another alpha decay and transmutes into Astatine-217. Lastly, Astatine-217 emits a final alpha particle to make Bismuth-213. Each emission decreases the mass number by four and the atomic number by two, as detailed in the sequential nuclear equations.

Key Takeaways from Alpha Decay Series

• Alpha decay leads to new elements with lower atomic and mass numbers.
• The decay series depicts a chain of transformations towards stability.
• The number of alpha decays gives insight into the distance from stability the original nucleus was.

These step-by-step transformations show the pathway Actinium-225 takes to approach a stable configuration, giving us a clear understanding of one of the essential mechanisms in nuclear decay.

Nuclei are made up of neutrons and protons, and the stability of a nucleus is greatly influenced by the ratio of these particles. The neutron-to-proton ratio (n/p ratio) is a critical factor in understanding why some isotopes are stable and others are radioactive and undergo decay processes.

A stable n/p ratio ensures that the strong nuclear force, which holds the nucleus together, sufficiently counters the electrostatic repulsion between protons. When a nucleus has too many or too few neutrons relative to protons, it becomes unstable and can decay to reach a more stable configuration. For lighter elements, the ratio of stability is approximately 1:1, while for heavier elements, this ratio grows larger to accommodate the increased electrostatic repulsion among the greater number of protons.

In the context of the given exercise, Actinium-225 begins with an n/p ratio of about 1.528, which reduces through alpha decay series to approximately 1.566 for Bismuth-213. Although the ratio of Bismuth-213 suggests a slight increase, it's crucial to recognize the overall trend towards stability within the context of heavier elements where a higher n/p ratio is typical for stable nuclei.

The band of stability is a concept that illustrates the range of neutron-to-proton ratios where nuclides are considered stable and do not undergo radioactive decay. This band is often represented on a graph with the number of neutrons versus the number of protons.

Inside the band, nuclides have a balanced n/p ratio that favors stability; those outside the band are radioactive and tend to decay toward stability. Light nuclides (with lower atomic numbers) are stable with n/p ratios near 1:1, implying an equal number of neutrons and protons. As the atomic number increases, the ratio shifts, and more neutrons are needed relative to protons to counter the growing proton-proton repulsion.

Implications of the Band of Stability

• Nuclei with high n/p ratios tend to undergo beta decay to convert a neutron into a proton.
• Nuclei with low n/p ratios may engage in positron emission or electron capture to convert a proton into a neutron.
• Understanding the band of stability can predict the type of decay an unstable nuclide might undergo.

Bismuth-213's n/p ratio inches closer to the band, signifying that, despite radioactivity and inherent instability, it is moving nearer to a balance conducive to stability among isotopes of similar mass numbers.