Problem 84 - Chemical Principles

Question

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?

Step-by-Step Solution

Answer

First decay: \( _{89}^{225}\text{Ac} \rightarrow _{87}^{221}\text{Fr} + _{2}^{4}\text{He} \), second decay: \( _{87}^{221}\text{Fr} \rightarrow _{85}^{217}\text{At} + _{2}^{4}\text{He} \), third decay: \( _{85}^{217}\text{At} \rightarrow _{83}^{213}\text{Bi} + _{2}^{4}\text{He} \). The n/p ratio of Bismuth-213 (final product) is closer to the band of stability than Actinium-225.

Step 1: Write the first decay equation

Identify the atomic number and mass number of Actinium-225 and then apply the emission of an \(\alpha\) particle (\textsc{Helium-4 nucleus}). An \(\alpha\) particle has 2 protons and 2 neutrons, so subtract these from Actinium-225 to get the first decay product. The equation for the first decay is: \[ _{89}^{225}\text{Ac} \rightarrow _{87}^{221}\text{Fr} + _{2}^{4}\text{He} .\]

Step 2: Write the second decay equation

Perform the same subtraction for the next \(\alpha\)-decay starting from the new isotope Francium-221. The second decay equation is: \[ _{87}^{221}\text{Fr} \rightarrow _{85}^{217}\text{At} + _{2}^{4}\text{He} .\]

Step 3: Write the third decay equation

Repeat the process for the third \(\alpha\) particle emission from the product of the second decay. The third decay equation is: \[ _{85}^{217}\text{At} \rightarrow _{83}^{213}\text{Bi} + _{2}^{4}\text{He} .\]

Step 4: Calculate neutron-to-proton (n/p) ratio for Actinium-225

Calculate the ratio of neutrons to protons for Actinium-225. The number of protons (atomic number) is 89, and the number of neutrons is the mass number minus the atomic number, which is \(225 - 89 = 136\). So, the initial n/p ratio is \(136/89\).

Step 5: Calculate n/p ratio for Bismuth-213

Calculate the ratio of neutrons to protons for the final product, Bismuth-213. It has 83 protons, so it has \(213 - 83 = 130\) neutrons. The final n/p ratio is \(130/83\).

Step 6: Compare n/p ratios and stability

Compare the initial and final n/p ratios. For Actinium-225, the n/p ratio is \(136/89 \: \approx \: 1.528\). For Bismuth-213, the n/p ratio is \(130/83 \: \approx \: 1.566\). Isotopes with n/p ratios that are closer to 1 are generally more stable, so the n/p ratio of the final product Bismuth-213 is closer to the band of stability.

Key Concepts

Actinium-225 Alpha Decay Series

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.

Neutron-to-Proton Ratio

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.

Band of Stability in Nuclear Chemistry

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.