Problem 24 - Chemistry: An Atoms-Focused Approach

Question

Calculate the neutron-to-proton ratios for each of the following and predict the decay pathways of the following radioactive isotopes: \((\mathrm{a})^{238} \mathrm{U} ;(\mathrm{b})^{186} \mathrm{Re} ;(\mathrm{c})^{86} \mathrm{Y}\).

Step-by-Step Solution

Answer

Question: Calculate the neutron-to-proton (n/p) ratios for the radioactive isotopes: (a) $^{238}\mathrm{U}$, (b) $^{186}\mathrm{Re}$, and (c) $^{86}\mathrm{Y}$. Based on these ratios, predict the decay pathways for each isotope. Answer: - (a) $^{238}\mathrm{U}$: The n/p ratio is 1.5870. This isotope will undergo beta(-) decay to decrease the number of neutrons and increase the number of protons, making it more stable. - (b) $^{186}\mathrm{Re}$: The n/p ratio is 1.4800. This isotope will most likely undergo beta(-) decay to reach a more stable configuration. - (c) $^{86}\mathrm{Y}$: The n/p ratio is 1.2051. This isotope will likely undergo beta(-) decay to decrease the number of neutrons and increase the number of protons, making it more stable.

Step 1: Determine the Atomic Numbers (Z) of each element using a Periodic Table

Refer to a periodic table and look for the atomic numbers of U, Re, and Y. The atomic numbers (Z) are: - Uranium (U): Z = 92 - Rhenium (Re): Z = 75 - Yttrium (Y): Z = 39

Step 2: Calculate the number of neutrons (N) for each isotope

Use the mass number (A) and atomic number (Z) to find the number of neutrons (N) in each isotope using the formula A = Z + N. - \((\mathrm{a}) ^{238} \mathrm{U}\): N = A - Z = 238 - 92 = 146 - \((\mathrm{b}) ^{186} \mathrm{Re}\): N = A - Z = 186 - 75 = 111 - \((\mathrm{c}) ^{86} \mathrm{Y}\): N = A - Z = 86 - 39 = 47

Step 3: Calculate the neutron-to-proton (n/p) ratios for each isotope

Divide the number of neutrons (N) by the number of protons (Z) to find the n/p ratios. - \((\mathrm{a}) ^{238} \mathrm{U}\): n/p = 146/92 = 1.5870 - \((\mathrm{b}) ^{186} \mathrm{Re}\): n/p = 111/75 = 1.4800 - \((\mathrm{c}) ^{86} \mathrm{Y}\): n/p = 47/39 = 1.2051

Step 4: Predict the decay pathways for each isotope based on their n/p ratios

For stable nuclei, the general rule is that lighter elements (Z < 20) should have an n/p ratio close to 1, while heavier elements (Z > 20) should have an n/p ratio ranging from 1 to 1.49. Therefore: - \((\mathrm{a}) ^{238} \mathrm{U}\): The n/p ratio of 1.5870 is greater than the stable range for heavy nuclei. Thus, this isotope will undergo beta(-) decay to decrease the number of neutrons and increase the number of protons, making it more stable. - \((\mathrm{b}) ^{186} \mathrm{Re}\): The n/p ratio of 1.4800 is within the stable range for heavy nuclei. However, it is still close to the upper limit of the stable range, and this isotope will most likely undergo beta(-) decay to reach a more stable configuration. - \((\mathrm{c}) ^{86} \mathrm{Y}\): The n/p ratio of 1.2051 is slightly above the stable range for light to medium nuclei (n/p ≈ 1). This isotope will likely undergo beta(-) decay to decrease the number of neutrons and increase the number of protons, making it more stable.

Key Concepts

Neutron-to-Proton Ratio

The neutron-to-proton ratio, also known as the n/p ratio, is crucial in determining the stability of a nucleus. It is calculated by dividing the number of neutrons by the number of protons in an atom. This ratio helps predict the radioactive behavior and potential decay pathways of isotopes.

In general:

• Lighter nuclei (with an atomic number, Z, less than 20) tend to have an n/p ratio close to 1.
• Heavier nuclei (Z greater than 20) often require a higher n/p ratio, around 1.3 to 1.5, to achieve stability.

When the n/p ratio deviates from these ranges, the nucleus is typically unstable and will likely undergo radioactive decay to reach a more stable state. This deviation guides predictions of decay types, such as beta decay, as seen in the isotopes of Uranium, Rhenium, and Yttrium.

Radioactive Isotopes

Radioactive isotopes, or radioisotopes, are unstable forms of elements that emit radiation over time. This instability is often due to an imbalance in the neutron-to-proton ratio within the nucleus. Various isotopes undergo decay at different rates, releasing energy in the form of particles or electromagnetic waves.

Understanding radioactive isotopes involves knowing:

• The original number of protons and neutrons in the nucleus.
• The decay process, which can transform the isotope into a more stable form, often a different element.
• The half-life, which is the time it takes for half of a sample of the isotope to decay.

In exercises where we examine isotopes like ^{238} ext{U}, ^{186} ext{Re}, and ^{86} ext{Y}, calculating the neutron count and the n/p ratio helps determine their stability and potential decay processes.

Beta Decay

Beta decay is a type of radioactive decay where a neutron in the nucleus is transformed into a proton, or vice versa. In this process, a beta particle, which is either an electron or a positron, is emitted from the nucleus.

This mechanism occurs when:

• The n/p ratio is too high, resulting in beta-minus ( ^{(-)}) decay, where a neutron converts into a proton, emitting an electron and an antineutrino.
• The n/p ratio is too low, leading to beta-plus ( ^{(+)}) decay, where a proton turns into a neutron, emitting a positron and a neutrino.

In the context of isotopes like ^{238} ext{U}, ^{186} ext{Re}, and ^{86} ext{Y}, beta-minus decay is typically expected. This is because these isotopes usually have a higher n/p ratio than what is considered stable, indicating an excess of neutrons that need to convert into protons to reach stability.