Question

\({ }_{13} \mathrm{Al}^{27}\) is a stable isotope. \({ }_{13} \mathrm{Al}^{29}\) is expected to disintegrate by (a) \(\alpha\) emission (b) \(\beta\) emission (c) positron emission (d) proton emission

Short answer

Disintegration is expected by \( \beta \) emission (option b).

Step-by-step solution

Understand the Problem

We need to determine the type of disintegration for the isotope \( _{13}\text{Al}^{29} \). Given that \( _{13}\text{Al}^{27} \) is stable, we aim to find out which disintegration process helps \( _{13}\text{Al}^{29} \) reach stability.

Evaluate Stability of \( _{13}\text{Al}^{27} \)

First, note that Aluminum-27 is stable. It has 13 protons and 14 neutrons, meaning a neutron-to-proton ratio (n/p) of \( \frac{14}{13}\approx 1.08\). This is a common trait of stability for lighter elements, where the n/p ratio is near 1.

Assess Neutron-to-Proton Ratio for \( _{13}\text{Al}^{29} \)

The isotope \( _{13}\text{Al}^{29} \) has 13 protons and 16 neutrons, giving an n/p ratio of \( \frac{16}{13} \approx 1.23 \). This ratio is higher than that of stable Aluminum-27, suggesting an excess of neutrons.

Determine Likely Disintegration Process

Since \( _{13}\text{Al}^{29} \) has too many neutrons, it needs to convert a neutron into a proton to achieve a more favorable n/p ratio. This is typically achieved through \( \beta \)-emission, where a neutron converts to a proton, emitting a beta particle (an electron).

Confirm with Options

Among the provided options: (a) \( \alpha \) emission, (b) \( \beta \) emission, (c) positron emission, and (d) proton emission, option (b) \( \beta \) emission is the process that decreases the neutron count while increasing the proton count, leading towards the stable n/p ratio.

Key concepts

Isotope Stability

Isotopes are variants of elements that have the same number of protons but different numbers of neutrons. This variety in neutron numbers leads to differences in mass and occasionally impacts stability. Stable isotopes, like \(_{13}\text{Al}^{27}\), have an optimal arrangement of protons and neutrons that prevents radioactive decay. However, certain isotopes become unstable when this balance is disrupted.

Aluminum-27, for instance, has 13 protons and 14 neutrons, which provides a neutron-to-proton (n/p) ratio of approximately 1.08. This n/p ratio is characteristic of stability in lighter elements, typically hovering around the value of 1. Stability is crucial because it defines whether an isotope will remain unchanged or decay over time. So, when examining isotopes, determining their stability helps predict their behavior and possible decay pathways.

Neutron-to-Proton Ratio

The neutron-to-proton ratio (/p) is a significant factor in identifying isotope stability. This ratio is derived by dividing the number of neutrons by the number of protons in an isotope.

For \(_{13}\text{Al}^{27}\), the n/p ratio is around 1.08, a range that supports stability for lighter elements. In contrast, \(_{13}\text{Al}^{29}\) has a higher n/p ratio of approximately 1.23 due to its additional neutrons, making it prone to instability.

When an isotope has too many neutrons, the nucleus may become unstable, often requiring certain decay processes to adjust this ratio. The mismatch in the n/p ratio can signal that an isotope is likely to undergo transformations, such as through beta decay. Understanding the n/p ratio is essential for predicting which isotopes are stable and which are susceptible to radioactive decay.

Beta Emission

Beta emission is a common decay process that helps isotopes achieve a more stable neutron-to-proton ratio. When an isotope like \(_{13}\text{Al}^{29}\) has an excess of neutrons, a neutron in the nucleus may transform into a proton. This process results in the emission of a beta particle, which is essentially a high-speed electron.

The conversion of a neutron into a proton during beta emission effectively reduces the neutron count and increases the proton count. This transformation helps the isotope move toward a more stable n/p ratio, akin to that of \(_{13}\text{Al}^{27}\).

Beta emission is an important mechanism because it not only affects the balance within an isotope's nucleus but also changes the chemical identity of the element. By increasing the number of protons, the atomic number steps up by one, leading to the formation of a new element. Understanding beta emission is critical for recognizing how unstable isotopes reach stability and how chemical elements evolve.