Problem 23 - Chemistry and Chemical Reactivity

Question

(a) Which of the following nuclei decay by \(_{-1}^{0} \beta\) decay? $$^{1} \mathrm{H} \quad^{23} \mathrm{Mg} \quad^{32} \mathrm{P} \quad^{20} \mathrm{Ne}$$ (b) Which of the following nuclei decay by \(^{0}_{+1} \beta\) decay? $$^{235} \mathrm{U} \quad^{35} \mathrm{Cl} \quad^{38} \mathrm{K} \quad^{24} \mathrm{Na}$$

Step-by-Step Solution

Answer

(a) \(^{32} \mathrm{P}\) decays by \(_{-1}^{0} \beta\) decay; (b) \(^{38} \mathrm{K}\) decays by \(^{0}_{+1} \beta\) decay.

Step 1: Understanding Beta Decay Types

Beta decay comes in two types: \(_{-1}^{0} \beta\)-decay (beta-minus) and \(^{0}_{+1} \beta\)-decay (beta-plus). In beta-minus decay, a neutron is transformed into a proton, resulting in an increase in the atomic number. In beta-plus decay, a proton is transformed into a neutron, resulting in a decrease in the atomic number.

Step 2: Identifying Beta-minus Decay Candidates

For \(_{-1}^{0} \beta\) decay, the nucleus must have an excess of neutrons. Isotope \(^{32} \mathrm{P}\) is known to undergo beta-minus decay. It decays because it has a neutron excess for its atomic number, transforming a neutron into a proton. \(^{1} \mathrm{H}\), \(^{23} \mathrm{Mg}\), and \(^{20} \mathrm{Ne}\) do not typically experience beta-minus decay.

Step 3: Identifying Beta-plus Decay Candidates

For \(^{0}_{+1} \beta\) decay, the nucleus must have an excess of protons. \(^{38} \mathrm{K}\) is known to undergo beta-plus decay by converting a proton into a neutron to reach a more stable state. The other isotopes, \(^{235} \mathrm{U}\), \(^{35} \mathrm{Cl}\), and \(^{24} \mathrm{Na}\), do not typically undergo beta-plus decay.

Key Concepts

beta-minus decay

Beta-minus decay is a process where a neutron transforms into a proton within an atomic nucleus. This process emits a beta particle, which is an electron, and an antineutrino. Often, nuclei that undergo beta-minus decay have an excess of neutrons, causing the neutron-to-proton ratio to be higher than desired for stability.
This type of decay results in the atomic number increasing by one, while the atomic mass remains unchanged. For example, in beta-minus decay, phosphorus-32 \(_{15}^{32} \mathrm{P}\) transforms into sulfur-32 \(_{16}^{32} \mathrm{S}\), increasing its atomic number from 15 to 16.
This transformation helps the nucleus achieve a more stable state by balancing the neutron-to-proton ratio.

beta-plus decay

In beta-plus decay, a proton is converted into a neutron. This process emits a positron, which is the antimatter equivalent of an electron, along with a neutrino. Nuclei that undergo beta-plus decay generally have more protons than neutrons, leading to an unstable configuration.
When beta-plus decay occurs, the atomic number decreases by one, though the atomic mass remains constant. An example is the decay of potassium-38 \(_{19}^{38} \mathrm{K}\), which is undergoing this transformation into argon-38 \(_{18}^{38} \mathrm{Ar}\), where the transition results in the reduction of its atomic number from 19 to 18.
Beta-plus decay assists in reaching a more stable, balanced state by adjusting the excess of protons.

nuclear stability

Nuclear stability refers to the resistance of a nucleus to change. A stable nucleus does not spontaneously change its composition, because the forces holding it together are sufficient to overcome internal disruptions. Key to understanding nuclear stability is knowing that a balanced ratio of neutrons to protons is essential for a nucleus to remain stable.
Unstable nuclei have an inappropriate neutron-to-proton ratio, leading to nuclear decay processes, such as beta decay, aiming to achieve balance. Through such decay, a nucleus seeks equilibrium, moving from a higher energy state to a lower, more stable one.

• High stability: Balanced forces between nuclear particles.
• Low stability: Leads to decay processes, like beta decay.

Understanding the balance between forces in a nucleus helps predict which isotopes will undergo beta decay.

neutron-to-proton ratio

The neutron-to-proton ratio is a critical aspect determining nuclear stability. For a nucleus to be stable, there needs to be a proper balance between the number of neutrons and protons. This balance is contingent on the size and mass of the nucleus.
Light elements, with fewer protons, usually have a neutron-to-proton ratio close to 1:1. Larger nuclei often require more neutrons for every proton to maintain stability. This is due to the increasing repulsive electromagnetic force between protons as their numbers grow.
Deviations in this balance lead to various decay processes:

• Excess neutrons result in beta-minus decay.
• Excess protons lead to beta-plus decay.

This ratio is a guiding principle in predicting the type of decay an isotope might undergo, ultimately guiding it towards a stable structure.