Question

Each of the following nuclei undergoes either beta decay or positron emission. Predict the type of emission for each: (a) \({ }_{38}^{90} \mathrm{Sr}\) (b) \({ }_{38}^{85} \mathrm{Sr}\) (d) sulfur-30. (c) potassium- 40 ,

Short answer

(a) Sr-90 will undergo beta decay. (b) Sr-85 will undergo beta decay. (c) Sulfur-30 will undergo positron emission. (d) Potassium-40 can undergo both beta decay and positron emission, but beta decay occurs more frequently (89% of the time).

Step-by-step solution

(a) Sr-90

Strontium-90 has 38 protons and 90-38=52 neutrons. The neutron/proton ratio is 52/38=1.37, which is higher than the stable ratio for lighter elements (around 1), indicating that it has too many neutrons relative to protons. Thus, beta decay is more likely to occur for Sr-90, converting a neutron into a proton and stabilizing the nucleus.

(b) Sr-85

Strontium-85 has 38 protons and 85-38=47 neutrons. The neutron/proton ratio is 47/38=1.24, which is closer to the stable ratio for lighter elements. However, it still has slightly more neutrons than stable nuclei with this mass number, so beta decay is more likely to occur for Sr-85, converting a neutron into a proton and stabilizing the nucleus.

(c) Sulfur-30

Sulfur-30 has a atomic number of 16 (since sulfur is the 16th element in the periodic table), so it has 16 protons and 30-16=14 neutrons. The neutron/proton ratio is 14/16=0.875, which is lower than the stable ratio for lighter elements, indicating that it has too few neutrons relative to protons. Thus, positron emission is more likely to occur for Sulfur-30, converting a proton into a neutron and stabilizing the nucleus.

(d) Potassium-40

Potassium-40 has an atomic number of 19 (since potassium is the 19th element in the periodic table), so it has 19 protons and 40-19=21 neutrons. The neutron/proton ratio is 21/19=1.11, which is slightly higher than the stable ratio for lighter elements, but still close. In this case, both beta decay and positron emission can stabilizing the nucleus, with beta decay leading to the formation of Calcium-40 and positron emission leading to the formation of Argon-40. In reality, both types of decay occur, but positron emission occurs around 11% of the time and beta decay around 89% of the time.

Key concepts

Beta Decay

Beta decay is a type of radioactive decay in which a neutron in an unstable nucleus is transformed into a proton. This process increases the number of protons while decreasing the number of neutrons. As a result, an electron, known as a beta particle, and an antineutrino are emitted from the nucleus.
Some important characteristics of beta decay include:

• Occurs in nuclei with a high neutron-to-proton ratio.
• Helps in lowering the neutron count relative to protons, thus stabilizing the nucleus.
• Common in heavier elements where stable nuclei have higher neutron-to-proton ratios.

In the case of Strontium-90 and Strontium-85, both have relatively high neutron-to-proton ratios, meaning beta decay is a likely occurrence to achieve stability by converting surplus neutrons into protons. As beta decay reduces the relative number of neutrons, it pushes these isotopes toward a more energetically favorable state.

Positron Emission

Positron emission is another form of radioactive decay, but it involves the conversion of a proton into a neutron. This conversion leads to the release of a positron and a neutrino. Positrons are the antimatter equivalent of electrons, carrying a positive charge rather than a negative one. The emission of a positron helps to reduce the proton number in a nucleus.
Key points about positron emission include:

• Tends to occur in nuclei with a low neutron-to-proton ratio.
• Reduces the count of protons, contributing toward nuclear stability.
• More frequent in lighter elements where stable nuclei typically have a lower neutron-to-proton ratio.

Sulfur-30 exemplifies a situation where positron emission is likely to happen, as it possesses a neutron-to-proton ratio that is lower than ideal for achieving stability. By decreasing its proton number, the nucleus becomes more balanced.

Neutron-to-Proton Ratio

The neutron-to-proton ratio is a crucial factor in determining the stability of a nucleus. For light atoms with low atomic numbers, a neutron-to-proton ratio close to 1 is generally stable. As the atomic number increases, the ratio for stability also increases slightly due to the need for additional neutrons to offset the repulsive forces between the positively charged protons.
In decay processes:

• A high neutron-to-proton ratio favors beta decay to convert excess neutrons into protons.
• A low neutron-to-proton ratio, however, is more suited to positron emission, reducing the number of protons.
• The right balance can sometimes involve both processes, as seen in Potassium-40, where both beta decay and positron emission contribute to achieving a stable state.

Understanding the neutron-to-proton ratio is essential for predicting the type of decay a particular isotope may undergo, thereby revealing its path toward a more stable nuclear configuration.