Question

Each of the following nuclei undergoes either beta or positron emission. Predict the type of emission for each: (a) \({ }_{32}^{66} \mathrm{Ge}\), (b) \({ }_{45}^{105} \mathrm{Rh}\), (c) iodine-137, (d) cerium-133.

Short answer

(a) Germanium-66 (\({ }_{32}^{66} \mathrm{Ge}\)) undergoes beta-minus (β-) decay. (b) Rhodium-105 (\({ }_{45}^{105} \mathrm{Rh}\)) undergoes beta-minus (β-) decay. (c) Iodine-137 (\({ }_{53}^{137} \mathrm{I}\)) undergoes beta-minus (β-) decay. (d) Cerium-133 (\({ }_{58}^{133} \mathrm{Ce}\)) undergoes beta-minus (β-) decay. All the given nuclei undergo beta-minus (β-) decay due to the excess of neutrons in their nuclei.

Step-by-step solution

(a) \({ }_{32}^{66} \mathrm{Ge}\)

Given, Germanium (\({ }_{32}^{66} \mathrm{Ge}\)) has 32 protons (Z = 32) and (66 - 32) = 34 neutrons (N = 34). Since N > Z, there is an excess of neutrons, and therefore, it will undergo beta-minus (β-) decay.

(b) \({ }_{45}^{105} \mathrm{Rh}\)

Given, Rhodium (\({ }_{45}^{105} \mathrm{Rh}\)) has 45 protons (Z = 45) and (105 - 45) = 60 neutrons (N = 60). Since N > Z, there is an excess of neutrons, and therefore, it will undergo beta-minus (β-) decay.

(c) Iodine-137

Iodine-137 (\({ }_{53}^{137} \mathrm{I}\)) has 53 protons (Z = 53) and (137 - 53) = 84 neutrons (N = 84). Since N > Z, there is an excess of neutrons, and therefore, it will undergo beta-minus (β-) decay.

(d) Cerium-133

Cerium-133 (\({ }_{58}^{133} \mathrm{Ce}\)) has 58 protons (Z = 58) and (133 - 58) = 75 neutrons (N = 75). Since N > Z, there is an excess of neutrons, and therefore, it will undergo beta-minus (β-) decay. In conclusion, all the given nuclei (Germanium-66, Rhodium-105, Iodine-137, and Cerium-133) will undergo beta-minus (β-) decay due to the excess of neutrons in their nuclei.

Key concepts

Nuclear Chemistry

Nuclear chemistry is a fascinating field that explores the changes in the nucleus of atoms, which is the core where protons and neutrons are tightly bound. Whether for generating energy, dating ancient artifacts, or treating cancer, understanding how atomic nuclei transform through different processes is fundamental in nuclear chemistry. These changes often involve alterations in the number of protons and neutrons to attain stability, an aspect pivotal to the behavior of different isotopes and elements.

For example, an unstable nucleus might release energy through various types of radioactive decay. In studying nuclear chemistry, we discover the forces that hold the nucleus together and what happens when the balance between protons and neutrons is disturbed. This understanding is not just academic; it has real-world applications in medical treatments, energy production, and even in how we approach environmental challenges.

Radioactive Decay

Radioactive decay is a spontaneous process by which an unstable atomic nucleus loses energy by emitting radiation. This transformation leads to the change of one element into another and is of particular interest in nuclear chemistry. There are several types of decay, such as alpha, beta, and gamma decay, each characterized by the particles ejected from the nucleus. Beta decay, for instance, includes the emission of beta particles, which can be electrons or positrons.

The rate of decay is measured by the half-life, which is the time it takes for half of a sample of a radioactive substance to decay. This provides a predictable pattern for understanding how long certain elements will remain active and is crucial in applications such as nuclear medicine where radioactive tracers are used for diagnostics or treatment.

Neutron-Proton Ratio

The neutron-proton ratio within an atomic nucleus plays a critical role in determining the stability of an atom. Nuclides with a balance between neutrons and protons tend to be stable, while those with an imbalance tend to be unstable, often leading to radioactive decay to reach a more stable state. An excess of either particles prompts different decay processes.

For stable isotopes, this ratio is approximately 1:1 for lighter elements and becomes slightly greater for heavier elements, due to the need for more neutrons to counterbalance the increasing repulsive force between protons as their number grows. Understanding the neutron-to-proton ratio is key in predicting which type of decay a nuclide will undergo. Excess neutrons, as seen in the exercise examples, tend to result in beta-minus decay where a neutron is transformed into a proton, emitting an electron (beta particle) and an antineutrino in the process.

Positron Emission

Positron emission, also known as beta-plus decay, is a type of radioactive decay in which a proton in an atom's nucleus is transformed into a neutron and a positron is emitted along with a neutrino. The positron is the antiparticle of the electron, with the same mass but a positive charge.

This process occurs in nuclides where the neutron-proton ratio is too low, meaning there are too many protons. This imbalance can be addressed by the conversion of a proton into a neutron, thereby increasing the ratio towards more stability. Positron emission is employed in medical imaging, notably in Positron Emission Tomography (PET) scans, which can provide detailed images of metabolic processes in the body. It's a stark example of nuclear chemistry's applications in health and diagnostics, offering insights into the body at a molecular level.