Question

Each of the following nuclei undergoes either beta decay or positron emission. Predict the type of emission for each: (a) tritium, \({ }_{1}^{3} \mathrm{H},(\mathbf{b}){ }_{38}^{89} \mathrm{Sr}\), (c) iodine-120, (d) silver-102.

Short answer

a) Tritium (\({ }_{1}^{3}\mathrm{H}\)) - Beta decay b) Strontium-89 (\({ }_{38}^{89}\mathrm{Sr}\)) - Beta decay c) Iodine-120 (\({ }_{53}^{120}\mathrm{I}\)) - Beta decay d) Silver-102 (\({ }_{47}^{102}\mathrm{Ag}\)) - Positron emission

Step-by-step solution

Process Each Nucleus Individually

Let's analyze each nucleus and predict the type of emission:

a) Tritium (\({ }_{1}^{3}\mathrm{H}\))

Tritium has 1 proton and 2 neutrons. Since the neutron-to-proton ratio is higher than what is ideal for a stable nucleus, we would expect beta decay to occur.

b) Strontium-89 (\({ }_{38}^{89}\mathrm{Sr}\))

Strontium-89 has 38 protons and 51 neutrons. The neutron-to-proton ratio is slightly higher than what is considered stable. So, we predict that beta decay will occur in this nucleus.

c) Iodine-120#\({ }_{53}^{120}\mathrm{I}\)

Iodine-120 has 53 protons and 67 neutrons. The neutron-to-proton ratio is higher than the range for a stable nucleus, so we predict that beta decay will occur.

d) Silver-102#\({ }_{47}^{102}\mathrm{Ag}\)

Silver-102 has 47 protons and 55 neutrons. Its neutron-to-proton ratio is smaller than the stable range that is typical for this element. We predict that positron emission will occur in this nucleus as this process will decrease the number of protons by changing one of them into a neutron. In summary: a) Tritium (\({ }_{1}^{3}\mathrm{H}\)) - Beta decay b) Strontium-89 (\({ }_{38}^{89}\mathrm{Sr}\)) - Beta decay c) Iodine-120 (\({ }_{53}^{120}\mathrm{I}\)) - Beta decay d) Silver-102 (\({ }_{47}^{102}\mathrm{Ag}\)) - Positron emission

Key concepts

Beta Decay

Beta decay is a nuclear process by which an unstable nucleus achieves a more stable state. This transformation involves the emission of a beta particle, which can be either an electron (in beta-minus decay) or a positron (in beta-plus decay).

In the case of beta-minus decay, a neutron in an atomic nucleus is transformed into a proton, and an electron (the beta particle) is emitted alongside an antineutrino. The formula for this decay can be expressed as: \( n \rightarrow p^+ + e^- + \bar{u}_e \), where \( n \) is a neutron, \( p^+ \) is a proton, \( e^- \) is the emitted electron, and \( \bar{u}_e \) is the antineutrino.

This mechanism helps to balance the neutron-to-proton ratio by increasing the number of protons in the nucleus. An example of beta decay is seen in tritium \( (_1^3\text{H}) \), where a neutron is converted to a proton, resulting in helium-3 \( (_2^3\text{He}) \).

Positron Emission

Positron emission, also known as beta-plus decay, is another type of nuclear decay in which a proton within the nucleus is converted into a neutron while releasing a positron (the antimatter counterpart of an electron) and a neutrino. This process can be represented by the equation: \( p^+ \rightarrow n + e^+ + u_e \).

Positron emission occurs in proton-rich nuclides that have a lower neutron-to-proton ratio than the stable range. By converting a proton to a neutron, the nucleus moves towards a more stable state. A practical instance of positron emission is found in silver-102 \( (_{47}^{102}\text{Ag}) \), where a proton is transformed into a neutron, leading to palladium-102 \( (_{46}^{102}\text{Pd}) \).

This form of decay is critical in medical applications such as positron emission tomography (PET), a diagnostic imaging technique that utilizes radioactive substances emitting positrons.

Neutron-to-Proton Ratio

The neutron-to-proton ratio is a fundamental aspect in determining the stability of an atomic nucleus. A certain balance between the number of neutrons and protons is required for a nucleus to be stable. If this balance is disrupted, the nucleus may undergo radioactive decay to achieve stability.

Stable nuclei typically have a neutron-to-proton ratio that closely matches the line of stability on the nuclear chart. Lighter elements usually have a ratio close to 1:1, whereas heavier elements tend to have more neutrons than protons. When the ratio is too high (neutron-rich), beta decay can help to adjust the balance by converting neutrons into protons. Conversely, when the ratio is too low (proton-rich), processes like positron emission or electron capture can increase the relative number of neutrons.

Understanding the neutron-to-proton ratio is crucial when predicting the type of decay that an unstable nucleus might undergo. For instance, in the step-by-step solutions of our exercises, analyzing this ratio has allowed us to predict whether a nucleus would undergo beta decay or positron emission.