Question

How much kinetic energy is released and what is the daughter nucleus in the \(\beta^{+}\) decay of nitrogen- \(13 ?\)

Short answer

The daughter nucleus in the \(\beta^{+}\) decay of nitrogen-13 is \(^{13}_{6}C\), Carbon-13. The kinetic energy release is typically determined experimentally as it requires the precise mass difference, it can also be calculated using E=mc^2 if the mass values are known.

Step-by-step solution

Identify the Parent Nucleus

The parent nucleus is Nitrogen-13. We denote it as \(^{13}_{7}N\), where 13 is the mass number (sum of protons and neutrons) and 7 is the atomic number (number of protons).

Identify the Daughter Nucleus in \(\beta^{+}\) Decay

During \(\beta^{+}\) decay, a proton changes into a neutron, a positron (denoted by \(^{0}_{1}\beta^{+}\)), and a neutrino. Therefore, the atomic number decreases by one and the mass number remains unchanged. This results in the formation of Carbon-13 atom, which is denoted by \(^{13}_{6}C\). This is the daughter nucleus.

Calculate the Kinetic Energy Released

In this step, using the nuclear reactions we know that a small amount of mass is lost, which gets transformed into energy as per the equation \(E=mc^2\). This lost mass (or mass defect) is the difference between the mass of Nitrogen-13 and the sum of the mass of Carbon-13 and the positron. However, this mass difference is so small that it can be neglected, and the kinetic energy release is often measured directly as a part of the lab experiment. However, if the mass difference were given or could be found from reference values, we could calculate the energy released using \(E = mc^2\) where m is mass and c is the speed of light.

Key concepts

Kinetic Energy Release in Decay

In nuclear reactions, such as beta plus decay, energy is released in the form of kinetic energy. This is because, during the decay process, a small amount of the nucleus's mass is converted into energy. This is guided by the principle of mass-energy equivalence, as described by Einstein's equation, \(E=mc^2\). Here, \(E\) represents energy, \(m\) is the mass lost, and \(c\) is the speed of light.In the case of beta plus decay of Nitrogen-13, when a proton inside the nucleus is transformed into a neutron, the resulting positron and neutrino carry away the kinetic energy. While pinpointing the exact kinetic energy released can be complex without precise mass difference data, it is often measured directly. In practice, this release contributes to the kinetic energy required to power the resulting particles and features observably in lab experiments.

Daughter Nucleus Identification

In beta plus decay, the parent nucleus undergoes transformation where a proton is converted into a neutron, which results in a change in the element's identity. The atomic number, which is the count of protons, decreases by one because of this transformation. However, the mass number, representing the sum of protons and neutrons, remains unchanged.For Nitrogen-13, represented as \(^{13}_{7}N\), during \(\beta^{+}\) decay, one proton is converted into a neutron, resulting in the atomic number moving from 7 to 6. This establishes the daughter nucleus as Carbon-13, denoted as \(^{13}_{6}C\). Understanding the shift in atomic numbers is crucial to identify the daughter nucleus accurately, as it reflects the essential changes happening within the atom during decay.

Mass-Energy Equivalence in Nuclear Reactions

Mass-energy equivalence is a pivotal concept when it comes to nuclear reactions, such as beta plus decay. Einstein's famous equation \(E=mc^2\) illustrates that mass can be converted into energy. During nuclear reactions, the loss of a tiny amount of mass from the system results in a significant amount of energy being released.This concept applies to the beta plus decay of Nitrogen-13, where a small mass loss occurs when a proton changes into a neutron, accompanied by the emission of a positron and a neutrino. The mass lost is fractional yet results in the release of substantial energy that manifests as kinetic energy carried by the decay products. This principle not only helps calculate the potential energy release in theoretical contexts but also guides experimental setups where precise energy measurements are required.