Question

Oxygen- \(19 \beta^{-}\) decays. What is the daughter nucleus, and what may be said of the kinetic energy of the emitted \(\beta^{-}\) particle?

Short answer

The daughter nucleus is fluorine-19 (F(9,19)). The kinetic energy of the emitted beta particle cannot be determined from the information provided, but it is variable and carries away the difference in energy between the parent and daughter nucleus.

Step-by-step solution

Write down the nuclear equation for beta-minus decay

In the general equation for beta-minus decay, X(Z,A) -> Y(Z+1,A) + e− + \overline{v}e, replace X(Z,A) with the oxygen-19 nucleus. Oxygen-19 is written as O(8,19). The resulting equation is: O(8,19) -> Y(Z+1,A) + e− + \overline{v}e.

Identify the daughter nucleus

In the nuclear equation, Z is increased by 1 while A remains constant. Thus, the daughter nucleus has 9 protons and the same total number of nucleons, 19. The nucleus with atomic number 9 is fluorine. Hence, Y(Z+1,A) is replaced with F(9,19) and the complete nuclear equation is: O(8,19) -> F(9,19) + e− + \overline{v}e.

Determine the kinetic energy of the emitted beta particle

In any nuclear decay, energy is conserved. The kinetic energy of the emitted beta particle comes from the conversion of the mass of the neutron into energy according to Einstein's equation E=mc^2. However, the actual calculation of this energy requires knowledge of the masses of the neutron, proton, electron and anti-neutrino, which are not given in the problem. Therefore, it's not possible to calculate the kinetic energy of the beta particle from the information provided. In general however, it can be said that the kinetic energy of the emitted beta particle is variable and carries away the difference in energy between the parent nucleus and daughter nucleus so as to conserve energy.

Key concepts

Understanding Nuclear Equations

Nuclear equations are like the math of particles! They help us understand how elements change during radioactive decay. When beta-minus decay happens, a neutron transforms into a proton, increasing the element’s atomic number by one. The nucleus loses a beta particle (an electron) and an antineutrino. The nuclear equation for beta-minus decay from Oxygen-19 looks like this:

• For Oxygen-19: \(\text{O}(8,19) \to \text{F}(9,19) + e^- + \overline{v}_e\)

The key is that the atomic mass (A) stays the same, while the atomic number (Z) goes up by one. This change results in the formation of a new element, which leads us to our next concept.

Identifying the Daughter Nucleus

The daughter nucleus is the new nucleus formed after radioactive decay. In the case of Oxygen-19, it turns into Fluorine-19. But why? In beta-minus decay, like our example, the atomic number increases by one, but the mass number remains constant.

• Oxygen-19 starts with 8 protons. After decay, it has 9 protons - making it Fluorine-19!

We identify the daughter nucleus by knowing the new element based on the updated atomic number. So, every time a beta-minus decay happens, you can spot the daughter nucleus by looking one element up on the periodic table.

Kinetic Energy of Beta Particles

The kinetic energy of beta particles is a key part of beta decay. But what is it, and why does it matter? When an electron (beta particle) and an antineutrino are emitted, energy is released. This energy is shared between the beta particle and the antineutrino, which means beta particles have variable kinetic energy.

• The energy source is the mass difference between the parent and daughter nucleus, as described by Einstein's equation \(E = mc^2\).
• Even though exact calculations need more data, we know this energy helps preserve the balance (or conservation) of energy in the decay.

Understanding the kinetic energy helps scientists and students alike figure out the behavior of beta particles in various materials.