Identify the unknown isotope $X$ in the following decays. $\text{X}{\to }^{40}\text{K}+{\text{e}}^{+}+\nu$

Verified step by step guidance

1. Understand the problem: The given decay involves an unknown isotope (X) decaying into potassium-40 (40K), a positron (e+), and a neutrino (ν). This is a beta-plus (β+) decay, where a proton in the nucleus of the parent isotope is converted into a neutron, emitting a positron and a neutrino.
2. Apply the principle of conservation of nucleon number: The total number of nucleons (protons + neutrons) must remain the same before and after the decay. Since 40K has a mass number of 40, the unknown isotope X must also have a mass number of 40.
3. Apply the principle of conservation of charge: In β+ decay, the atomic number of the daughter nucleus decreases by 1 because a proton is converted into a neutron. Potassium (K) has an atomic number of 19, so the unknown isotope X must have an atomic number of 20 (19 + 1).
4. Identify the element with atomic number 20: Using the periodic table, the element with atomic number 20 is calcium (Ca). Therefore, the unknown isotope is calcium-40 (40Ca).
5. Summarize the result: The unknown isotope X is identified as calcium-40 (40Ca), which undergoes β+ decay to produce potassium-40 (40K), a positron, and a neutrino.

Key Concepts

Isotopes

Isotopes are variants of a particular chemical element that have the same number of protons but different numbers of neutrons. This results in different atomic masses for the isotopes of the same element. Understanding isotopes is crucial in nuclear physics, as they can undergo various types of decay, affecting their stability and the reactions they participate in.

Beta Plus Decay

Beta plus decay is a type of radioactive decay in which a proton in the nucleus of an atom is transformed into a neutron, emitting a positron (e+) and a neutrino (ν) in the process. This decay decreases the atomic number of the element by one while keeping the mass number constant. Recognizing this decay process is essential for identifying the parent isotope and the resulting daughter isotope.

Conservation of Energy and Momentum

In nuclear reactions, the conservation of energy and momentum principles state that the total energy and momentum before the decay must equal the total energy and momentum after the decay. This principle helps in analyzing decay processes and determining the properties of the emitted particles, such as the positron and neutrino in beta plus decay, ensuring that the reaction adheres to fundamental physical laws.