understand that nucleon number and charge are conserved in nuclear processes

Atoms, Nuclei and Radiation

Learning Objective

Understand that the total nucleon number (mass number, $A$) and the total charge (atomic number, $Z$) are conserved in all nuclear processes.

Key Concepts

• Nucleon number ($A$): total number of protons and neutrons in a nucleus.
• Charge number ($Z$): number of protons, which determines the element.
• Conservation laws: In any nuclear reaction, $$A_{\text{initial}} = A_{\text{final}}, \qquad Z_{\text{initial}} = Z_{\text{final}}.$$
• Only the arrangement of nucleons may change; the total counts remain the same.

Common Nuclear Processes

1. Alpha ($\alpha$) Decay

An $\alpha$ particle is a $^{4}_{2}\text{He}$ nucleus (2 protons, 2 neutrons). The parent nucleus loses these nucleons.

General form:

$$^{A}_{Z}\text{X} \;\rightarrow\; ^{A-4}_{Z-2}\text{Y} \;+\; ^{4}_{2}\text{He}.$$

Both $A$ and $Z$ are reduced by the same amounts on both sides of the equation, preserving total values.

2. Beta Minus ($\beta^{-}$) Decay

A neutron transforms into a proton, emitting an electron and an antineutrino:

$$n \;\rightarrow\; p^{+} \;+\; e^{-} \;+\; \bar{u}_{e}.$$

For the nucleus:

$$^{A}_{Z}\text{X} \;\rightarrow\; ^{A}_{Z+1}\text{Y} \;+\; e^{-} \;+\; \bar{u}_{e}.$$

The mass number $A$ stays the same, while $Z$ increases by 1, keeping total charge conserved when the emitted electron is included.

3. Beta Plus ($\beta^{+}$) Decay (Positron Emission)

A proton converts into a neutron, emitting a positron and a neutrino:

$$p^{+} \;\rightarrow\; n \;+\; e^{+} \;+\; u_{e}.$$

For the nucleus:

$$^{A}_{Z}\text{X} \;\rightarrow\; ^{A}_{Z-1}\text{Y} \;+\; e^{+} \;+\; u_{e}.$$

Again $A$ is unchanged; $Z$ decreases by 1, balanced by the positive charge of the emitted positron.

4. Gamma ($\gamma$) Emission

Excited nuclei release excess energy as a photon:

$$^{A}_{Z}\text{X}^{*} \;\rightarrow\; ^{A}_{Z}\text{X} \;+\; \gamma.$$

No change in $A$ or $Z$; only energy is carried away.

Conservation Demonstrated with Example Reactions

Example 1: Alpha Decay of Uranium‑238

Reaction:

$$^{238}_{92}\text{U} \;\rightarrow\; ^{234}_{90}\text{Th} \;+\; ^{4}_{2}\text{He}.$$

Check conservation:

	Mass Number ($A$)	Charge Number ($Z$)
Initial (Uranium)	238	92
Final (Thorium + Alpha)	234 + 4 = 238	90 + 2 = 92

Example 2: Beta Minus Decay of Carbon‑14

Reaction:

$$^{14}_{6}\text{C} \;\rightarrow\; ^{14}_{7}\text{N} \;+\; e^{-} \;+\; \bar{u}_{e}.$$

Check conservation:

	Mass Number ($A$)	Charge Number ($Z$)
Initial (Carbon)	14	6
Final (Nitrogen + Electron)	14 + 0 = 14	7 + (-1) = 6

General Procedure for \cdot erifying Conservation

1. Write the nuclear equation, including all emitted particles.
2. Identify $A$ and $Z$ for each reactant and product.
3. Sum $A$ values on the left‑hand side (LHS) and right‑hand side (RHS); they must be equal.
4. Sum $Z$ values (including charges of emitted leptons); they must also be equal.
5. If the sums differ, the equation is not balanced and must be corrected.

Common Pitfalls

• For $\beta^{-}$ decay, forgetting to include the electron’s $-1$ charge when checking $Z$.
• For $\beta^{+}$ decay, forgetting the positron’s $+1$ charge.
• Assuming gamma emission changes $A$ or $Z$ – it does not.
• Neglecting neutrinos/antineutrinos in the balance of charge (they are neutral, so they do not affect $Z$).

Suggested Diagram

Summary

In every nuclear process, the total number of nucleons ($A$) and the total charge ($Z$) remain unchanged. By carefully accounting for all particles—including emitted leptons and photons—students can verify that these conservation laws hold, providing a solid foundation for further study of nuclear reactions and radiation.