Nuclear decay can change which particles are present in a nucleus, but it cannot violate fundamental bookkeeping rules. These rules let you decide whether a proposed nuclear equation is physically allowed.

Conserved quantities in nuclear decay

A nuclear decay equation is like an accounting statement for particles. The particles after the decay may be different from the particles before the decay, but certain totals must remain the same. In AP Physics 2, the required totals are nucleon number, lepton number, and charge. When any one of these is not balanced, the decay equation is incomplete or impossible.

When checking a decay, you must count all particles shown on both sides of the equation. That includes the nucleus and any emitted particles.

In nuclear notation, the upper number of a nuclide, often written as $A$ in $^{A}_{Z}X$, represents nucleon number.

Nuclide notation labels the nucleus with mass number $A$ (total nucleons) and atomic number $Z$ (protons). This visual supports the idea that conservation of nucleon number and charge can often be checked by comparing the sums of the upper and lower indices across a nuclear equation. Source

During decay, the number of protons and neutrons separately may change, but the total number of nucleons must stay constant. This means the sum of the upper numbers on the left must equal the sum of the upper numbers on the right.

A different conserved quantity is needed for particles such as electrons, positrons, neutrinos, and antineutrinos.

For AP-level nuclear decay, the important idea is that nuclei, protons, neutrons, and photons do not carry lepton number. Electrons and neutrinos count as leptons, while positrons and antineutrinos count as antileptons. If a decay produces a lepton, the full equation must still keep the total lepton number unchanged.

You must also track ordinary electric charge.

A proton has charge $+1$, an electron has charge $-1$, a positron has charge $+1$, and neutrons and neutrinos have charge $0$. In a nuclear equation, the lower number of a nuclide helps track the nucleus’s positive charge, but emitted particles must be included too. A balanced nuclear equation must have the same total charge on both sides.

For quick bookkeeping, these common particles are especially useful:

• Proton: nucleon number $1$, lepton number $0$, charge $+1$

• Neutron: nucleon number $1$, lepton number $0$, charge $0$

• Electron: nucleon number $0$, lepton number $+1$, charge $-1$

• Positron: nucleon number $0$, lepton number $-1$, charge $+1$

• Neutrino: nucleon number $0$, lepton number $+1$, charge $0$

• Antineutrino: nucleon number $0$, lepton number $-1$, charge $0$

Conservation statements

These three rules can be written in compact form.

These equalities do not tell you how a nucleus decays, but they do tell you whether the listed particles are consistent with the conservation laws.

How to check a nuclear decay equation

Step 1: Check nucleon number

Add the nucleon numbers of all particles on the reactant side and compare that total to the product side. Because emitted electrons, positrons, neutrinos, and photons are not nucleons, they contribute $0$ to nucleon number. If the totals differ, the equation cannot represent a valid nuclear decay.

Step 2: Check charge

Now add the charges of everything shown. For nuclei, the charge is the lower number in nuclide notation. For emitted particles, use their particle charges. A decay can change the charge of the nucleus itself, but only if the rest of the products balance the total charge.

Step 3: Check lepton number

This step is often the one students forget. Count leptons as $+1$, antileptons as $-1$, and all non-leptons as $0$.

This schematic writes the $\beta^-$ decay process explicitly as $n^0 \rightarrow p^+ + e^- + \bar{\nu}_e$. It highlights that the antineutrino is required for lepton-number accounting even though it carries no electric charge. Source

If an electron appears among the products, for instance, you cannot ignore lepton number. The total on the left and right must still match. This is why some nuclear equations require an additional neutrino or antineutrino to be complete.

The diagram illustrates $\beta^-$ decay: a neutron transforms into a proton while emitting an electron and an electron antineutrino. It is a compact visual example of how charge and lepton number remain conserved only when the (anti)neutrino is included as a product. Source

Step 4: Decide what the result means

If nucleon number, charge, and lepton number are all conserved, the decay equation is allowed by these conservation rules. That does not automatically guarantee that the decay will occur. Conservation laws are a test of consistency, not a full prediction of nuclear behavior.

Common points of confusion

Nucleon number is not the same as the number of protons

A decay may change one type of nucleon into another, so the number of protons alone does not have to stay constant. What stays constant is the total number of protons plus neutrons.

Charge conservation is not enough by itself

Some incorrect equations balance charge and nucleon number but fail lepton number. On AP problems, always check all three rules, not just the two numbers attached to the nucleus.

Missing particles matter

A nuclear equation may look almost balanced but still be incomplete if a particle has been omitted. In that case, use the three conservation rules as clues to identify what type of particle must be present. If no possible added particle fixes all three quantities, the proposed decay is not valid.

Neutral particles still matter

A particle with zero charge can still affect conservation checks. Neutrons affect nucleon number. Neutrinos affect lepton number. Never assume that “neutral” means “irrelevant.”

The rules apply to the whole system

Do not balance only the nucleus. Balance the entire decay equation, including every emitted particle. Conservation laws apply to the full interaction from initial state to final state.