Nuclear reactions can transform matter dramatically, but they are never arbitrary. A proposed reaction is physically possible only if several conservation laws are satisfied at the same time.

Why these conservation laws matter

A nuclear reaction changes a system of nuclei and particles from an initial state to a final state. Even though the particles involved may be different after the reaction, some total quantities must remain unchanged. These are called conservation laws.

In AP Physics 2, the key idea is that a nuclear reaction is not judged only by whether it “looks reasonable.” It must satisfy conservation of nucleon number, conservation of energy, mass-energy equivalence, and conservation of momentum. If even one of these fails, the reaction cannot occur as written.

These laws are especially important because nuclear reactions often involve very large energy changes.

Binding energy per nucleon versus mass number shows a broad peak near iron, indicating unusually high stability there. Reactions that move nuclei toward higher binding energy per nucleon (toward the peak) can release energy, consistent with $\Delta E = \Delta m c^2$ when total rest mass decreases. Source

The energy released or absorbed can be much larger than in chemical reactions, so careful accounting is essential.

Conservation of nucleon number

What must stay constant

The first check in a nuclear reaction is the total nucleon number before and after the interaction.

In a valid nuclear reaction, the total number of nucleons in all reactants must equal the total number of nucleons in all products. The nucleons can be rearranged into different nuclei, but they cannot simply appear or disappear.

This does not mean each individual nucleus keeps the same nucleon number. Instead, the sum over the whole reaction must remain constant. A large nucleus may split into smaller nuclei, or smaller nuclei may combine into a larger nucleus, but the overall nucleon count must match.

Because of this, nucleon number is one of the fastest ways to test whether a proposed nuclear equation is even possible.

The figure expresses nuclear-equation balancing as two simultaneous constraints: matching the total mass number (nucleon number) and matching the total atomic number (charge) on both sides. This provides a fast, reliable method to detect impossible reactions or determine an unknown product in a proposed reaction. Source

If the total on the left side differs from the total on the right side, the reaction is impossible as written.

Conservation of energy and mass-energy equivalence

Rest mass is part of the energy budget

A nuclear reaction must also obey conservation of energy. However, in nuclear physics, energy must be understood broadly. It includes not only kinetic energy, but also energy associated with mass.

This means the total rest mass of the reactants does not have to equal the total rest mass of the products.

A bound nucleus has less total mass than the same protons and neutrons when they are separated, because forming the nucleus releases binding energy. The mass difference (mass defect) corresponds to energy via $\Delta E = \Delta m c^2$, so nuclear energy accounting must include rest-mass changes. Source

Instead, a decrease in total rest mass corresponds to an increase in other forms of energy, and an increase in total rest mass requires energy input from elsewhere in the system.

If the reactants have greater total rest mass than the products, the difference can appear as kinetic energy of the products or other emitted energy. If the products have greater total rest mass, then the reaction can occur only if enough energy is supplied initially.

This is why mass is not separately conserved in nuclear reactions, even though total energy is conserved. The correct rule is that the total amount of energy, including mass energy, remains constant.

A reaction can therefore pass the nucleon-number test and still be impossible if the energy accounting fails. In nuclear physics, balancing particle counts alone is not enough.

Conservation of momentum

Direction matters as well as amount

A nuclear reaction must also satisfy conservation of momentum. Momentum is a vector quantity, so both magnitude and direction matter.

This law applies to the entire isolated system. The total momentum before the reaction must equal the total momentum after the reaction.

If the initial system is at rest, then the total initial momentum is zero. In that case, the vector sum of all final momenta must also be zero. The products may move, but their momenta must balance overall.

Momentum conservation is crucial because energy conservation alone cannot determine whether a reaction is possible. A proposed set of products might have the correct total energy, but if their momenta cannot add up to the initial total momentum, the reaction cannot occur that way.

Momentum conservation also affects how the released energy is shared among products. Different particles may leave with different speeds, but their momenta must still fit the vector balance required by the initial state.

Evaluating a proposed nuclear reaction

To decide whether a nuclear reaction is possible, apply the conservation laws together:

• Check that total nucleon number is the same before and after.

• Check that total energy is conserved.

• Include mass-energy equivalence, so rest mass is treated as part of the total energy.

• Check that total momentum is the same before and after.

These are not separate optional tests. They must all hold simultaneously.

A reaction is physically allowed only when every required conserved quantity matches between the initial and final states. In practice, these conservation laws let physicists reject impossible reactions, identify missing products, and interpret measured nuclear data.