Writing and Balancing Nuclear Equations

The two conservation rules

A nuclear equation uses an arrow to show a nuclear transformation, for example:

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

The equation is correct only if two quantities are conserved — equal on the left and right sides:

1. Conservation of mass number: the sum of all A values on the left equals the sum of all A values on the right.
2. Conservation of atomic number (charge): the sum of all Z values on the left equals the sum of all Z values on the right.

These two rules replace the atom-counting you use in chemistry. Because nuclear reactions change protons into neutrons and vice versa, the elements themselves are not conserved — and that is expected, not an error.

Writing an alpha decay equation

In alpha decay the parent ejects an alpha particle, $^{4}_{2}He$. Therefore the daughter always has a mass number 4 lower and an atomic number 2 lower than the parent.

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

Check: $A$: $238 = 234 + 4$ ✓. $Z$: $92 = 90 + 2$ ✓.

The daughter's identity follows from its Z: Z = 90 is thorium, so the product must be $^{234}_{90}Th$. A frequent textbook example is radium-226 decaying to radon-222: $^{226}_{88}Ra \rightarrow ^{222}_{86}Rn + ^{4}_{2}He$.

Writing a beta-minus decay equation

In beta-minus decay, a neutron converts into a proton and ejects an electron, $^{0}_{-1}e$. The mass number does not change (a neutron becomes a proton, so the nucleon count is the same), but the atomic number increases by one.

$$^{14}_{6}C \rightarrow ^{14}_{7}N + ^{0}_{-1}e$$

Check: $A$: $14 = 14 + 0$ ✓. $Z$: $6 = 7 + (-1)$ ✓.

Because Z increases by one, the daughter is the next element up the periodic table: carbon-14 becomes nitrogen-14. Note how the negative Z of the electron makes the charge sum work — this is the detail students most often forget.

Positron emission, electron capture, and gamma

Two related processes complete the picture. In positron emission (beta-plus decay), a proton converts into a neutron and ejects a positron, $^{0}_{+1}e$. The mass number is unchanged and Z decreases by one:

$$^{22}_{11}Na \rightarrow ^{22}_{10}Ne + ^{0}_{+1}e$$

In electron capture, the nucleus absorbs one of its own inner electrons. A proton plus the captured electron becomes a neutron, so Z decreases by one with no emitted particle:

$$^{7}_{4}Be + ^{0}_{-1}e \rightarrow ^{7}_{3}Li$$

Note that electron capture is written with the electron on the left side of the arrow — it is a reactant, not a product.

Finally, gamma emission is written with a $^{0}_{0}\gamma$ term that contributes 0 to both sums. For example, an excited technetium-99 nucleus relaxes: $^{99m}_{43}Tc \rightarrow ^{99}_{43}Tc + ^{0}_{0}\gamma$. The 'm' marks the excited (metastable) state; A and Z are unchanged.

Finding a missing particle

When a piece of the equation is missing, let the conservation laws find it. Suppose radon-222 decays to polonium-218 plus an unknown particle:

$$^{222}_{86}Rn \rightarrow ^{218}_{84}Po + ^{?}_{?}?$$

1. Mass number: $222 - 218 = 4$, so the unknown has $A = 4$.
2. Atomic number: $86 - 84 = 2$, so the unknown has $Z = 2$.
3. Identify: $A = 4$, $Z = 2$ is $^{4}_{2}He$ — an alpha particle.

The completed equation is $^{222}_{86}Rn \rightarrow ^{218}_{84}Po + ^{4}_{2}He$. This difference method works for any missing particle: subtract the known products from the parent on both the A and Z lines.

A checklist for balanced equations

Work through these checks before calling any nuclear equation finished:

1. Sum A on each side separately. Every reactant and product counts, including small particles like $^{1}_{0}n$ and $^{0}_{-1}e$.
2. Sum Z on each side separately. Remember that beta particles carry $Z = -1$ and positrons $Z = +1$.
3. Verify each element matches its Z. If you wrote $^{234}_{90}Th$, confirm that element 90 is actually thorium.
4. Check the physical sense. Alpha decay lowers A by 4 and Z by 2; beta-minus raises Z by 1 with A unchanged. If your equation does something else, re-examine the decay mode.

With these four checks, balancing nuclear equations becomes a routine arithmetic exercise rather than a guessing game.

Variables and units

Common mistakes

Real-world applications

• Predicting the products of an unknown radioactive decay

• Identifying decay modes from parent-daughter data in decay series

• Verifying proposed nuclear reactions in research and forensics

• Determining what radiation type a source emits by completing its equation