Writing and balancing nuclear decay equations

Key ideas

A nuclear decay equation shows a parent nucleus breaking down into a daughter nucleus plus the emitted particle. Every decay equation must be balanced:

• The mass numbers ($A$, top numbers) on both sides must add up to the same total.
• The atomic numbers ($Z$, bottom numbers) on both sides must add up to the same total.

Alpha decay

An alpha particle ($^{4}_{2}\text{He}$) is emitted from the nucleus. The general form is:

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

• $X$ — the parent nuclide
• $Y$ — the daughter nuclide
• The daughter's mass number drops by 4 and atomic number drops by 2, because it has lost 2 protons and 2 neutrons.

Beta-minus decay

A neutron in the nucleus converts into a proton, releasing an electron (the beta particle):

$n \rightarrow p + \beta^{-}$

So the decay equation is:

$^{A}_{Z}X \rightarrow \, ^{A}_{Z+1}Y + \, ^{0}_{-1}\beta$

• The mass number stays the same — a neutron becomes a proton, so the total nucleon count is unchanged.
• The atomic number increases by 1, because there is now one more proton.
• The beta particle is written as $^{0}_{-1}\beta$: mass number 0 (negligible mass) and "atomic number" $-1$ (its charge), which balances the equation.

Gamma decay

• Gamma emission usually happens after an alpha or beta decay, when the daughter nucleus is left with excess energy.
• Emitting a gamma ray does not change $A$ or $Z$ — the nucleus loses energy, not particles, so it does not appear as a separate line in the mass/atomic number balance.

Practice question

Carbon-14 ($^{14}_{6}\text{C}$) undergoes beta-minus decay. Write the balanced decay equation (the daughter is nitrogen, $Z = 7$).

Worked solution: mass number is unchanged, atomic number increases by 1: $^{14}_{6}\text{C} \rightarrow \, ^{14}_{7}\text{N} + \, ^{0}_{-1}\beta$.