Balance a simple nuclear equation

To balance a nuclear equation, make the upper numbers (nucleon numbers) add up to the same total on both sides, and make the lower numbers (proton numbers, which also track charge) add up to the same total on both sides. Decay equations of this kind appear in short calculations and “complete the equation” questions in atomic and nuclear models.

You should already be able to tell atomic and nuclear changes apart, because only the nucleus changes here.

What do the symbols mean?

A nucleus is written as a symbol with two numbers: nucleon number A at the upper left and proton number Z at the lower left. For example, uranium-238 is ²³⁸₉₂U.

Two particles need to be recognised.

• Alpha particle: ⁴₂He, a helium nucleus (some texts write it as ⁴₂α).
• Beta particle (beta-minus): ⁰₋₁e, a fast electron from the nucleus.

What happens in each decay?

Alpha decay: the nucleus loses 2 protons and 2 neutrons. A goes down by 4 and Z goes down by 2.

Beta decay: a neutron changes into a proton and an electron, and the electron leaves. A stays the same and Z goes up by 1.

Gamma emission: A and Z do not change.

Worked example

Write the equation for uranium-238, ²³⁸₉₂U, decaying by alpha emission, and identify the new nucleus. Then do the same for a beta decay of carbon-14.

Part 1, alpha.

Start: ²³⁸₉₂U → ^A_ZX + ⁴₂He

Upper numbers: 238 = A + 4, so A = 234. Lower numbers: 92 = Z + 2, so Z = 90.

Z = 90 is thorium. The equation is:

²³⁸₉₂U → ²³⁴₉₀Th + ⁴₂He

Check: upper 234 + 4 = 238, lower 90 + 2 = 92. Both balance.

Part 2, beta.

Start: ¹⁴₆C → ^A_ZX + ⁰₋₁e

Upper: 14 = A + 0, so A = 14. Lower: 6 = Z + (−1), so Z = 7.

Z = 7 is nitrogen. The equation is:

¹⁴₆C → ¹⁴₇N + ⁰₋₁e

Check: upper 14 + 0 = 14, lower 7 + (−1) = 6. Both balance.

The mistake to watch for

Mistaken answer: ²³⁸₉₂U → ²³⁶₉₀Th + ⁴₂He

The student removed 2 from both numbers, thinking “an alpha particle has 2 protons”.

The lower number is right, but the upper number is wrong. Check the upper: 236 + 4 = 240, not 238.

The alpha particle’s nucleon number is 4, since it holds 2 protons and 2 neutrons, so A must drop by 4. The habit that prevents this is to write both sums, upper and lower, every time.

Check yourself

1. Polonium-210 is ²¹⁰₈₄Po. It emits an alpha particle. Write the equation.

2. Strontium-90 is ⁹⁰₃₈Sr and decays by beta emission. What is the new nucleus?

3. A nucleus X decays by beta emission to ¹⁴₇N. What are the nucleon number and proton number of X?

Where this leads next

After equations, move on to comparing radiation properties descriptively. Then test yourself in the mixed practice set.

Students often get the arithmetic right and still lose marks because the symbols are written untidily. Reviewing written working is a natural part of online one-to-one Physics tuition.