Decay Equations
Balancing nuclear equations is like balancing a checkbook — what goes in must equal what comes out. In chemical equations, you balance atoms. In nuclear equations, you balance two quantities: mass number (A) and charge (Z).
If you can add and subtract single-digit numbers, you can solve every nuclear equation the MCAT throws at you. The trick is just being systematic about which numbers go where.
Conservation Laws
Two quantities are always conserved in nuclear reactions:
- Mass number (A) - the total number of nucleons (protons + neutrons) on the left side equals the total on the right side.
- Atomic number (Z) - the total charge (proton count) on the left side equals the total on the right side.
Common Particles in Nuclear Equations
Before we practice, here are the particles you will encounter, with their A and Z values:
| Particle | Symbol | A | Z |
|---|---|---|---|
| Proton | 1 | 1 | |
| Neutron | 1 | 0 | |
| Electron (β-minus) | 0 | -1 | |
| Positron (β-plus) | 0 | +1 | |
| Alpha particle | 4 | 2 | |
| Gamma photon | 0 | 0 |
Practice: Identifying Unknown Products
Example 1: Alpha Decay of Radium-226
Balance A: 226 = A + 4, so A = 222
Balance Z: 88 = Z + 2, so Z = 86
Z = 86 is radon (Rn). Answer:
Example 2: Beta-Minus Decay of Iodine-131
Balance A: 131 = A + 0, so A = 131
Balance Z: 53 = Z + (-1), so Z = 54
Z = 54 is xenon (Xe). Answer:
Example 3: Identify the Missing Particle
Balance A: 210 = 206 + A, so A = 4
Balance Z: 84 = 82 + Z, so Z = 2
A = 4, Z = 2 is an α particle ().
Decay Chains
Unstable nuclei rarely reach stability in a single step. A parent nucleus decays to a daughter, which may also be unstable and decay further, forming a decay chain. The chain continues until a stable nucleus is produced.
The most famous example is the uranium-238 decay chain, which involves 14 steps (8 α decays and 6 β decays) before reaching stable lead-206. You do not need to memorize the entire chain, but you should be able to work through any individual step.
Tracking Multiple Decays
If a nucleus undergoes two α decays and one β-minus decay:
- Total change in A: 2 x (-4) = -8
- Total change in Z: 2 x (-2) + 1 x (+1) = -3
So if you start with and apply two alphas and one β-minus, you end at A = 230, Z = 89 (actinium-230).
Electron Capture
One additional process worth knowing: in electron capture, the nucleus absorbs an inner-shell electron, converting a proton to a neutron. The effect on Z and A is identical to β-plus decay (Z decreases by 1, A unchanged), but no positron is emitted. Instead, the atom emits X-rays as outer electrons fill the vacancy left by the captured electron.