7.2 Radioactivity, Decay & Half-Life

Radioactive decay and half-life are included as public, textbook-level quantitative science. Their inclusion is editorial; current public Navy sources do not identify them as NAPT topics or disclose their frequency.

The Three Main Types of Radioactive Decay

Alpha Decay

In alpha decay, an unstable nucleus ejects an alpha particle — a tightly bound cluster of 2 protons and 2 neutrons, identical to a helium-4 nucleus. Because the parent nucleus loses 2 protons and 2 neutrons:

• Mass number (A) decreases by 4
• Atomic number (Z) decreases by 2

General form: a parent nucleus with mass number A and atomic number Z produces a daughter nucleus with mass number (A − 4) and atomic number (Z − 2), plus an alpha particle. For example, Uranium-238 (Z = 92, A = 238) alpha-decays into Thorium-234 (Z = 90, A = 234) plus an alpha particle (Z = 2, A = 4). Check the balance: mass numbers 234 + 4 = 238 ✓; atomic numbers 90 + 2 = 92 ✓.

Beta Decay

In the most common form of beta decay (beta-minus decay), a neutron inside the nucleus converts into a proton, an electron (the beta particle, ejected from the nucleus), and an antineutrino. Because a neutron turns into a proton:

• Mass number (A) stays the same (one nucleon converted to another nucleon — total nucleon count is unchanged)
• Atomic number (Z) increases by 1

For example, Carbon-14 (Z = 6, A = 14) beta-decays into Nitrogen-14 (Z = 7, A = 14) plus a beta particle. Mass numbers: 14 = 14 ✓. Charge: the beta particle carries a charge of −1, and 7 + (−1) = 6 ✓.

Gamma Decay

After an alpha or beta decay, the daughter nucleus is often left in an excited, higher-energy state. It sheds that excess energy by emitting a gamma ray — a high-energy photon, not a particle with mass or charge. Gamma decay changes neither the mass number nor the atomic number; it only releases energy. Gamma decay almost always accompanies alpha or beta decay rather than occurring entirely on its own.

Notice the pattern that shows up repeatedly on decay-equation questions: every decay equation must balance on both sides — the sum of mass numbers on the left must equal the sum on the right, and the same is true for atomic (charge) numbers. If a question asks you to identify a missing daughter product, balance the equation the same way you would balance a chemical equation.

Half-Life: The Core Concept

Radioactive decay of any single atom is random — you cannot predict exactly when one specific atom will decay. But across a large sample of identical atoms, decay is statistically predictable, and that predictability is captured in one number: the half-life (T½) — the time required for half of a radioactive sample (or its activity) to decay.

Half-life follows an exponential decay pattern, not a linear one. The formula:

N = N₀ × (1/2)^(t / T½)

where N₀ is the starting quantity, N is the quantity remaining, t is the elapsed time, and T½ is the half-life. When t is a whole-number multiple of the half-life, this simplifies to a useful local practice rule: after n half-lives, the fraction remaining is (1/2)ⁿ.

Worked Half-Life Problems

Problem 1: Iodine-131 has a half-life of about 8 days. A hospital starts with an 80-gram sample. How much remains after 24 days?

• Number of half-lives: 24 days ÷ 8 days/half-life = 3 half-lives
• Fraction remaining: (1/2)³ = 1/8
• Amount remaining: 80 g × 1/8 = 10 grams

Problem 2: A radioactive sample with a 5-year half-life starts at 200 grams. How much remains after 20 years?

• Number of half-lives: 20 years ÷ 5 years/half-life = 4 half-lives
• Fraction remaining: (1/2)⁴ = 1/16
• Amount remaining: 200 g × 1/16 = 12.5 grams

The pattern to internalize: find how many half-lives fit into the elapsed time, then repeatedly cut the sample in half that many times (or use (1/2)ⁿ directly). Elapsed time that is not a whole-number multiple of the half-life requires the full exponential formula; this guide begins with whole half-life multiples for clarity.

Well-Known Half-Life Examples