7.2 Radioactivity, Decay & Half-Life

Some combinations of protons and neutrons are stable indefinitely; others are not. An unstable nucleus — one with an unfavorable ratio of protons to neutrons, or simply too much internal energy — will spontaneously emit particles or energy to move toward a more stable configuration. This process is called radioactive decay, and it is directly relevant to the NAPT because decay equations and half-life calculations are among the most concretely testable nuclear-science topics: unlike many conceptual topics, they have a definite right numerical answer.

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 rule worth memorizing for the NAPT: 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, but the NAPT's worked examples typically use whole half-life multiples.

Well-Known Half-Life Examples