The Nucleus & Radioactivity

You’ll be able to

Describe the composition of the nucleus and the meaning of isotopes
Identify alpha, beta, and gamma decay and apply conservation of nucleons and charge
Use half-life to determine how much of a radioactive sample remains

Inside the nucleus

The nucleus contains protons (positive) and neutrons (neutral), together called nucleons. The number of protons — the atomic number Z — defines the element; the total number of nucleons is the mass number A. Atoms of the same element with different neutron counts are isotopes. Holding the like-charged protons together against their electric repulsion is the strong nuclear force, which is extremely powerful but acts only over tiny nuclear distances. When that balance fails, the nucleus is unstable and radioactive.

Three kinds of decay

Unstable nuclei shed energy through radioactive decay, and two quantities are always conserved: total charge and total nucleon number. In alpha (α) decay the nucleus emits a helium nucleus (2 protons + 2 neutrons), so A drops by 4 and Z by 2. In beta (β⁻) decay a neutron turns into a proton plus an emitted electron, so Z rises by 1 while A is unchanged. In gamma (γ) decay the nucleus sheds excess energy as a high-energy photon, changing neither A nor Z. Einstein’s E = mc² accounts for the energy released as a tiny loss of mass.

Half-life & mass–energy

remaining fraction = (1/2)ⁿ, n = t / t₁/₂ · E = m · c²

The half-life t₁/₂ is the time for half of a sample to decay; after n half-lives, a fraction (1/2)ⁿ remains. E = mc² converts the mass lost in a reaction into released energy.

Worked example

A radioactive sample has a half-life of 5.0 years. What fraction of the original sample remains after 15 years?

1.Count the number of half-lives: n = t / t₁/₂ = 15 / 5.0 = 3.
2.After each half-life, half remains, so multiply ½ three times: (1/2)³.
3.Compute: (1/2)³ = 1/8.

Answer: 1/8 of the original sample remains (the other 7/8 has decayed) after 3 half-lives.

Checkpoint

In alpha decay, a nucleus emits an alpha particle (a helium-4 nucleus). How do its mass number A and atomic number Z change?

Watch out

Balance both the top (mass number A) and the bottom (atomic number Z) numbers in every decay equation. The sums of A and of Z must be equal on the two sides — that is conservation of nucleons and of charge, and it pins down the unknown product.

Checkpoint

A radioactive isotope has a half-life of 5.0 years. After 15 years, what fraction of the original sample remains?

On the exam

For half-life problems, first find n = t / t₁/₂, then the surviving fraction is (½)ⁿ. Watch that decay is exponential, not linear — after 2 half-lives one-quarter remains, not zero.

Answer the 2 checkpoints as you read.