Atoms that fall apart: radioactive decay and the half-life clock

How to use this simulation

1. Pick an isotope under “Isotope”. U-238 and Ra-226 decay by alpha, C-14 and Co-60 by beta-minus, and Tc-99m by gamma emission.
2. Watch the top of the stage. The nucleus shakes, then throws out a particle. Its mass number A (small number on top) and atomic number Z (small number below) change as the daughter appears.
3. Read the equation under the nucleus. It is rebuilt from the decay rule every time you change isotope, and the readings give the change in A and in Z.
4. Look at the lower half. Each dot in the grid is one atom of a hundred. As time runs, dots hollow out at random moments while the orange staircase falls along the blue curve N = N₀(½)^(t/T).
5. Drag the “Time” slider to scrub through time. At exactly 1, 2 and 3 half-lives the blue curve passes through ½, ¼ and ⅛. The staircase wobbles around it, because decay is random.
6. Press reset or let the animation loop. Every loop uses a new random pattern of decays, but the crowd always lands close to the same curve.

Atoms that fall apart

Most atoms you meet are stable. Their nuclei just sit there for ever. But some nuclei, usually very heavy ones or ones with too many or too few neutrons, are unstable. Sooner or later they change, and in changing they throw out energy or small particles. That is radioactivity, discovered by Henri Becquerel in 1896.

Radioactivity comes from the nucleus, not the electron cloud, so heating the sample, crushing it or mixing it into a compound does not change how fast it decays. That is why it is such a reliable clock.

Reading A and Z

Every nuclide is written with its symbol, its mass number A at the top left and its atomic number Z at the bottom left. A is the number of protons plus neutrons. Z is the number of protons, which decides the element. For example, uranium-238 is written ²³⁸₉₂U: 238 nucleons, of which 92 are protons and 146 are neutrons.

In a nuclear change, two quantities are always conserved: the total A and the total Z. Add up the numbers on the left side and on the right side. If they match, the equation is balanced. That is the whole trick behind nuclear equations.

Alpha decay: A drops by 4, Z drops by 2

An alpha particle is a helium nucleus, two protons and two neutrons stuck together, written ⁴₂He or α. A heavy nucleus sheds one when it has too many nucleons to hold together. Take U-238: it loses 4 from A and 2 from Z, leaving Th-234.

The equation is ²³⁸₉₂U → ²³⁴₉₀Th + ⁴₂He. Check it in the table: the left side has A = 238 and Z = 92; the right side has 234 + 4 = 238 and 90 + 2 = 92. Balanced.

Beta-minus decay: A stays, Z goes up by 1

Inside a nucleus with too many neutrons, one neutron turns into a proton and an electron is thrown out at high speed. That electron is the beta-minus particle, written β⁻ or ⁰₋₁e. A hardly changes, since a neutron became a proton, but Z rises by 1, so the atom becomes the next element along. C-14 (Z = 6) turns into N-14 (Z = 7).

The equation is ¹⁴₆C → ¹⁴₇N + ⁰₋₁e. A tiny, nearly invisible antineutrino also leaves. It has A = 0 and Z = 0, so it never upsets the balance, which is why most school equations leave it out. The electron’s charge is −1, which is why its Z is written −1 and why the sum works: 7 + (−1) = 6.

Gamma emission: same nucleus, less energy

After alpha or beta decay, the new nucleus is often left excited, like a ringing bell. It settles by releasing a photon of gamma radiation, which has no mass and no charge. So A and Z do not change at all. Technetium-99m (the small m means metastable, a long-lived excited state) is the standard example: ⁹⁹ᵐ₄₃Tc → ⁹⁹₄₃Tc + ⁰₀γ.

Cobalt-60 is a good case where both happen. First ⁶⁰₂₇Co → ⁶⁰₂₈Ni + ⁰₋₁e, which leaves Ni-60 excited, and then it releases gamma rays. Those gamma rays, not the beta electrons, are what hospitals use for radiotherapy.

The three rays compared

The three radiations differ in charge, mass and, most importantly for safety, in how far they get through matter.

Building a balanced nuclear equation

The method is the same every time. Write the parent nuclide. Decide the type of decay. Apply the rule to find the daughter, then write the particle that leaves. Finally check that the A values and the Z values add up on both sides.

Some nuclei decay in a chain. Starting with U-238, six steps in a row give Rn-222, and after every step both A and Z are known without looking anything up. Use the element symbol that matches the new Z.

• Alpha: subtract 4 from A and 2 from Z.
• Beta-minus: add 1 to Z, leave A.
• Gamma: change nothing, but the nucleus loses energy.
• Always find the new element from its Z in the periodic table.

Half-life: the clock inside every sample

The half-life, T, is the time for half of the radioactive atoms in a sample to decay. After one half-life half remain. After two, half of that half: a quarter. After three, an eighth. Every half-life takes away half of whatever is left, so the sample shrinks quickly at first and then ever more slowly, without ever quite reaching zero.

Different isotopes have wildly different half-lives. Uranium-238 takes about four and a half billion years. Technetium-99m takes about six hours. This is the reason isotopes are chosen for different jobs.

One atom is random, a crowd is not

Look at the grid in the simulation. Each atom has the same chance of decaying in any given moment, but no one knows which atom will go next. A single atom could last a microsecond or a million years. Yet with a hundred atoms, the staircase of the actual count hugs the smooth curve. With a billion billion atoms, it is indistinguishable from it.

That is why the half-life of U-238 (4.468 billion years) is useful even though no one can say when a given uranium atom will decay. It is the same as flipping coins: one flip is unpredictable, but a thousand flips give very nearly half heads. After 4.54 billion years, about the age of the Earth, the formula says 49.4 percent of the uranium-238 that formed with the planet is still there.

Carbon-14 dating

Living things take in carbon all their lives, including a tiny, steady share of radioactive carbon-14. When they die, the intake stops and the carbon-14 starts to decay with a half-life of 5,730 years. Measuring what fraction of the original carbon-14 is left gives the time since death.

The age comes from rearranging the decay law: t = T × log₂ (N₀ / N). Each row of the table below applies that to a different fraction.

Try it in the simulation

Five experiments for the simulation.

• Choose U-238 and watch the numbers: A goes from 238 down to 234, Z from 92 down to 90. Then pick C-14 and see A stay put while Z climbs by one.
• Choose Tc-99m. The nucleus changes its energy but not its numbers. Both readings say there is no change.
• Drag the time slider to 1 half-life. About 50 of the 100 dots should be hollow. Not exactly 50, though, and that wobble is the randomness.
• Let the loop run a few times. Each pass has a different staircase, but all of them hug the blue curve.
• Scrub to 3 half-lives and compare the dots left with ⅛ of 100. Then to 5 half-lives, where only about three remain.

Solved problems

Six common exam problems. Every number below is computed with the same code as the simulation.

Mistakes to avoid

These cost marks again and again.

• Thinking half-life means “after two half-lives it is all gone”. Two half-lives leave a quarter, not nothing.
• Changing A in beta decay. A stays the same; only Z goes up by 1.
• Writing the beta particle with Z = +1. The electron is ⁰₋₁e.
• Forgetting to check the sums. Add A and Z on both sides before you finish.
• Using the wrong element. After you find Z, look up the symbol in the periodic table.
• Saying a bigger sample has a longer half-life. The half-life depends only on the isotope.

Where radioactivity is used, and staying safe

Hospitals use technetium-99m to image the heart, bones and thyroid because its six-hour half-life is long enough for a scan but short enough to leave the body soon. Cobalt-60 gamma rays treat cancer. Smoke detectors contain a tiny alpha source. Nuclear power stations release energy from uranium, and Bangladesh’s Rooppur plant belongs to this family. Geologists date rocks with uranium and carbon-14 dates bones, wood and cloth.

The same rays that treat cancer can cause it, so people who work with sources follow three habits: keep the time near the source short, keep the distance as large as possible, and put shielding between you and it. Alpha sources need little shielding but must never be swallowed, and gamma sources need thick lead.

Exam corner

For HSC chemistry, physics and admission tests, keep these in your pocket.

• Alpha: A − 4, Z − 2. Beta-minus: A same, Z + 1. Gamma: nothing changes.
• Balance an equation by adding A and Z on each side separately.
• N = N₀ (½)^(t/T): count the half-lives first, then halve that many times.
• For dating, use t = T × log₂ (N₀ / N) or step through the half-lives.
• Know the ranking of penetration: alpha, then beta, then gamma, from the weakest to the strongest.

Frequently asked questions