Radioactive Decay and Half-Life Calculations for the MCAT

Radioactive decay is a first-order process: in each half-life, half of whatever parent nuclei remain will decay, no matter how many you start with. That single fact gives you the working formula N = N₀(1/2)^t/t_1/2, and most half-life questions reduce to counting how many half-lives fit into the elapsed time. The other half of the topic is bookkeeping: every decay equation must conserve mass number (A) and atomic number (Z), and each decay mode changes them in a fixed way.

The decay modes and what they do to A and Z

Write a nuclide as ^A_ZX, where A is the number of protons plus neutrons and Z is the number of protons. A decay equation balances when the A values add up on both sides and the Z values add up on both sides.

The logic behind the beta modes is worth owning rather than memorizing. In beta-minus decay a neutron inside the nucleus turns into a proton, so Z goes up by one while A stays the same. The emitted electron is created in that conversion; it does not come from the electron cloud. Beta-plus decay and electron capture both turn a proton into a neutron, so Z goes down by one. Since stable heavy nuclei need more neutrons than protons (up to about 1.5 neutrons per proton), a nucleus that is neutron-rich fixes itself with beta-minus decay, and one that is neutron-poor uses positron emission or electron capture.

To balance an unknown, write the two conservation sums separately. Suppose ²¹⁰₈₄Po emits an alpha particle. Mass: 210 = 4 + A, so A = 206. Charge: 84 = 2 + Z, so Z = 82, which is lead. The product is ²⁰⁶₈₂Pb.

Penetration and ionization

The three classic emissions sort in opposite orders for penetration and ionization. Alpha particles are heavy and doubly charged, so they ionize strongly and are stopped by a sheet of paper or the outer layer of skin. Beta particles get through paper but are stopped by a few millimeters of aluminum. Gamma rays carry no charge and no mass, so they penetrate furthest and need thick lead or concrete for shielding. A useful consequence: an alpha emitter is relatively harmless outside the body but very damaging if swallowed or inhaled, because all of its energy is deposited in a small volume of tissue.

Why decay follows half-lives

Each unstable nucleus has the same fixed probability of decaying per unit time, set by the decay constant k. With N nuclei present, the number decaying per second is proportional to N:

rate = kN, which integrates to N = N₀e^-kt

Setting N = N₀/2 and solving gives the half-life:

t_1/2 = ln 2 / k = 0.693 / k

Because ln 2 is a constant, the half-life does not depend on how much material you have, on temperature, or on chemical form. A gram of iodine-131 and a kilogram of it both lose half their nuclei in about 8.0 days. Combining the two equations gives the version most problems use:

N / N₀ = (1/2)ⁿ, where n = t / t_1/2

Two related quantities show up in harder questions:

• Activity A = kN, the number of decays per second, measured in becquerels (1 Bq = 1 decay per second). Older problems use curies, 1 Ci = 3.7 × 10¹⁰ Bq. Since activity is proportional to N, it falls with the same half-life as the nuclei themselves.
• Mean lifetime τ = 1/k = t_1/2 / 0.693, about 1.44 half-lives. It is the average time a nucleus survives before decaying.

A fraction table worth knowing cold

The ten half-lives row is a handy anchor: 2¹⁰ = 1024 is close to 1000, so after ten half-lives roughly one part in a thousand survives. Read questions carefully for "remaining" versus "decayed", since the two columns are easy to swap.

Worked examples

Example 1: whole numbers of half-lives

A hospital pharmacy measures 800 MBq of fluorine-18 (half-life about 110 minutes) at 8:00 a.m. What is the activity at 11:40 a.m.?

1. Elapsed time: 3 hours 40 minutes = 220 minutes.
2. Number of half-lives: n = 220 / 110 = 2.
3. Activity: 800 × (1/2)² = 800 / 4 = 200 MBq.

Example 2: a fractional number of half-lives

Technetium-99m has a half-life of about 6.0 hours. What fraction of a sample is left after 9.0 hours?

1. n = 9.0 / 6.0 = 1.5.
2. Fraction = (1/2)^1.5 = 1 / (2 × √2) = 1 / 2.83, about 0.35, or 35%.

You can sanity-check this without a calculator: after one half-life 50% remains and after two 25% remains, so 1.5 half-lives must land between them. Because the curve is exponential, the answer sits a little below the midpoint of 37.5%.

Example 3: solving for time

How long until a sample of iodine-131 (half-life 8.0 days) falls to 10% of its starting amount?

1. Set (1/2)ⁿ = 0.10, so 2ⁿ = 10.
2. n = log 10 / log 2 = 1 / 0.301 = 3.32 half-lives.
3. t = 3.32 × 8.0 days, about 27 days.

Bracket check: three half-lives leave 12.5% and four leave 6.25%, so 10% must fall between 24 and 32 days. On a multiple-choice question, the bracket alone usually picks the answer.

Example 4: radiocarbon dating

A wood sample has 12.5% of the carbon-14 found in living wood. Carbon-14 has a half-life of about 5730 years. Since 12.5% = (1/2)³, three half-lives have passed, and the age is 3 × 5730 = about 17,200 years. Living organisms keep exchanging carbon with the environment, so the clock starts at death, when uptake stops.

Example 5: comparing activities

Two samples contain the same number of nuclei: one of technetium-99m (half-life 6 hours) and one of iodine-131 (half-life 8.0 days, which is 192 hours). Which is more active, and by how much? Activity is kN and k = 0.693 / t_1/2, so at equal N the activity ratio is the inverse of the half-life ratio: 192 / 6 = 32 times more active for Tc-99m. A short half-life means a hotter sample, not a weaker one.

Common traps

• Thinking a sample is gone after two half-lives. Two half-lives leave 25%, not zero. Decay approaches zero but never jumps to it.
• Subtracting half the original amount each time. Each half-life removes half of what is left, not half of the starting amount. The amounts go 100, 50, 25, 12.5, not 100, 50, 0.
• Expecting half-life to change with sample size, temperature or pressure. It does not. Nuclear decay is insensitive to ordinary chemical conditions.
• Getting the beta direction backward. Beta-minus raises Z (neutron becomes proton). Beta-plus and electron capture lower Z.
• Placing the beta electron in the electron cloud. It is made in the nucleus during the neutron-to-proton conversion.
• Forgetting that gamma changes nothing. A and Z are unchanged. Only the energy state of the nucleus drops.
• Linking "most penetrating" with "most ionizing". They are opposite ends: alpha ionizes most and penetrates least, gamma the reverse.
• Mixing time units. Convert the elapsed time and the half-life into the same unit before dividing.

One-minute recap

Balance every nuclear equation by conserving A and Z separately. Alpha: A drops 4, Z drops 2. Beta-minus: Z up 1. Beta-plus and electron capture: Z down 1. Gamma: no change. Decay is first order, so N = N₀(1/2)^t/t_1/2 and t_1/2 = 0.693/k. Count half-lives first, then halve that many times. For non-integer cases, bracket between whole half-lives or use logs. Activity A = kN falls with the same half-life, and at equal N a shorter half-life means higher activity.