Half-Life

Imagine 1000 popcorn kernels in a hot pan. After one minute, roughly half have popped — about 500 unpopped left. Another minute, half of the remaining pop — now about 250. After another minute, ~125. Each “round” cuts the remaining count in half.

Radioactive decay works exactly the same way. After each half-life, exactly half the remaining unstable nuclei have decayed. The MCAT loves this topic because the math is dead simple (just halving), and the underlying physics ties to carbon-14 dating, drug half-lives in biology, medical imaging tracers, and any first-order kinetic process you learned about in chemistry.

The simulation below lets you watch this process in real time. Each circle represents a single radioactive atom with an independent, random chance of decaying each moment - yet despite the randomness, the overall curve follows a smooth, predictable exponential. Hit “Start Decay” and watch the law of large numbers in action.

The Half-Life Equation

The half-life ($t_{1/2}$) is the time it takes for half of a radioactive sample to decay. After n half-lives, the fraction of the original sample remaining is:

The Half-Life Table

This table is worth memorizing - it covers nearly every MCAT half-life question:

| Half-lives (n) | Fraction remaining | Percent remaining |
|----------------|-------------------|-------------------|
| 0 | 1 | 100% |
| 1 | $\frac{1}{2}$ | 50% |
| 2 | $\frac{1}{4}$ | 25% |
| 3 | $\frac{1}{8}$ | 12.5% |
| 4 | $\frac{1}{16}$ | 6.25% |
| 5 | $\frac{1}{32}$ | 3.125% |

Solving Half-Life Problems

Step 1: Find the Number of Half-Lives

Divide the total time by the half-life: n = t / $t_{1/2}$

Step 2: Apply the Fraction

Multiply the initial amount by ($\frac{1}{2}$)^n.

Example

A sample contains 800 mg of iodine-131 ($t_{1/2}$ = 8 days). How much remains after 24 days?

n = 24 / 8 = 3 half-lives

Amount remaining = 800 x $(1/2)^3$ = 800 x $\frac{1}{8}$ = 100 mg

First-Order Kinetics

Radioactive decay is a first-order process, which means the rate of decay is proportional to the amount of radioactive material present. The more atoms you have, the more decays per second - but the fraction that decays per unit time stays constant.

This is the same first-order kinetics you learned in chemistry. The decay constant (λ) relates to the half-life:

Semi-Log Plots

On a normal (linear) graph, radioactive decay produces a curved exponential line that swoops downward. But if you plot the natural log of the amount remaining (ln N) versus time, you get a straight line with slope = -λ.

Activity

The activity of a radioactive sample is the number of decays per second, measured in becquerels (Bq) or curies (Ci). Activity follows the same half-life pattern as the number of atoms:

Activity = λ x N

Since N decreases by half each half-life, so does the activity. A freshly prepared sample is most active, and the activity drops exponentially over time.