Radioactive samples change unpredictably one nucleus at a time, but large groups follow precise statistical patterns. The decay constant connects that randomness to measurable changes in remaining nuclei and estimated ages.

Decay constant and half-life

Each radioactive isotope has its own decay constant, which describes how quickly the nuclei in a sample tend to decay.

The decay constant is a property of the isotope itself, not of the sample size. A larger value of $\lambda$ means nuclei decay more quickly, while a smaller value means they decay more slowly.

A closely related idea is half-life, the time required for half of the undecayed nuclei in a sample to remain.

Half-life and decay constant describe the same process in different ways. Half-life is often easier to visualize, while the decay constant is especially useful in mathematical models.

Because $\ln 2$ is a constant, half-life and decay constant are inversely related. If the half-life is short, the decay constant is large. If the half-life is long, the decay constant is small.

Predicting remaining nuclei over time

For a large sample, the number of undecayed nuclei does not decrease by the same amount each second. Instead, it decreases by the same fraction over equal time intervals. That is why radioactive decay follows an exponential pattern.

Decay curve showing the percent remaining as a function of the number of half-lives. The labeled points (50%, 25%, 12.5%, …) make it easy to see that each equal half-life interval multiplies the remaining amount by $\tfrac{1}{2}$, which is the defining feature of exponential decay. Source

In a real sample, the exact nucleus that decays next cannot be predicted. However, when many nuclei are present, the overall behavior becomes highly regular, so the exponential model gives reliable predictions for the number remaining.

This equation shows that the number remaining depends on both the isotope, through $\lambda$, and the elapsed time. The equation can also be written using half-life when that form is more convenient.

These two forms are equivalent. The half-life form is especially useful when the elapsed time is an exact whole-number multiple of the half-life.

Useful interpretations include:

• If one half-life has passed, $N=\dfrac{N_0}{2}$.

• If two half-lives have passed, $N=\dfrac{N_0}{4}$.

• If three half-lives have passed, $N=\dfrac{N_0}{8}$.

Age problems are often easiest when the fraction remaining matches one of these simple patterns. Recognizing a familiar fraction can save time and reduce algebra.

Because the decrease is exponential, the sample never reaches zero in the mathematical model. It becomes smaller and smaller, but the fraction remaining keeps shrinking rather than dropping to zero at a fixed time.

Finding the age of a material

These same ideas can be used for dating, which means determining how long it has been since a material started with a known initial amount of a radioactive isotope.

To determine age, the key information is the initial amount and the amount remaining now. If those are known, then the elapsed time can be found from the decay model. In AP Physics 2 problems, the initial amount is typically given directly or can be inferred from the way the question is stated.

This equation is the decay law rearranged to solve for time. In many AP Physics 2 situations, the easiest path is to recognize how many half-lives have passed from the fraction remaining, then multiply by the half-life.

When using these ideas, the same ratio idea applies even if the amount is measured in mass rather than by counting nuclei directly, as long as the measurement refers to the same radioactive isotope throughout. The fraction remaining is what matters.

What to emphasize on AP Physics 2 problems

Most questions on this subsubtopic focus on interpreting ratios and choosing the most efficient equation. Common reasoning patterns include:

• A decay constant gives the rate tendency of the isotope, not a fixed number of nuclei lost each second.

• The number of remaining nuclei decreases exponentially, not linearly.

• Half-life stays constant for a given isotope, even though the number of decays during each half-life gets smaller as the sample shrinks.

• If the initial amount is known, the current amount can be used to find the age.

• If only the current amount is known, the age cannot be determined without more information about the starting amount or the fraction that remains.

A good check on any answer is whether it matches the physical trend. Older samples should have fewer parent nuclei remaining, and isotopes with larger decay constants should reach the same fraction remaining in less time.