How does radiometric dating calculate the numerical age of a rock or mineral?

Radiometric dating determines the numerical age of rocks and minerals by measuring the ratio of radioactive isotopes (parent atoms) to their stable decay products (daughter atoms) within a sample.12

Because radioactive isotopes decay at a constant, known rate—characterized by a specific half-life—the accumulation of daughter atoms serves as a geologic clock. The fundamental relationship is expressed as:12

$$t=\frac{1}{\lambda }\ln (1+\frac{D}{P})$$

In this equation, $t$ is the time elapsed, $\lambda$ is the decay constant, $D$ is the number of daughter atoms, and $P$ is the number of parent atoms.2

To ensure an accurate age, several conditions must be met:
* The system must have remained closed, meaning no parent or daughter atoms were added or lost since the rock or mineral formed.2
* Scientists must account for any initial daughter atoms present at the time of formation.23
* The decay constant must be precisely known.2

Advanced techniques, such as the isochron method, allow researchers to validate these assumptions by analyzing multiple samples to ensure internal consistency.23