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Argon-40 is a stable isotope with 18 protons and 22 neutrons. Radiogenic argon-40 accumulated from potassium-40 is a daughter product used in potassium-argon dating.

Let $K(t)$ be the surviving potassium-40 population and $A_r(t)$ the accumulated radiogenic argon-40. The total nuclear decay constant $\lambda$ sets $K(t)=K_0{e}^{-\lambda t}$. Only fraction $b=0.11$ of lost potassium atoms produces argon, so under the stated retention and zero-initial-argon assumptions,Dividing by the present potassium abundance eliminates the unknown initial population:With $A_r/K=0.3$,Thus the inferred potassium-argon dating age is about $2.5$ billion years, consistent with the proposed $2.5\times 10^9$-year age to the accuracy of the supplied inputs. Omitting the branching fraction would seriously underestimate the age. At exactly $2.5$ billion years, the model predicts $A_r/K\simeq 0.307$, consistent with the rounded measured $0.3$.

This dates closure of the mineral to argon loss and potassium exchange under the assumptions, not necessarily its original crystallization or the first formation of the Earth's crust. The depth alone is not an age measurement. No uncertainty or argon-retention history is supplied, so the agreement is a consistency check rather than proof of a particular geological event.

Potassium-40 is a long-lived radioactive isotope with 19 protons and 21 neutrons. It can decay to calcium-40 through beta-minus decay, or to argon-40 through electron capture and a rare beta-plus decay. The argon-producing branch supports potassium-argon dating.

With no initial radiogenic argon-40 and complete daughter retention, the surviving potassium-40 population is $N_K=N_{K,0}{e}^{-\lambda t}$ and the accumulated daughter population is $N_{\mathrm{Ar}}=b(N_{K,0}-N_K)$. Here $\lambda$ is the total nuclear decay constant and $b$ is the argon-producing branching fraction. Their ratio gives the displayed age. The result measures closure of the mineral's isotope system under those assumptions.

TIPP 2021 presentation: indico.cern.ch/event/981823/contributions/4295324/attachments/2250469/3818551/KDK_TIPP_2021_pub.pdf

Its potassium-40 decay scheme gives the approximate atomic mass-energy differences of $1.311$ MeV to calcium-40 and $1.505$ MeV to argon-40, and distinguishes capture to an excited argon level from the rare positron branch. The positron kinetic-energy endpoint additionally subtracts $2m_e$ from the atomic mass difference.