Radioactive Dating Formula

The formula assumes that at the time of formation (t=0), there were no daughter atoms present (D₀ = 0), and the system has remained closed to both parent and daughter migration. It is 'general' because it applies to any parent-daughter system where the daughter is stable and there is no initial daughter. By measuring the current ratio of daughter to parent atoms (D/P) in a sample, and knowing the decay constant (λ), one can calculate the elapsed time. This is the basis for isochron dating and is used in systems like Rb-Sr, Sm-Nd, and U-Pb, though the latter often accounts for multiple decay chains.

The formula derives from the decay law: P = P₀ e^{-λt} and D = P₀ - P (since each parent decay produces one daughter). Substituting P₀ = P + D gives D = (P + D)(1 - e^{-λt}), which rearranges to e^{λt} = 1 + D/P. Thus, the logarithm of (1 + D/P) is correct. Using ln(D/P) would only be valid if D >> P (i.e., very old samples where most parent has decayed), but for young samples or those with low daughter production, it leads to significant underestimation of age. The correct formula handles all ranges, from zero age (D/P → 0, ln(1) = 0) to very old age.

If initial daughter is present, the age equation becomes t = (1/λ) ln(1 + (D - D₀)/P). Since D₀ is often unknown, a single sample cannot yield a reliable age. Isochron dating uses multiple samples from the same closed system with different parent/daughter ratios; plotting D/P vs. a reference isotope yields a line whose slope gives the age and whose intercept gives the initial daughter ratio. This eliminates the need to assume D₀ = 0, making isochron dating much more robust. The simple formula provided is a special case assuming D₀ = 0, which is valid for some systems (e.g., meteorites, but rarely for terrestrial rocks).

The decay constant λ is the probability per unit time that a given parent nucleus will decay, with units of s⁻¹ (or yr⁻¹). The half-life t₁/₂ is the time required for half of the parent atoms to decay, related by t₁/₂ = ln(2)/λ ≈ 0.693/λ. In the dating formula, λ is used directly; if you only know the half-life, you must calculate λ first. For example, for ¹⁴C (half-life 5730 yr), λ = 0.693/5730 ≈ 1.21×10⁻⁴ yr⁻¹. Always use consistent time units for λ and t (e.g., years if λ is in yr⁻¹).

The U-Pb system is powerful because two independent clocks coexist. For each chain, the same formula applies separately: t = (1/λ₂₃₈) ln(1 + ²⁰⁶Pb*/²³⁸U) and t = (1/λ₂₃₅) ln(1 + ²⁰⁷Pb*/²³⁵U), where the asterisk denotes radiogenic lead (corrected for initial lead). If the sample has remained closed, both ages will agree (concordia). If not, they will disagree, revealing lead loss or initial lead contamination. The formula with D/P works for each chain independently, but in practice, geochronologists often use the more complex concordia-discordia plots that account for common lead and multiple decay steps.

For a branching decay, the total decay constant λ = λ₁ + λ₂, where λ₁ and λ₂ are the partial decay constants for each branch. If you measure only daughter 1 (e.g., ⁴⁰Ar from ⁴⁰K), the accumulation equation is D₁ = (λ₁/λ) P₀ (1 - e^{-λt}). Using P = P₀ e^{-λt}, the age formula becomes t = (1/λ) ln(1 + (λ/λ₁)(D₁/P)). This differs from the simple formula by the factor λ/λ₁. For ⁴⁰K-⁴⁰Ar dating, the branching ratio is λ_Ar/λ_total ≈ 0.104, so the age equation is t = (1/λ) ln(1 + (λ_total/λ_Ar)(⁴⁰Ar*/⁴⁰K)), often written as t = (1/λ) ln(1 + 9.54 × ⁴⁰Ar*/⁴⁰K) when using certain constants. Always use the correct branching-corrected formula when applicable.

Using the half-life relation, t = (t₁/₂ / ln 2) × ln(1 + D/P). For a sample to be dated, D/P must be measurable. For young samples (relative to t₁/₂), D/P is small, and the age is approximately t ≈ (D/P) / λ, i.e., linearly proportional to D/P. For old samples, D/P can become large, and ln(1 + D/P) approaches ln(D/P), so t ≈ t₁/₂ × log₂(D/P). To date a sample as old as several half-lives, D/P must be > 1; if D/P is very small (e.g., < 0.01), the age may be indistinguishable from zero within analytical error. This is why dating methods choose isotopes whose half-lives match the expected age range of the sample (e.g., ¹⁴C for 10-50,000 years, U-Pb for billions of years).

The age uncertainty is derived from error propagation: σ_t = (1/λ) × (1/(1 + D/P)) × sqrt( (σ_D/P)^2 ), where σ_D/P is the relative error in the D/P ratio. For old samples with high D/P, the factor 1/(1+D/P) ≈ 1/(D/P), making the age uncertainty roughly proportional to the relative error in D/P divided by λ. In practice, mass spectrometry can achieve D/P ratios with precision of 0.1-1%, yielding age uncertainties of about 0.1-1% for well-behaved systems. For young samples with low D/P, the uncertainty is larger relative to age. Isochron dating improves precision by using multiple data points and regression analysis, often achieving errors of < 1 Ma for rocks billions of years old.

For ¹⁴C (t₁/₂ = 5730 yr), after about 50,000 years (≈ 8.7 half-lives), the remaining parent fraction P/P₀ = (1/2)^(50,000/5730) ≈ 2.4×10⁻³, meaning only 0.24% of the original ¹⁴C remains. At this point, the D/P ratio is ≈ (1 - 0.0024)/0.0024 ≈ 415. The formula gives t = (1/λ) ln(1 + 415) = (1/λ) ln(416) ≈ 8.72 half-lives ≈ 50,000 years. However, the small amount of remaining ¹⁴C is extremely difficult to measure accurately, and any contamination with modern carbon (even 0.1%) can significantly bias the result. Beyond 50,000 years, the background noise and contamination dominate, making the age unreliable. This is a practical limit, not a theoretical one, and is why other isotopes (e.g., K-Ar, U-Pb) are used for older samples.

A system can become open through: (1) loss of daughter atoms (e.g., ⁴Ar escapes from minerals due to heating), (2) loss or gain of parent atoms (e.g., uranium leaching), (3) introduction of initial daughter that is not accounted for, (4) metamorphic recrystallization that resets the clock. Detection methods include: (a) comparing ages from multiple decay systems (e.g., U-Pb concordia), (b) isochron plots that reveal scatter from closed-system behavior, (c) mineral separates that show different ages (indicating partial resetting), (d) step-heating techniques (e.g., ⁴⁰Ar/³⁹Ar) that release argon from different crystal sites and can reveal thermal events. If a system is open, the simple formula yields a meaningless 'age' that may represent a mixture or a cooling age, not the time of formation.

In practice, geologists measure isotope ratios (e.g., ⁸⁷Sr/⁸⁶Sr, ⁸⁷Rb/⁸⁶Sr) rather than absolute atom numbers. The formula is rewritten in terms of these ratios: (⁸⁷Sr/⁸⁶Sr)_measured = (⁸⁷Sr/⁸⁶Sr)_initial + (⁸⁷Rb/⁸⁶Sr)(e^{λt} - 1). This is a linear equation; a set of co-magmatic samples with variable Rb/Sr ratios yields an isochron line whose slope gives e^{λt} - 1, from which t is derived. No absolute abundances are needed, only accurate ratio measurements. Reference materials like standards (e.g., NIST SRM 987 for Sr) are used to calibrate the mass spectrometer and correct for mass fractionation, ensuring the ratio accuracy. The age is then calculated using the measured slope and the known decay constant.

Yes, but with modifications. For short-lived isotopes like tritium (t₁/₂ = 12.3 yr), the formula t = (1/λ) ln(1 + D/P) is valid if the initial concentration is known from precipitation records. However, the input function (initial D/P) is often time-varying, so age is determined by comparing the measured tritium concentration to the known historical atmospheric input (bomb-pulse dating). For ³⁶Cl (t₁/₂ = 301,000 yr), the formula works for groundwater residence times up to ~1 million years, assuming no significant addition of chloride from other sources. In all cases, the simple formula is the starting point, but hydrologists often use more complex models that account for mixing, recharge, and dilution.

A model age is calculated using the formula t = (1/λ) ln(1 + D/P) assuming a specific initial daughter ratio (often zero or a fixed value). It is highly sensitive to that assumption and can be wrong if initial daughter was present. An isochron age is determined from multiple samples with variable P/D ratios, using regression to determine both the slope (age) and the intercept (initial daughter ratio) without assuming D₀. This eliminates the need for a priori knowledge of initial conditions and provides a statistical assessment of the data's consistency. Isochron ages are therefore much more reliable and are the standard for dating ancient igneous and metamorphic rocks, as they can detect open-system behavior through scatter in the data.

In a complex thermal history, the simple formula gives a 'cooling age' that represents the time when the mineral cooled below its closure temperature (the temperature at which the daughter isotope becomes trapped). For example, in K-Ar dating, the closure temperature for argon in biotite is ~300°C; the age reflects the time since the rock cooled below that temperature, not necessarily its crystallization age. To model multiple events, geochronologists use numerical thermal models (e.g., diffusion modeling) that solve the diffusion equation for the specific isotope and mineral, fitting the measured age spectra (e.g., from ⁴⁰Ar/³⁹Ar step-heating) to infer the full thermal evolution. The simple formula is just a snapshot; more advanced techniques reconstruct the history.