Radiometric Isochron Dating Equation
Calculates the absolute geological crystallization age t of minerals without assuming initial daughter isotope concentration D0, using the slope of an isochron plot.
\[ \frac{D_t}{D_i} = \frac{D_0}{D_i} + \frac{P_t}{D_i} \left( e^{\lambda t} - 1 \right) \]
Plain Text Formula:
(D_t / D_i) = (D_0 / D_i) + (P_t / D_i) * (exp(lambda * t) - 1)
📐 Variable Definitions & Parameters
| Symbol | Variable Name | Standard Unit | Physical Definition |
|---|---|---|---|
| \[ t \] | Geological Age | years or Ma | Time elapsed since rock cooled below closure temperature |
| \[ P_t \] | Parent Isotope Abundance | atoms | Unstable radioactive parent remaining today (e.g. ⁸⁷Rb, ⁴⁰K, ²³⁸U) |
| \[ D_t \] | Daughter Isotope Abundance | atoms | Radiogenic daughter isotope measured today (e.g. ⁸⁷Sr, ⁴⁰Ar, ²⁰⁶Pb) |
| \[ D_i \] | Stable Reference Isotope | atoms | Non-radiogenic invariant isotope (e.g. ⁸⁶Sr) |
| \[ \lambda \] | Decay Constant | 1/year | Decay constant: ln(2) / half_life |
💡 Worked Engineering Example
Problem Statement:
In Rubidium-Strontium dating (half-life = 48.8 Ga, lambda = 1.42e-11 yr⁻¹), an isochron slope is measured at m = 0.0355. Find sample age.
Step-by-Step Solution:
- Slope m = exp(lambda * t) - 1 ==> exp(lambda * t) = 1.0355
- lambda * t = ln(1.0355) = 0.03488
- t = 0.03488 / 1.42e-11 = 2,456,600,000 years = 2,457 Ma
🎯 Result: The sample dates to 2.46 billion years ago (Early Paleoproterozoic era).