The Arrhenius Equation: Proposed 1889 by Svante Arrhenius, a chemist from Sweden. Arrhenius performed experiments that correlated chemical reaction rate constants with temperature. After observing that many chemical reaction rates depended on the temperature, Arrhenius developed this equation to characterize the temperature-dependent reactions. Arrhenius Equation

This formula for the temperature dependence of reaction rates can be used to model the temperature-variance of diffusion coefficients, population of crystal vacancies, creep rates, and many other thermally-induced processes/reactions. Arrhenius Equation

plot of 1/T vs. ln(k) is useful to get empirically: – Y-intercept: ln(A)  get the pre-exponential factor, A – Slope: -Ea/R  get activation energy Ea Arrhenius Plot

Apply general concept to geochronology, diffusion in minerals, closure temperature theory 101

Modes of Diffusion Radiogenic isotopes: diffusive loss of daughter product Assume volume diffusion is rate-limiting process (no recrystallization, we just care about transition from open to closed system behavior)

Chemical Diffusion (Fick’s First Law) Where F C =chemical flux D=diffusivity C=concentration x=spatial coordinate

Changes in Concentration With Diffusion (Fick’s Second Law) is a description of changes in the distribution of a species in a system with time (t) as a consequence of diffusion. This equation has a number of solutions for different geometries.

Dependence of D on P and T Based on many experimental studies of natural systems, D has been found to depend exponentially on pressure and temperature through the so-called “Arrhenius relationship”: WhereD=Diffusivity at conditions of interest D o =Pre-exponential constant E=Activation energy P=Pressure V=Activation volume R=Gas constant T=Temperature

Reasonable Simplification In virtually all cases of interest, E >> V and we can simplify the Arrhenius relationship to: WhereD o =Diffusivity at infinite T

Estimating Diffusivity of Chemical Species in Silicate Minerals Empirical Studies Combination of heat flow modeling with estimation of fractional loss of an element or specific isotope Estimation of fractional loss of a species from individual crystals based on previously known thermal history Experimental Studies Theoretical Estimation Comparison of crystal structures of minerals for which isotopic diffusivities are known with those for which they are not

Farley, K.A., 2000, Helium diffusion from apatite: General behavior as illustrated by Durango fluorapatite, Journal of Geophysical Research 105,

Closure temperature

The Transition from Open- to Closed-System Behavior Dodson, M. H., 1973, Closure Temperature in Cooling Geochronological and Petrological Systems. Contributions to Mineralogy and Petrology 40, Consider the accumulation of daughter isotope in a sample:

The Transition from Open- to Closed-System Behavior (2) For an open system from which the daughter isotope is free to escape, there is no accumulative path. If we assume that the escape of daughter occurs through the process of diffusion, it will occur at a rate that is proportional to temperature. As a system cools, we might expect it to have an early history of purely open-system behavior, and a later history of purely closed-system behavior:

Closure Temperature Concept The “closure temperature” – T c – was defined by Dodson as the temperature of a system at the time of its measured date.

Bulk Closure Temperature Equation Where T c = closure temperature D 0, E=diffusion parameters R=gas constant A=geometric term (55 for a sphere, 27 for a cylinder, 8.7 for a plane sheet) a=effective diffusion dimension dT/dt=cooling rate