Geospeedometry

Geospeedometry makes use of the temperature dependence of the chemical diffusion of elements between zones of a mineral grain. According to Fick's second law, the concentration of an element in one dimension across a diffusive interface is given by the partial differential equation

${\displaystyle {\operatorname {\partial } \!C \over \operatorname {\partial } \!t}={{\operatorname {\partial } } \over {\operatorname {\partial } \!x}}\left(D{\operatorname {\partial } \!C \over \operatorname {\partial } \!x}\right)}$

where

• D is the diffusion coefficient, or diffusivity (m² s⁻¹) of the element in the medium
• x is the position (length)
• t is time
• C is the concentration of the element; C(x,t) is a function dependent on both length and time.

For a one-dimensional interface at x = 0 with a fixed concentration C₀,

${\displaystyle C\left(x,t\right)=C_{0}\mathrm {erfc} \left({\frac {x}{2{\sqrt {Dt}}}}\right)}$

where erfc is the complementary error function.

The diffusivity of an element in a medium is given by the temperature-dependent Arrhenius equation

${\displaystyle D=D_{0}e^{-E_{A}/RT}}$

where

• D₀ is the maximum diffusivity at infinite temperature; also known as the pre-exponential factor (m² s⁻¹)
• E_A is the activation energy for diffusivity (J mol⁻¹)
• R is the gas constant (J mol⁻¹ K⁻¹)
• T is absolute temperature (K)

Given an experimentally determined D₀ for a given element in a given substance, an empirical diffusion profile can be used to calculate the peak temperature or duration of a thermal pulse experienced by the substance. In cases of varying temperature (e.g. during cooling) D becomes a function of time and the simple analytical solution may no longer be applicable.