Partition Coefficients

1. Partition Coefficients Lecture 26

2. The Partition Coefficient • Geochemists find it convenient to define a partitionor distribution coefficientof element i between phases α and β: • Where one phase is a liquid, the convention is the liquid is placed on top: • Incompatible elements are those with Ds/l≪ 1. Compatibleelements are those with Ds/l ≥ 1. These terms refer to partitioning between silicate melts and phases common to mantle rocks (peridotite). It is this phase assemblage that dictates whether lithophile trace elements are concentrated in the Earth’s crust, hence the significance of these terms.

3. Thermodynamic Basis • The chemical potentials of element i in phases α and β are • At equilibrium: • Since trace elements obey Henry’s Law, we can replace the activity coefficient with h.. • The left hand side is ∆G˚, so that • and • where C is simply a constant converting concentration units (usually ppm) to mole fraction. • Thus the distribution coefficient is a kind of equilibrium constant.

4. Relationship among distribution coefficients • In a system with three phases, α, β, and γ, if α and β are in equilibrium and α and γare in equilibrium, then β and γmust also be in equilibrium. It follows that: Dα-β=Dα-γ/Dα-β • This relationship has practical use. For example, if we can determine the partition coefficient for an element between pyroxene and melt and between garnet and pyroxene, we can then calculate the garnet–melt partition coefficient for this element.

5. Temperature and Pressure Dependence • In ideal solutions, the temperature dependence of the partition coefficient is the same as that of the equilibrium constant: • ∆G can be expanded into entropy and enthalpy terms: • In ideal solution, and assuming again that ∆V is independent of temperature and pressure, the pressure dependence is also the same as that of the equilibrium constant: • We would predict a strong pressure dependence when the ionic radius of an element differs greatly from that of the available crystal lattice site. Thus, for example, we would predict the partition coefficient for K between pyroxene and melt would be strongly pressure-dependent since the ionic radius of K is 150 pm and is much larger that the size of the M2 site in clinopyroxene, which is normally occupied by Ca, with a radius of about 100 pm. Conversely, where the size difference is small (e.g., Mn (83 pm) substituting for Fe (78 pm)), we would expect the pressure dependence to be smaller. • In non-ideal solutions, the T and P dependencies will be more complex because the activity coefficients (the Henry’s Law coefficients) will also depend on T and P. • Bottom line: partition coefficient is temperature and pressure dependent.

6. Importance of Ionic Size and Charge • Much of the interest in trace elements in igneous processes centers on the elements located in the lower left portion of the periodic table (K, Rb, Cs, Sr and Ba; the rare earths, Y, Zr, Nb, Hf and Ta). • One reason for this focus of attention is that these elements are all lithophile and therefore present at relatively high abundance in the Earth’s crust and mantle. There is another reason, however: their chemical behavior is comparatively simple: their behavior in igneous systems is mainly (not entirely) a function of ionic size and charge. • The other trace elements that receive the most attention from igneous geochemists are the first transition series elements. Though their electronic structures and bonding behavior are considerably more complex, charge and size are also important. Many of these elements, particularly Ni, Co, and Cr, have partition coefficients greater than 1 in many Mg–Fe silicate minerals. Hence the term “compatible elements” often refers to these elements. Ionic radius (picometers) vs. ionic charge contoured for clinopyroxene/liquid partition coefficients. Cations normally present in clinopyroxene M1 and M2 sites are Ca2+, Mg2+, and Fe2+, shown by ✱ symbols. Elements whose charge and ionic radius most closely match that of the major elements have the highest partition coefficients

7. Quantitative Treatment • Consider the substitution reaction: Mℓ+ CaMgSi2O6 ⇄ Caℓ+ MMgSi2O6 • The Gibbs free energy change of this reaction can be expressed as: • The first term is ∆G for transferring an M2+ ion from the melt to the crystal lattice and simultaneously transferring a Ca2+ ion from the lattice site to the liquid. • The second term is the ∆G associated with the melting of diopside, and governs the distribution of Ca between diopside and the liquid. The distribution coefficient for element M then depends on these two components of free energy:

8. Strain Theory • According to the lattice strain energy theory, ∆Gexchangeis dominated by the energy associated with the lattice strain resulting from M2+being a different size than Ca2+. Because the melt (at least at low pressure) has a far less rigid structure and is more compressible than the solid, any strain in the melt is essentially negligible compared with the strain in the solid. In other words: • The strain energy, ∆Gstrain, may be calculated as: • where r0 is the optimal radius of the lattice site, rM is the ionic radius of M, NA is Avogadro’s Number, and E is Young’s modulus, which is the ratio of stress applied to the resulting strain (change in dimension) and has units of pressure. It is a property of the material and is related to compressibility.

9. Blundy & Wood Model • We can now substitute this expression into • and obtain • where • Where charge is different from the ion normally occupying the site, we much consider the coupled substitution. For example, we can balance charge by substituting bot La3+ and Na+ into the Ca2+ site in diopside. In this case, our relationship is: • The point is we can build a theoretical framework to predict how partitioning will depend on ionic size and charge (and the T and P dependency).

10. Blundy & Wood Model We still depend heavily on experimental determination of partition coefficients. But by experimentally determining a few, we can use the Blundy & Wood model to predict others (as well as pressure dependence).

11. Dependence on Composition Ol-liquid partition coefficient for Zn parameterized as a function of non-bridging oxygens in the melt and T. pyx-liquid partition coefficient for Sm as a function of liquid and solid composition

12. Rare Earth Partition Coefficients

13. Crystal Field Theory • Because transition metal outer electron orbitals are directional, partitioning depends on site geometry. Furthermore, they are not completely filled, so there is a choice as to which orbit an electron will go. • There is a trade of in energy costs between placing 2 electrons in 1 orbit and placing them in geometrically unfavorable orbits for a given lattice site geometry. • It can be quantified as a site preference energy, the difference between the Crystal Field Stabilization Energy(CFSE) of octahedral and tetrahedral sites.

14. Crystal Field Theory Transition metal partitioning depends on the distribution of octahedral and tetrahedral sites in the mineral and the liquid (and well as charge and size) Thus partitioning is more complex.