Equation of the tangent to a curve

When minerals fractionate from a crystallizing magma, mass balance requires that, in a linear plot of Cr vs Ni, the points representing the composition of the parental 

In the present case, the question is whether Reunion olivine-rich rocks belong to a cumulate control line of the basaltic rocks. Through a simple log-log regression, we find that the basalt trend (= the liquid line of descent) can be approximated by a power law, a form justified in Sections 1.5 and 9.3. 

The slope of the linear array defined by the olivine-rich lavas (olivine control line) with more than 100 ppm Ni is found by linear regression to be 

Thermodynamic stability. Let us consider a liquid binary mixture of two species, a and b, crystallizing to a solid solution. Let G be the free enthalpy (Gibbs) function of each phase, either solid or liquid. Calculate the change AG in molar free enthalpy when a liquid of molar composition XUqb crystallizes to a solid of molar composition 

At first sight, the answer would be AG = AGo = G,iq(Anqb)— GU so. ), but we will show that this is wrong. We assume that dn = dna+dnb moles are transferred from the liquid to the solid phase. Let us assign the symbol fi to chemical potentials, e.g., /iUqa for species a in the liquid. Then 

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