Vapour pressures

For silicate materials, vaporisation reactions can be generalised as:

where Mx+nOx+n2 is the metal oxide species in the silicate melt, MxOx2 is the stable gas species with metal oxidation state of x, balanced by n4 moles of O₂ gas for a congruent reaction (i.e., in which the gas(es) produced have the same composition as the reactant(s)).

At equilibrium, the partial pressure (assuming ideal gas behaviour such that f = p), is given by:

where,

for which ΔG_(r) is the Gibbs Free Energy of the vaporisation reaction, R the gas constant and T_s the equilibrium temperature set by the magma ocean at its surface in Kelvin. At low pressures in which volume changes are insignificant relative to the standard state (here taken at 1 bar),

In which ΔH is the enthalpy change and ΔS the entropy change of reaction. Substituting this equation into eq. (12), one obtains:

From experimental assessment of the equilibrium vapour pressures of metal-bearing gas species above metal oxides or metal oxide components in silicate melts, ΔS is near-constant^15,70, which results in slopes in logp_i vs. T space that are sub-parallel for almost all elements (Supplementary Figure 1). Element volatilities are defined by their 50% condensation (or evaporation) temperature in the solar nebular gas¹⁵:

Here, f_vap = the fraction in the vapour (=0.5 for 50% condensation). The fO₂ is that of a cooling solar nebula⁷¹, while the total pressure is 10^-4 bar in keeping with convention^19,29. The T_c⁵⁰ of a given fictive element is defined by holding ΔS constant and changing the value of ΔH in eq. (16). Their properties are shown in Supplementary Table 1.

The partial pressure is determined as a function of temperature, element activity, fO₂ of the atmosphere and the stoichiometry of the vaporisation reaction, n (eq. 11). This links the T_c⁵⁰ of an element with its general volatility behaviour under planet-forming conditions. Because most elements evaporate according to n = 2 stoichiometries¹⁵, including all five major cation-forming elements in the BSE composition (Fe, Mg, Si, Ca, and Al), we adopt n = 2 for all fictive elements (Supplementary Table 1). Hence, relative fractionation by volatility is insensitive to fO₂, which is not the case for the alkalis (n = 1), or for the Group VI metals (Cr, Mo, and W) for which n < 0¹⁵. However, this simplification is apt to establish the overall trend of element depletion, and not to account for anomalies in the abundances of certain elements. Moreover, it is independent of any eventual revisions to T_c⁵⁰ values.

The sum of the partial pressures dictate the total pressure at any given T and fO₂:

since p(MxOx2) depends on (fO₂)^-n/4, congruent evaporation (eq. 11) demands that oxygen fugacity be equal to n/4 times the sum of the partial pressures of the metal oxide species. Because n = 2 in our treatment (Supplementary Table 1), then:

Thus, the value of fO₂ is solved iteratively until the term 0.5P_T is equal to ½ the sum of the individual partial pressures of each component at a given temperature. Here, we constrain element activities by fitting our value of P_T to that calculated for evaporation of silicate mantles⁷². To do so, we assign an activity to each fictive element as a function of its T_c⁵⁰ value. As per planetary materials, the three most abundant elements (Fe, Mg and Si) have T_c⁵⁰ ~ 1300 – 1400 K. We use a Gaussian distribution to model a peak element activity about 1350 K. The rate at which element activity declines is found by minimisation to the P_T determined in ref. 72 over the range 1800 < T (K) < 3000, using the objective function:

By iterating the value of c:

We obtain a best-fit value to the total pressure above the bulk silicate Earth⁷² (Supplementary Figure 2) at c = 100, yielding the initial activities, (xγ)_initial, shown in Supplementary Table 1.