  Despite its importance to questions of lunar origin, the chemical composition
of the Moon is not precisely known. In recent years, however, the isotopic
composition of lunar samples has been determined to high precision and found to
be indistinguishable from the terrestrial mantle despite widespread isotopic
heterogeneity in the Solar System. In the context of the giant-impact
hypothesis, this level of isotopic homogeneity can evolve if the proto-lunar
disk and post-impact Earth undergo turbulent mixing into a single uniform
reservoir while the system is extensively molten and partially vaporized. In
the absence of liquid-vapor separation, such a model leads to the lunar
inheritance of the chemical composition of the terrestrial magma ocean. Hence,
the turbulent mixing model raises the question of how chemical differences
arose between the silicate Earth and Moon. Here we explore the consequences of
liquid-vapor separation in one of the settings relevant to the lunar
composition: the silicate vapor atmosphere of the post-giant-impact Earth. We
use a model atmosphere to quantify the extent to which rainout can generate
chemical differences by enriching the upper atmosphere in the vapor, and show
that plausible parameters can generate the postulated enhancement in the
FeO/MgO ratio of the silicate Moon relative to the Earth's mantle. Moreover, we
show that liquid-vapor separation also generates measurable mass-dependent
isotopic offsets between the silicate Earth and Moon and that precise silicon
isotope measurements can be used to constrain the degree of chemical
fractionation during this earliest period of lunar history. An approach of this
kind has the potential to resolve long-standing questions on the lunar chemical
composition.
