  Volcanic gases supplied a large part of Earth's early atmosphere, but
constraints on their flux are scarce. Here we model how C-O-H outgassing could
have evolved through the late Hadean and early Archean, under the conditions
that global plate tectonics had not yet initiated, all outgassing was
subaerial, and graphite was the stable carbon phase in the melt source regions.
The model fully couples numerical mantle convection, partitioning of volatiles
into the melt, and chemical speciation in the gas phase. The mantle oxidation
state (which may not have reached late Archean values in the Hadean) is the
dominant control on individual species' outgassing rates because it affects
both the carbon content of basaltic magmas and the speciation of degassed
volatiles. Volcanic gas from mantles more reduced than the iron-w\"ustite
mineral redox buffer would contain virtually no CO2 because (i) carbonate ions
dissolve in magmas only in very limited amounts, and (ii) almost all degassed
carbon takes the form of CO instead of CO2. For oxidised mantles near the
quartz-fayalite-magnetite buffer, we predict median CO2 outgassing rates of
less than approximately 5 Tmol/yr, still lower than the outgassing rates used
in many Archean climate studies. Relatively weak outgassing is due in part to
the redox-limited CO2 contents of graphite-saturated melts, and also to a
stagnant lid regime's inefficient replenishment of upper mantle volatiles. Our
results point to certain chemical and geodynamic prerequisites for sustaining a
clement climate with a volcanic greenhouse under the Faint Young Sun.
