Photoevaporation shapes the observed radii of small exoplanets and constrains the underlying distributions of atmospheric and core masses. However, the diversity of atmospheric chemistries corresponding to these distributions remains unelucidated. We develop a first-principles carbon–hydrogen–oxygen–sulfur–silicon (CHOSSi) outgassing model that accounts for nonideal gas behavior (via fugacities) at high pressures, as well as the tendency for water and hydrogen to dissolve in melt (via solubility laws). We use data-driven radius valley constraints to establish the relationship between the atmospheric surface pressures and melt temperatures of sub-Neptunes. Sub-Neptunes with less massive rocky cores retain less of their primordial hydrogen envelopes, which leads to less heat retention and diminished melt temperatures at the surfaces of these cores. Lower melt temperatures lead thermodynamically to the dominance of carbon-, oxygen-, sulfur-, and silicon-bearing molecules over molecular hydrogen, which naturally produce a diversity of mean molecular weights. Our geochemical outgassing calculations robustly predict a gradient of mean molecular weight across the radius valley, where the strength of this gradient is primarily driven by the oxygen fugacity of the molten cores and not by the carbon enrichment (or “metallicity”) of the atmosphere. Smaller sub-Neptunes are predicted to have less hydrogen-dominated atmospheres. Establishing the precise relationship between the observed and outgassed chemistries requires an understanding of how convection near the core interacts with large-scale atmospheric circulation (driven by stellar heating) near the photosphere, as well as the influence of photochemistry.