We examined the density, bulk sound (compressional) velocity, and Grüneisen parameter of liquid pure Fe, Fe 100 H 28 (0.50 wt % H), Fe 88 H 40 (0.81 wt % H), and Fe 76 H 52 (1.22 wt % H) at Earth's outer core pressure and temperature (P-T) conditions (~100 to 350 GPa, 4000 to 7000 K) based on first-principles molecular dynamics calculations. The results demonstrate that the thermodynamic Grüneisen parameter of liquid iron alloy decreases with increasing pressure, temperature, and hydrogen concentration, indicating a relatively small temperature gradient in the outer core when hydrogen is present. Along such temperature profile, both the density and compressional velocity of liquid iron containing~1 wt % hydrogen match seismological observations. It suggests that hydrogen could be a primary light element in the core, although the shear velocity of the inner core is not reconciled with solid Fe-H alloy and thus requires another impurity element. 2. Computational Method First-principles molecular dynamics (FPMD) calculations using pseudopotentials within the density-functional theory were performed with 128 atoms in fixed cubic cells corresponding to~100-350 GPa with different concentrations of hydrogen: pure Fe (0 wt % H), Fe 100 H 28 (0.50 wt %H), Fe 88 H 40 (0.81 wt %H), and Fe 76 H 52