Atmospheric scale height H = RT/(Mg), where R is the universal gas constant, T is temperature, M is the mean molar mass of the gas, and g is gravitational acceleration. Scale height is inversely proportional to molar mass, so a lighter gas (like H2) produces a much larger scale height than a heavier gas mixture (like air).
Earth's real atmosphere (mean molar mass ≈29 g/mol) has a scale height of about 8 km. If the atmosphere were composed of H2 (molar mass...
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