How much does Earth's atmosphere weigh?

Air pressure is the weight of the air overhead, so pressure times area gives a force. The units won't let you call that force a mass, and a second pass with the scale height of the atmosphere gets within 1% of the measured value.

First pass: pressure times area

Sea-level pressure and Earth's surface area are built in as atm and earth_surface_area.

pressure = atm
area = earth_surface_area
force = pressure * area
force as N

5.17 × 10¹⁹ newton {force}

Pascals times square meters is newtons, a force. It's tempting to stop here and call this the atmosphere's mass, but force as kg fails with "Cannot convert Pa m^2 to kg: incompatible dimensions". To turn a weight into a mass, divide by the gravitational acceleration g.

first_pass = force / g as kg

first_pass =5.27 × 10¹⁸ kg {mass}

Second pass: the ground isn't at sea level

atm is the pressure at sea level, but 29% of Earth's surface is land, averaging 0.6 to 1 km above the sea. Less air sits above high ground. How much less depends on how quickly pressure falls with height.

Pressure falls by a factor of e every scale height, which is set by how hard gravity pulls air down against its thermal motion: H = RT / (Mg). Air's molar mass is 29 g/mol, and the lower atmosphere averages 250 to 290 K.

air_molar_mass = 29 g/mol
air_temperature = 250 to 290 K
scale_height = R * air_temperature / (air_molar_mass * g) as km

scale_height =7.87 [7.32, 8.47] km {length}

That's 7.87 [7.32, 8.47] km, about the height of Everest. Over land, the pressure is lower by a factor of e^−h/H.

land = 29%
land_height = 0.6 to 1 km
pressure_factor = (1 - land) + land * exp(-land_height / scale_height)
mass = first_pass * pressure_factor

mass =5.126 × 10¹⁸ [5.086 × 10¹⁸, 5.158 × 10¹⁸] kg {mass}

How close is it?

Trenberth and Smith put the atmosphere's mass at 5.148 × 10¹⁸ kg (Journal of Climate, 2005). The first pass gave 5.27 × 10¹⁸ kg, 2% high. With the land correction it's 5.126 × 10¹⁸ [5.086 × 10¹⁸, 5.158 × 10¹⁸] kg.