How to Calculate the Total Mass of Earth's Atmosphere Using Physics Modelling?

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The discussion focuses on calculating the total mass of Earth's atmosphere using the density formula p = p(o) exp(-r/r(o)), where p(o) is 1.3 kg/m³ and r(o) is 8 km. The user attempts to integrate this expression with respect to volume to find the total mass, considering the Earth's radius (R(earth) = 6.4 x 10^6 m) and the significant difference in scale between the height r and the Earth's radius. The challenge lies in effectively managing the integration and the approximation of mass within the atmosphere.

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Spoony
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so I am tearing my hair out with this one...

the density of the Earth's atmostsphere vaires with height r above the Earth's surface as

p = p(o) exp(-r/r(o))

where p(o) the density of the air at ground level, is 1.3 kg/m^3 and r(o) the scale height is 8km
use this to estimate an approximate value for the total mass of the atmostsphere, given most of the mass lies within a ddistance above the ground that is much smaller than the radius of the Earth

R(earth) = 6.4x10^6

So my attemot at a solution
1) I've got that p = mass/volume so intergrating the expresion with respect to volume gives the total mass of the Earth r = v^(1/3).
then R(earth) >> r
but then i get stuck.

2) I've tried to divide the equation through by V but still that leaves me with
r(earth) >> r.

the problem is deailing with the much greater than part of r(earth) >> r, i don't know how to approach it.

thanks guys :)
 
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No one even got a clue on how to solve this?
 
What is the mass of the air that lies between a height r and a height r+dr?
 

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