Atmospheric Density, exponential decay derivation

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The discussion centers on the derivation of atmospheric density using the equation dP/dz = -gρ, where P represents pressure, z is the distance, g is the acceleration due to gravity, and ρ is the density. Participants confirm that when treating the atmosphere as an isothermal ideal gas with a constant mean molecular mass, the density can be expressed as ρ = ρ0 e^(-z/h), with h representing the scale height. The scale height is identified as approximately 8.5 km, although the exact value is not specified for the problem at hand.

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tony_cruz
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I've read my lecture notes about 100x but can't even begin to see where this derivation can come from. A previous derivation was the equation
dP/dz = -gρ
(P = pressure, z = distance, g= acc due to grav, ρ = density)

If atmosphere can be treated as an isothermal ideal gas of constant mean molecular mass m, show that density drops exponentially with height,
ρ= [ρ0]e^-z/h
where h is a constant*



later i worked through to find it was the scale height ~8.5km but the value is unknown for this question.
 
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