Atmospheric Density, exponential decay derivation

[tony_cruz]

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.

 

[tony_cruz]

bump 2pm deadline ...

 

[ogb p]

The derivation of the exponential decay of atmospheric density with height comes from the ideal gas law and the assumption of isothermal conditions. The ideal gas law states that the pressure of a gas is directly proportional to its density and temperature, and inversely proportional to its volume. In a constant temperature environment, this can be written as P = ρRT, where ρ is the density, R is the gas constant, and T is the temperature.

Using this equation and the previous derivation of dP/dz = -gρ, we can rearrange to solve for ρ and substitute it into the ideal gas law. This gives us ρ = (P/RT) and plugging this into the previous equation gives us:

dP/dz = -g(P/RT)

We can then rearrange this equation to isolate P and integrate both sides with respect to z:

∫dP/P = -g/R∫dz

ln(P) = -gz/R + C

Where C is a constant of integration. We can then exponentiate both sides to eliminate the natural log:

P = e^(-gz/R) * e^C

Since e^C is just another constant, we can combine it with the constant of integration to get a new constant, A. This gives us the final equation of:

P = Ae^(-gz/R)

Since density is directly proportional to pressure, we can substitute ρ for P and get:

ρ = Ae^(-gz/R)

This is the exponential decay equation for atmospheric density with height, where h = R/g is the scale height. This means that for every increase of h in height, the density decreases by a factor of e. This derivation assumes that the atmosphere is an isothermal ideal gas, meaning that the temperature remains constant throughout the atmosphere. However, in reality, the temperature does change with height and this can affect the accuracy of this equation. But for a rough estimate, this derivation provides a good understanding of the exponential decay of atmospheric density.