Pressure At Surface And Scale Height

[bobbles22]

Please can someone tell me if my thinking here is right...

I've got a planet with an atmospheric pressure at 6km of 0.5 P₀ and at 8km of 0.4 P₀ (P₀ = pressure at the surface).

I want to work out the scale height of the atmosphere.

Given scale height = λ

and for height above surface = z

P(z)=P(0)e^(-z/λ)

I could rearrange to show the pressure at the surface as:

P(0)=P(z)/e^(-z/λ)

I could then use the relative pressure, assume P(0)=1 (as it will cancel out shortly) and height from each of the know quantities and set them equal to each other like this:

0.4/e^(-8000/λ) = 0.5/e^(-6000/λ)

A little mutliplication...

0.4 e^(-6000/λ) = 0.5 e^(-8000/λ)

Take the Log of both sides...

(-6000/λ) log 0.4 = (-8000/λ) log 0.5

But know I'm left with the λ cancelling out if I multiply both sides by λ. I'm sure I've gone wrong here somewhere. Probably something very simple. Can anyone advise? Have I made a simple mistake in my working or have I gone completely off the reservation and need to start again? I just need to end up with λ = xxx metres.

Thank you.

 

[tiny-tim]

hi bobbles22!

0.4 e^(-6000/λ) = 0.5 e^(-8000/λ)

Take the Log of both sides...

(-6000/λ) log 0.4 = (-8000/λ) log 0.5

nooo …

(-6000/λ) + log 0.4 = (-8000/λ) + log 0.5 ​