## Pressure At Surface And Scale Height

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.

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