The Exponential Atmosphere: How Pressure and Density Vary with Altitude

[phrygian]

Homework Statement

a. Consider a horizontal slab of air whose thickness (height) is dz. If this slab is at rest, the pressure holding it up from below must balance both the pressure from above and the weight of the slab. Use this fact to find an expression for dP/dz, the variation of pressure with altitude, in terms of the density of air.

b. Use the ideal gas law to write the density of air in terms of pressure, temperature, and the average mass m of the air molecules. Show, then, that the pressure obeys the differential equation:

dP/dz = - P(mg)/(kT)

Homework Equations

The Attempt at a Solution

R= rho

P_(z+dz) + mg - P_(z) = 0

dP = mg

m = R A dz

dp/dz = R A mg



I don't think that that is right for a because I am completely stuck on how to get it to work with b...
thanks for the help

 

[tiny-tim]

Hi phrygian!

(have a rho: ρ )

P_(z+dz) + mg - P_(z) = 0

dP = mg

m = R A dz

hmm … you're getting rather confused …

start again, and specify what size of slice you're dealing with …

that way, you'll not only impress the examiner, you'll also avoid confusing yourself!

in this case, a slice of height dz and area A …

so its mass is … ?

and the forces top and bottom are … ?

 

[D H]

P_(z+dz) + mg - P_(z) = 0

Look at your units. Pressure has units of force/area, m*g has units of force. This equation, as written, doesn't make sense. It is a good idea to always make sure you have consistent units in an expression.