Ideal gas law concerning the exponential atmosphere

[pentazoid]

Homework Statement

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 the altitudem in terms of the density of air

Homework Equations

PV=NkT
PV=nRT

The Attempt at a Solution

I don't know how to begin this problem. I guess the first thing I should do is determined the forces acting on the slab, which I guess would be the weight of the slab and the pressure above. Fnet=Pressure-mg=0 => P=mg. The slab also has some potential energy since it moves at a height of dz. not sure what to do with the air density term but I thin P_f=P₀+rho*g*h. Not sure what else to do.

 

[Delphi51]

I'm thinking of a slab of air with area A on top and on the bottom. Pressure on the bottom is P, on the top P + dP.

There are THREE Forces on this - the mg you mention plus a force due to the pressure on the top and another due to the pressure on the bottom. Pressure is not a force - it is defined as force per unit area so you must include the A.

The air density will simply appear in the answer because the mass of the slab depends on it.

 

[pentazoid]

I'm thinking of a slab of air with area A on top and on the bottom. Pressure on the bottom is P, on the top P + dP.

There are THREE Forces on this - the mg you mention plus a force due to the pressure on the top and another due to the pressure on the bottom. Pressure is not a force - it is defined as force per unit area so you must include the A.

The air density will simply appear in the answer because the mass of the slab depends on it.

should I calculated the volume of the slab in order to calculate the mass

 

[Delphi51]

Yes, you will need an expression for the volume. No numbers of course, just letters.

 

[pentazoid]

Yes, you will need an expression for the volume. No numbers of course, just letters.

would the volume be V=A*dz and now the mass therefore is m= $$\rho$$_air*V. How will this relate to an expression for dP/dz?

 

[pentazoid]

Scroeder Textbook doesn't give you the answer to the problems like most physics texts should. Anyway , I think I have the correct answer: PA-P(0)A=mg, m=rho*V, V=A*dz

A's go away and I am left with this expression: P-P(0)=rho*g*dz. dP=P-P(0) and there fore , rho*g=dP/dz correct?