Ideal gas law concerning the exponential atmosphere

In summary, the problem is to find an expression for dP/dz, the variation of pressure with altitude, for a horizontal slab of air at rest. This can be done by considering the forces acting on the slab, including the weight and pressures from above and below. The air density will appear in the final expression due to the slab's mass depending on it. The volume of the slab can be expressed as V=A*dz and the mass as m=rho*V. This leads to the equation dP=P-P(0)=rho*g*dz and therefore, rho*g=dP/dz.
  • #1
pentazoid
146
0

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 Pf=P0+rho*g*h. Not sure what else to do.
 
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  • #2
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.
 
  • #3
Delphi51 said:
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
 
  • #4
Yes, you will need an expression for the volume. No numbers of course, just letters.
 
  • #5
Delphi51 said:
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= [tex]\rho[/tex]air*V. How will this relate to an expression for dP/dz?
 
  • #6
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?
 

1. What is the ideal gas law?

The ideal gas law is a mathematical equation that describes the relationship between the pressure, volume, temperature, and number of moles of an ideal gas. It is represented by the formula PV = nRT, where P is the pressure, V is the volume, n is the number of moles, R is the gas constant, and T is the temperature in Kelvin.

2. How does the ideal gas law relate to the exponential atmosphere?

The ideal gas law can be used to describe the behavior of an ideal gas in a constant pressure system, such as the Earth's atmosphere. As the altitude increases, the pressure decreases exponentially, and the ideal gas law can be used to calculate the change in pressure based on the change in altitude.

3. What are the assumptions made in the ideal gas law?

The ideal gas law assumes that the gas is in a state of thermodynamic equilibrium, the gas particles have no volume and do not interact with each other, and all collisions between particles are perfectly elastic. It also assumes that the gas is at a low enough density that intermolecular forces can be ignored.

4. How is the ideal gas law helpful in studying the atmosphere?

The ideal gas law is helpful in studying the atmosphere because it provides a simple yet accurate way to calculate the behavior of an ideal gas at different altitudes. It can be used to predict changes in pressure, volume, and temperature as well as determine the density of the gas. This information is important in understanding the properties and behavior of the Earth's atmosphere.

5. Are there any limitations to using the ideal gas law in studying the atmosphere?

While the ideal gas law is a useful tool in studying the atmosphere, there are some limitations to its application. It assumes that the gas is ideal, meaning that the gas particles have no volume and do not interact with each other, which is not always the case in the real atmosphere. Additionally, the ideal gas law does not take into account the effects of gravity, which can be significant at high altitudes.

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