# Mass, Density and Volume of atmosphere

• Any Help
In summary, the problem involves calculating the mass of 15 km of the Earth's atmosphere above a 1m2 area at sea level. The density of the atmosphere can be approximated by an exponential equation, ρ(h)=ρ0e^(-h/h0), where ρ0 = 1.3 kg/m3 and h0 = 8.2 km. The attempts at solving the problem involved substituting the height in the equation and using the expanded equation to integrate and calculate the mass. However, both attempts were incorrect due to errors in defining and integrating for the incremental volume. The correct solution involves using the formula for incremental mass, dM = Aρ(h)dh, where A is the area,
Any Help

## Homework Statement

:
[/B]
The density of the Earth's atmosphere varies with altitude, and can be approximated by an exponential: ρ(h)=ρ0e^(-h/h0) where ρ0 = 1.3 kg/m3 (the approximate density at sea level) and h0 = 8.2 km (this is determined empirically). Calculate the mass of 15 km of the atmosphere above a 1m2 area at sea level.

## Homework Equations

Mass= density. volume volume= area . hight

## The Attempt at a Solution

I tried to solve it in two different ways but both gives me incorrect answer, The correct answer is 8900kg
My First attempt
: I substitute the height in the equation then i got p(h)=1.3xe^(-15/8.2) p(h)=0.19936 kg/m^3
Mass= p(h) x volume =0.19936x area x height = 0.19936x1x15x10^3=2990.4kg but the answer is not correct?!
My second attempt: I expand the equation to get mass=V.p(h)=h.A.p0.e^(-h/h0) then integrate it to get mass= [h^2/2 . A.p0.e^(-h/h0)+ h.p0/h0 .e^(-h/h0)] from h=0 to h=15000
mass= 23478038kg also not correct!

Any Help said:
My second attempt: I expand the equation to get mass=V.p(h)=h.A.p0.e^(-h/h0) then integrate it to get mass= [h^2/2 . A.p0.e^(-h/h0)+ h.p0/h0 .e^(-h/h0)] from h=0 to h=15000
mass= 23478038kg also not correct!
It would be easier to follow your work if you defined all your variable names before using them.
So you have mass = V.p(h). V being the volume and p(h) being the density at the current height h.

You proceed to equate V to h.A where A is the surface area under a column of height h. But that's wrong. If you are integrating, the total height h does not contribute to the volume of an incremental element. Only the incremental height ##dh## contributes. You need to be adding up the masses of incremental volumes in the column.

Keeping track of units could have shown you that the result was dimensionally inconsistent.

jbriggs444 said:
It would be easier to follow your work if you defined all your variable names before using them.
So you have mass = V.p(h). V being the volume and p(h) being the density at the current height h.

You proceed to equate V to h.A where A is the surface area under a column of height A. But that's wrong. If you are integrating, the total height h does not contribute to the volume of an incremental element. Only the incremental height ##dh## contributes. You need to be are adding up the masses of incremental volumes in the column.

Keeping track of units could have shown you that the result was dimensionally inconsistent.
you mean that i should not integrate ? but in my first attempt i didn't integrate and the answer was also incorrect

Any Help said:
you mean that i should not integrate ? but in my first attempt i didn't integrate and the answer was also incorrect
No, I mean that you need to be careful what you integrate.

When you integrate, you are essentially adding up the masses of a whole bunch of incremental volumes starting at the base of the column and working your way to the top. What is the formula for the mass of an incremental volume of height ##dh## at altitude ##h## with area ##A## and density ##\rho(h)##?

Last edited:
jbriggs444 said:
No, I mean that you need to be careful what you integrate.

When you integrate, you are essentially adding up the masses of a whole bunch of incremental volumes starting at the base of the column and working your way to the top. What is the formula for the mass of an incremental volume of height ##dh## at altitude ##h## with area ##A## and density ##p(h)##?
mass=area.p0.e^(-dh/h0)dh right?

Any Help said:
mass=area.p0.e^(-dh/h0)dh right?
No, that is not it.

Try it one step at a time. Do not substitute for the density yet. What is the formula for the incremental mass in terms of ##dh##, ##h##, ##A## and ##\rho(h)##?

Last edited:
jbriggs444 said:
Try it one step at a time. Do not substitute for the density yet. What is the formula for the incremental volume in terms of dhdhdh, hhh, AAA and ρ(h)ρ(h)\rho(h)?
the formula for volume for each dh is Area.dh and that for the density p(h)=p0.e^(-dh/h0)
right till now?

Any Help said:
the formula for volume for each dh is Area.dh and that for the density p(h)=p0.e^(-dh/h0)
right till now?
No. That is not the formula for density.

Any Help
jbriggs444 said:
No. That is not the formula for density.
p(h)= -p(0)/h0 .e^(-dh/h0) ??

Any Help said:
p(h)= -p(0)/h0 .e^(-dh/h0) ??
By now you have seen and liked the complete solution that was posted by another helper and subsequently removed.

The problem with the above formula for density is that it is not a function of h. "h" and "dh" are different things.

Any Help
jbriggs444 said:
By now you have seen and liked the complete solution that was posted by another helper and subsequently removed.

The problem with the above formula for density is that it is not a function of h. "h" and "dh" are different things.
yeah i put h in the equation instead of dh and that what made all my wrong going wrong

## 1. What is the mass of the Earth's atmosphere?

The mass of the Earth's atmosphere is approximately 5.1480 x 10^18 kilograms.

## 2. How is the density of the atmosphere calculated?

The density of the atmosphere is calculated by dividing the mass of the atmosphere by its volume. This can also be expressed as the mass of air per unit volume.

## 3. What factors affect the density of the atmosphere?

The density of the atmosphere is affected by temperature, pressure, and humidity. As temperature increases, the density decreases. As pressure increases, the density also increases. And as humidity increases, the density decreases.

## 4. What is the relationship between mass and volume in the atmosphere?

The mass and volume of the atmosphere are directly proportional. This means that as the mass of the atmosphere increases, so does its volume. Similarly, as the mass decreases, the volume also decreases.

## 5. How does the density of the atmosphere change with altitude?

The density of the atmosphere decreases as altitude increases. This is because the air becomes less compressed at higher altitudes, resulting in a decrease in mass per unit volume.

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