Question

Estimate the total number molecules in the atmosphere of the earth, take the radius of the earth to be 8000 km ?

Answer 1

Usually we take radius of earth as `6,371\ km`

Explanation:

We know that atmosphere of Earth consists of gaseous layers which surround the Earth and is retained by Earth's gravitational pull.

Dry air contains about `78%` nitrogen, `21%` oxygen, `0.93%` argon, `0.04%` of carbon dioxide, and small amounts of other gases, by volume. Air also contains a variable amount of water vapor. We observe that almost `99%` of atmosphere consists of diatomic gases. Molar mass of air is taken as `29.0\ g cdot mol^-1`

To estimate the total number of molecules in the atmosphere, we begin with finding the mass of the atmosphere.

Taking the earth as a sphere of given radius 8000 km, its surface area is given as

`A=4pir^2`
`A=4pi(8000xx1000)^2\ m^2`
`A=8.0425xx10^14\ m^2`

We know that

`1` Atm Pressure = `101,325` Pascals,
`"Pressure" = "Force" / "Area"`
`"Force" = "Pressure"xx "Area"`

`:.` Force due to `1` Atm pressure

`F= (101,325) xx( 8.0425 xx 10^14)`
`=>F= 8.1490 xx 10^19\ N`

Mass of atmosphere `M= F / g`
where `g` is acceleration due to gravity and `=9.81\ ms^-2`

`:.M = (8.1490 xx 10^19) / 9.81 = 8.3068 xx 10^18\ kg`

We know that `1\ mol` of anything contains `6.0221xx10^23` molecules. We get an estimate of number of molecules in the atmosphere as

`( 8.3068 xx 10^18) xx (1/ (29/10^3) )xx6.0221xx10^23`
`= 1.725 xx 10^44 "molecules"`