How many molucles are there in 1 g of air at normal condtions. The radius of the earth is about 6400 km
Answers
There are approximately 1044 molecules in the atmosphere. Let's calculate (cracks knuckles).
We are given no parameters, so we have to use some well known properties of the atmosphere.
What do we know about it? First of all, the pressure of the atmosphere at sea level is 100,000 Pa.
Practically the whole thing consists of molecules of nitrogen and oxygen, with an average molar mass of 0.029 kg/mole.
So if we knew the mass of the atmosphere, we could easily calculate the number of molecules.
How to find the mass of the atmosphere? Its “lower” bound is the radius of earth, equal to 6400 km on average. What is the upper bound of the atmosphere? Modern planes can fly at altitudes of 30 km, that means there is enough air there. But at 200 km, there is practically no air, as the ISS (International Space Station) is stationed at that altitude. The air density decreases exponentially as a function of height. So basically the whole mass (99.99%) of the atmosphere below 70 km, which is like 1% of Earth’s radius.
Now, how does the free fall acceleration change in this range? As we know from Newton’s Law of gravitation, it changes as
g(h)=GM(R+h)2=g(1+h/R)2≈g(1−2hR)
That is, the change in free fall acceleration is only about 2 %. This change is small enough that we can consider it constant for our purpose.
The pressure at sea level is equal to the weight of the atmosphere divided by the surface area of earth.
P=Mg4πR2
The mass of the atmosphere is
M=4PπR2g
Dividing the mass by the average molar mass, we obtain the molar mass. Then we get the number of molecules by multiplying the molar mass by Avogadro’s number:
N=4πR2Pμg
Inserting all the numbers in SI units, we get N≈1044.