Mean distance between air molecules

[songoku]

Homework Statement

Which of the following is closest to the mean distance between air molecules at room temperature and pressure (298 K and 101325 Pa)? Assume air is made of 79% N2 and 21% O2 by moles.
A 0.4 nm
B 4 nm
C 40 nm
D 400 nm
E 4 μm

Homework Equations

PV = nRT

The Attempt at a Solution

My idea is to find the volume and take the cube root of volume to get the mean distance between air molecules. However, I don't know how to find the n (number of moles)

 

[mfb]

What is the volume of 1 mole at room temperature/pressure?

You don't need V and n separately: The answer does not depend on the size of the room. Only their ratio is relevant.

 

[songoku]

What is the volume of 1 mole at room temperature/pressure?

24 dm³?

You don't need V and n separately: The answer does not depend on the size of the room. Only their ratio is relevant.

I don't get this hint. I need to find the ratio of volume and number of moles? What is the relation of the ratio to the distance between molecules?

 

[mfb]

You know how many molecules are in one mole of gas, and you know the volume this mole occupies. How much volume is there per molecule?

The distances are not fixed, of course, but you can get something like a typical distance if you imagine all molecules arranged in a regular pattern, e.g. each molecule gets its own cube. That relates a volume to a distance.

 

[songoku]

You know how many molecules are in one mole of gas, and you know the volume this mole occupies. How much volume is there per molecule?

Volume per molecule = 24 x 10^-3 / (6.02 x 10²³)

The distances are not fixed, of course, but you can get something like a typical distance if you imagine all molecules arranged in a regular pattern, e.g. each molecule gets its own cube. That relates a volume to a distance.

I can image a molecule (sphere) located inside a cube with the sphere touches the inner of the cube but I still don't know how it relates to the mean distance between molecule.

What should I find actually to get the mean distance?

Thanks

 

[mfb]

Imagine a room filled with 1m x 1m x 1m cubes. Put an atom in the center of each cube. You now have 1 atom per cubic meter. The shortest distance between adjacent atoms is 1 meter.
1 cubic meter per atom leads to a distance of 1 meter between atoms.

How can you apply the same approach to your (smaller) volume per atom?

 

[songoku]

Imagine a room filled with 1m x 1m x 1m cubes. Put an atom in the center of each cube. You now have 1 atom per cubic meter. The shortest distance between adjacent atoms is 1 meter.
1 cubic meter per atom leads to a distance of 1 meter between atoms.

How can you apply the same approach to your (smaller) volume per atom?

Ah I see. I just need to take the cube root of 24 x 10^-3 / 6.02 x 10²³ and it is the answer.

So the information about percentage of nitrogen and oxygen doesn't matter at all?

Thanks

 

[haruspex]

It's not quite clear what "mean distance between molecules" means. It sounds more like mean distance to nearest neighbour than the distance you would get in a cubic lattice. But that gives a rather smaller answer, a bit under 2nm. See https://en.m.wikipedia.org/wiki/Mean_inter-particle_distance#Ideal_gas

 

[mfb]

So the information about percentage of nitrogen and oxygen doesn't matter at all?

It matters if you start with the mean density of air, then you need the average mass per molecule.

@haruspex: Atoms are not arranged in a cubic lattice of course, but the question asks for a rough estimate, where we don't care about prefactors close to 1.

 

[haruspex]

Atoms are not arranged in a cubic lattice of course, but the question asks for a rough estimate, where we don't care about prefactors close to 1

Yes, I appreciate that only a rough estimate is needed to answer the question. Just clarifying to the OP that the real answer is a bit different.

 

[songoku]

Sorry for late reply

Thank you very much