Average Distance Between Neighboring N2 Molecules at NTP

[Mandavi]

Homework Statement

What is the average distance between neighboring molecules if N₂ molecule is assumed to be spherical?(Average volume available to a molecule of N₂ gas at NTP
is 3.72*10^-20 cm^3,which has been calculated by dividing 22.4 litres by Avogadro's number)

Homework Equations

Volume occupied by an ideal gas at NTP is 22.4 litres.
Avogadro's number is 6.022*10^23.

The Attempt at a Solution

Going by the information in the brackets,we have considered that each and every part of the volume is occupied by an N₂ molecule,then how are we supposed to find out the average distance between them?Is there a formula for it?If so,please do explain it along with its conditions.
Thank you.

 

[Borek]

Simple geometry. Imagine that molecules are evenly spread, and each one is inside its own cube...

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[Mandavi]

Simple geometry. Imagine that molecules are evenly spread, and each one is inside its own cube...

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But,even so,the cubes would all be immediately adjacent to each other.How will we find the distance between them?

 

[Ygggdrasil]

Let the gas molecule be at the center of the cube. What is the distance between the centers of the cubes?

 

[Mandavi]

the length of the cube.

 

[Mandavi]

the length of the cube which can be obtained by calculating the cube root of the volume.right?

 

[Ygggdrasil]

Yes. That is correct.

 

[Mandavi]

But,the answer coming this way is different from the answer given!
As per the cube root of volume,answer is coming as 3.33*10^-7 cm,but the answer given is 2.07*10^-7 cm.

 

[Borek]

Answer given is radius of the sphere of a given volume, but it is very wrong. First - if the distance between molecules equals R, volume available to the molecule is not that of a sphere with radius R, as part of this sphere belongs to the other molecule. Second - spheres don't occupy whole available volume, but only about 74% (see http://en.wikipedia.org/wiki/Kepler_conjecture).

Also note, that shape of the molecules is in this case completely meaningless, as the distance between molecules is much larger than the molecule itself, so shape is in no way connected with the volume available to the molecule.

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methods

 

[Mandavi]

Answer given is radius of the sphere of a given volume, but it is very wrong. First - if the distance between molecules equals R, volume available to the molecule is not that of a sphere with radius R, as part of this sphere belongs to the other molecule. Second - spheres don't occupy whole available volume, but only about 74% (see http://en.wikipedia.org/wiki/Kepler_conjecture).

Also note, that shape of the molecules is in this case completely meaningless, as the distance between molecules is much larger than the molecule itself, so shape is in no way connected with the volume available to the molecule.

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Exactly!Thanks a zillion!