What is the volume occupied by a molecule?

[Fabrizio Vassallo]

Hello, I'm trying to calculate what is the effective volume of a molecule of N2 in the most precise way possible (I'm a high school student so "in the most precise way possible" is probably not that precise considering my lack of advanced mathematical knowledge). What I want to know is, if I had incredibly good eyes and could see the molecule, as I can see a basketball in my hands, how much would that molecule of 2 atoms of nitrogen occupy? I've attempted this by using the following formulas:

Molar mass of N2: 28.0134 g/mole
1 mol = 6.022x10²³ molecules/mole
Therefore 1 molecule = (28.0134 g/mole) / (6.022x10²³ molecules/mole) = 4.6517x10^-23 grams = 4.6517x10^-26 kg

And then I took the average density of the atmosphere between 0 and 80 km that I calculated using a 1976 Technical Report by NASA, which resulted to be 0.13523 kg/m³ and multiplied that density by the percentage of nitrogen present in the atmosphere, which is 78.084% to get the density of N₂ in the atmosphere, which resulted in 0.10559 kg/m³. I then divided the mass in kg of a single molecule that I calculated before (4.6517x10^-26) by that density of N₂ to get the volume occupied by one molecule of N₂.

I know that this result is wrong, somewhere, because what I'm seeking to know is what is the ratio of occupied space to free space in the atmosphere, and when I calculate the amount of space that the N₂ molecules would occupy in total, it gives me just a bit less than the total volume of the atmosphere between 0 and 80km, and that can't be the case because you still have to fit the +20% of oxygen and other gases, and still should have plenty of free space. I know that if you put a gas in any place it will occupy all available space, but my aim is to work out how much free space there is between each of those molecules that form that gas.

Just to be clear, I'm considering the atmosphere to be at all times comprehended by the space there is between 0 km above the surface and 80 km above the surface. I know there's other ways of defining the atmosphere but this is the one that I chose for this exercise.

Thanks for reading, and if I wasn't very clear in my explanation please let me know and I'll try to rephrase or explain what I'm trying to do better so you can understand it. Thanks again :)

 

[Vagn]

Hello, I'm trying to calculate what is the effective volume of a molecule of N2 in the most precise way possible (I'm a high school student so "in the most precise way possible" is probably not that precise considering my lack of advanced mathematical knowledge). What I want to know is, if I had incredibly good eyes and could see the molecule, as I can see a basketball in my hands, how much would that molecule of 2 atoms of nitrogen occupy? I've attempted this by using the following formulas:

Molar mass of N2: 28.0134 g/mole
1 mol = 6.022x10²³ molecules/mole
Therefore 1 molecule = (28.0134 g/mole) / (6.022x10²³ molecules/mole) = 4.6517x10^-23 grams = 4.6517x10^-26 kg

And then I took the average density of the atmosphere between 0 and 80 km that I calculated using a 1976 Technical Report by NASA, which resulted to be 0.13523 kg/m³ and multiplied that density by the percentage of nitrogen present in the atmosphere, which is 78.084% to get the density of N₂ in the atmosphere, which resulted in 0.10559 kg/m³. I then divided the mass in kg of a single molecule that I calculated before (4.6517x10^-26) by that density of N₂ to get the volume occupied by one molecule of N₂.

I know that this result is wrong, somewhere, because what I'm seeking to know is what is the ratio of occupied space to free space in the atmosphere, and when I calculate the amount of space that the N₂ molecules would occupy in total, it gives me just a bit less than the total volume of the atmosphere between 0 and 80km, and that can't be the case because you still have to fit the +20% of oxygen and other gases, and still should have plenty of free space. I know that if you put a gas in any place it will occupy all available space, but my aim is to work out how much free space there is between each of those molecules that form that gas.

Just to be clear, I'm considering the atmosphere to be at all times comprehended by the space there is between 0 km above the surface and 80 km above the surface. I know there's other ways of defining the atmosphere but this is the one that I chose for this exercise.

Thanks for reading, and if I wasn't very clear in my explanation please let me know and I'll try to rephrase or explain what I'm trying to do better so you can understand it. Thanks again :)

You might want to look up something called the mean-free path (MFP), which is the average distance a molecule or particle travels between collisions.
For air, in typical atmospheric conditions, this turns out to be 65-66 nm.

 

[tech99]

A mole of gas occupies 22.5 litres at standard temp and pressure (Avogadro's Law of Gaseous Volumes).
There are 6.022140857×10^23 molecules in a mole of gas (according to Avogadro).
So I presume a molecule occupies 22.5/6x10^23 litres approx.

 

[Fabrizio Vassallo]

You might want to look up something called the mean-free path (MFP), which is the average distance a molecule or particle travels between collisions.
For air, in typical atmospheric conditions, this turns out to be 65-66 nm.

Hi! Thanks for your answer. I'll look up that concept and see how I can apply it. I've thinking of what I actually calculated with my formulas, and I have the impression that what my formulas calculated is the volume 'allotted' to each molecule, so like if someone else and I were in a huge house of 100 m³ we would have 50 m³ for each other, but that isn't what our bodies would occupy, it'd be the space that we have to move around, while our bodies would only occupy like 0.08m³ and the rest would be free space. I think that the mean-free path thing that you mentioned might help me get rid of that extra space, which is exactly what I want to do.
Thanks again, and if you have any more knowledge that might help me solve this please don't hesitate to comment :D

 

[Fabrizio Vassallo]

A mole of gas occupies 22.5 litres at standard temp and pressure (Avogadro's Law of Gaseous Volumes).
There are 6.022140857×10^23 molecules in a mole of gas (according to Avogadro).
So I presume a molecule occupies 22.5/6x10^23 litres approx.

Hello! I suppose you didn't read my description of the question. My question is how much space a molecule occupies effectively in a gas of varying density, pressure and temperature like the atmosphere (from 0 to 80 thousand meters).

 

[davenn]

if I had incredibly good eyes and could see the molecule, as I can see a basketball in my hands, how much would that molecule of 2 atoms of nitrogen occupy?

well, it's only 2 atoms, so you probably wouldn't see it without the most sophisticated microscope system.
there is a video on the net showing the moving of individual atoms...

... 2 bonded atoms are not going to take up much space/volume, are they ?Dave

 

[Fabrizio Vassallo]

well, it's only 2 atoms, so you probably wouldn't see it without the most sophisticated microscope system.
there is a video on the net showing the moving of individual atoms...

... 2 bonded atoms are not going to take up much space/volume, are they ?Dave

Well I don't know how much volume they might take up, that's why I'm asking. And I'm not implying that I am able to see that or would be able to, I'm imagining how that would look by comparing the N₂ molecule to a basketball so that I can explain where I'm coming from by trying to understand how much space is taken up by that molecule. I know it is incredibly small, but the point of doing this is to see how much space in the gas is actually taken up by the molecule itself and how much is just empty space.

 

[russ_watters]

Hello, I'm trying to calculate what is the effective volume of a molecule of N2 in the most precise way possible (I'm a high school student so "in the most precise way possible" is probably not that precise considering my lack of advanced mathematical knowledge). What I want to know is, if I had incredibly good eyes and could see the molecule, as I can see a basketball in my hands, how much would that molecule of 2 atoms of nitrogen occupy?

I don't like the answers so far, although they are probably driven by the direction of the question...

The defining feature of a gas is that the molecules are not bonded together, so the volume they take up is not fixed. They just bounce around, filling-up whatever volume is available to them. That's not the "volume of a molecule". It would work better for a solid, where the atoms are constrained...

The short answer is that the volume of a molecule is not clear-cut, because the parts of the molecule - the electron in particular - are not fixed in place. The electron "orbit" isn't even really an orbit, but a probability function.

The best I would think you can do is the distance between the nuclei and considering the molecule to be a dumbbell or cylinder. The bond length of nitrogen (center-to-center distance) is 71 angstroms or 7x10^-9m (7 millionths of a milliliter):
https://chem.libretexts.org/Bookshe...ls_of_Chemical_Bonding/Bond_Order_and_Lengths

https://chem.libretexts.org/Ancillary_Materials/Reference/Reference_Tables/Atomic_and_Molecular_Properties/A3:_Covalent_Radii

 

[davenn]

The defining feature of a gas is that the molecules are not bonded together

I didn't say the molecules were bonded together ( if that was a reference to my comment )
Rather the 2 atoms of N would be to make up the N₂ molecule ( singular)

would that not be correct ?D

 

[russ_watters]

I didn't say the molecules were bonded together ( if that was a reference to my comment )
Rather the 2 atoms of N would be to make up the N₂ molecule ( singular)

would that not be correct ?

Sure, but your main answer to the question was "...2 bonded atoms are not going to take up much space/volume..." which is a little vague...

 

[davenn]

Sure, but your main answer to the question was "...2 bonded atoms are not going to take up much space/volume..." which is a little vague...

true ... I would have to dig for that info

 

[Lord Jestocost]

A better way to estimate the apparent volume of a nitrogen molecule is to use the density of liquid nitrogen. Other methods can be found in
https://home.uni-leipzig.de/energy/pdf/freume2.pdf

 

[tech99]

Hello! I suppose you didn't read my description of the question. My question is how much space a molecule occupies effectively in a gas of varying density, pressure and temperature like the atmosphere (from 0 to 80 thousand meters).

If you find the volume occupied by one mole as I have described at standard temp and pressure, then find the volume occupied at any other temp and pressure by using the gas law.