Average Separation between molecules in hydrogen at STP (Alonso Finn Problem)

In summary, the conversation discusses the calculation of the number of molecules per 1 ##m^3## of Hydrogen gas using relative density data and Avogadro's number. The result is 2.685 x 10^25 molecules per 1 ##m^3## of Hydrogen gas. The question of the average separation or distance between molecules is raised, and it is noted that this would depend on the model used to arrange the molecules. The simplest model is a cubic lattice, where each molecule occupies a volume of ##D^3##, with D being the distance to the nearest neighbor. It is also mentioned that for stacked spheres, the average distance would
  • #1
agnimusayoti
240
23
Homework Statement
Using relative density data, estimate the average separation between molecules in hydrogen at STP (gas).
Relevant Equations
$$N (molecules) = n (mol) * N_A (molecules/mol)$$
The relative density data:
$${\rho_{H_{2},H_{2}O}}=\frac{\rho_{H_{2}}}{\rho_{H_{2}O}}=8.988\times10^{-5}$$
With avogadro number, thus I can obtain number of molecules per 1 ##m^3## of Hydrogen gas, that is:
$$N = \frac{{{\rho_{H_{2},H_{2}O}}}\times{\rho_{H_{2}O}}}{M_{r}}\times{N_{A}} $$
thus, I get
$$N = 2.685 \times 10^{25} \frac{molecules}{m^{3}}$$

The problem is I can't relate this result with the question which ask the average separation or average distance between the molecules.

Oh, btw, the solution mention 2 different geometries, that is ball and cubic.

Could you help me? Thanks guys!
 
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  • #2
How about converting to e.g. ##\ m^3/\text{molecule} \ ## :smile: ?

##\ ##
 
  • #3
agnimusayoti said:
the average separation between molecules
How do you think that is defined? Taken literally, you would have to sum the distances between all pairs of molecules in the volume. So it must be more in the sense of nearest neighbours. But it seems it's not what that would really be either.
From what I read, you have to pretend a particular model for how the molecules are arranged. The simplest is as a cubic lattice. In that, if the distance to nearest neighbours is D then each molecule effectively occupies a volume ##D^3##.
The "ball" model cannot represent any actual structure since spheres don't pack without gaps, so I assume this means merely that you take each molecule as occupying an octant of volume ##\pi D^3/6## (but that's just my guess).

Out of interest, the real average distance to the nearest neighbour in a random arrangement I calculate to be ##(\frac 3{4\pi\rho})^{\frac 13}\Gamma(\frac 43)\approx 0.55\rho^{-\frac 13}##.
 
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  • #4
What I think about average separation is the average distance to the nearest neighbour of the molecules. But, I can't relate the number of molecules in ##1 m^3## space that is ##2.685\times{10^{25}}## with the distance.
Then, I try to think like you said:
BvU said:
How about converting to e.g. ##\ m^3/\text{molecule} \ ## :smile: ?

##\ ##
and get one molecules occupies ##3.724\times{10^{-26}} m^{3}##.

but, what is the relationship between this result and cubic lattice and either the spherical models.

For example, if there is 2 molecule along 1 m line , then I know that the nearest neighbour separated by 1 m. Now, I use these lines to make a cubic with 1 m side. Therefore, in my 1 ##m^{3}## cubic, I have 8 molecules.

Now, I try to reverse the process of thinking:
suppose 8 molecules occupy ##1 m^{3}## cubic. Therefore, each of the molecule occupies ##0,125 m^3## or 1/8 part of the cubic. If I take cubic root, then I come with 0.5 m. Then, what is that mean?
 
  • #5
agnimusayoti said:
For example, if there is 2 molecule along 1 m line , then I know that the nearest neighbour separated by 1 m. Now, I use these lines to make a cubic with 1 m side. Therefore, in my 1 ##m^{3}## cubic, I have 8 molecules.

Now, I try to reverse the process of thinking:
suppose 8 molecules occupy ##1 m^{3}## cubic. Therefore, each of the molecule occupies ##0,125 m^3## or 1/8 part of the cubic. If I take cubic root, then I come with 0.5 m. Then, what is that mean?

I think my process of thinking relate with your explanation but still not clear yet.

haruspex said:
From what I read, you have to pretend a particular model for how the molecules are arranged. The simplest is as a cubic lattice. In that, if the distance to nearest neighbours is D then each molecule effectively occupies a volume ##D^3##.
 
  • #6
agnimusayoti said:
but, what is the relationship between this result and cubic lattice and either the spherical models
So in a cubic lattice the distance between molecules is this to the power ##\ \frac 1 3## (check with @haruspex post #3 !). For stacked spheres there is a little more distance.

Now, the exercise asks to provide an estimate. Would the average distance be very different if the molecules move like crazy instead of sitting still ?

##\ ##
 
  • #7
agnimusayoti said:
average separation is the average distance to the nearest neighbour
As I tried to explain, that is only meaningful for a given model for the spatial arrangement. Regular arrangements, like a cubic lattice, lead to several molecules at about the same distance from a given molecule. A cubic lattice gives you six. In a real gas, there will often be one quite a bit closer than the rest, leading to a rather smaller average distance.
In the end, it depends what you intend to do with the value when you have found it.
 
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  • #8
@agnimusayoti When having molecules on the corners of a cube, you need to be careful how you count them. You stack these like blocks on each other=at each corner of the cube, you have only 1/8 of a molecule assigned to the cube. The adjacent cubes make up the other 7/8. Thereby each cube has just one molecule=8 corners x 1/8.

Edit: The alternative is to put the molecules at the center of the cubes and stack them. Once again there is one molecule per block, and you will find the molecules (the centers of adjacent cubes) are one meter apart.
 
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  • #9
Charles Link said:
@agnimusayoti When having molecules on the corners of a cube, you need to be careful how you count them. You stack these like blocks on each other=at each corner of the cube, you have only 1/8 of a molecule assigned to the cube. The adjacent cubes make up the other 7/8. Thereby each cube has just one molecule=8 corners x 1/8.

Edit: The alternative is to put the molecules at the center of the cubes and stack them. Once again there is one molecule per block, and you will find the molecules (the centers of adjacent cubes) are one meter apart.
Hmm, could you refer a link or website that supply the image or drawing of this explanation?
 
  • #10
BvU said:
So in a cubic lattice the distance between molecules is this to the power ##\ \frac 1 3## (check with @haruspex post #3 !). For stacked spheres there is a little more distance.

Now, the exercise asks to provide an estimate. Would the average distance be very different if the molecules move like crazy instead of sitting still ?

##\ ##
Yeah, but I came with the different answer (see my #4 post) that is ##(1/8)^{\frac{1}{3}} = 0.5##. Actually, I begin with 1 m apart of molecules.
 
  • #11
agnimusayoti said:
Yeah, but I came with the different answer (see my #4 post) that is ##(1/8)^{\frac{1}{3}} = 0.5##. Actually, I begin with 1 m apart of molecules.
Here's another way to look at it.
Each molecule is at a corner of 8 1m cubes; each 1m cube has 8 molecules at its corners; so over a large region, there must be an average of 1 molecule per 1m cube.
 
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  • #12
agnimusayoti said:
Hmm, could you refer a link or website that supply the image or drawing of this explanation?
The chemistry book by Mahan has this type of diagram if I'm not mistaken. I think I've also seen it in the Solid State Physics book by Kittel. The designation "primitive cell" is used in the description.
 
  • #13
agnimusayoti said:
For example, if there is 2 molecule along 1 m line , then I know that the nearest neighbour separated by 1 m. Now, I use these lines to make a cubic with 1 m side. Therefore, in my 1 ##m^{3}## cubic, I have 8 molecules.

Now, I try to reverse the process of thinking:
suppose 8 molecules occupy ##1 m^{3}## cubic. Therefore, each of the molecule occupies ##0,125 m^3## or 1/8 part of the cubic. If I take cubic root, then I come with 0.5 m. Then, what is that mean?
Your analysis is wrong because you have have placed the molecules at the corners of a cube and ignored each molecule’s ‘territory’ outside the cube.

Try thinking of it this way.

Take a 1m³ cubic box.

Imagine the box partitioned into 2x2x2 = 8 smaller cubes; volume of each is 1m³/8 = 0.125m³.

Put a molecule at the centre of each of the smaller cubes.

You now have 8 molecules in 1m³.

The volume per molecule is V = 0.125m³.

The side-length of each smaller cube is ∛V = ∛(0.125m³). = 0.5m.

The centre-to-centre distance between adjacent smaller cubes is 0.5m. We can therefore take the separation between, molecules as 0.5m.
 
  • #14
Well, it is more confusing because there are 2 different answer depend on how I placed the molecules in the smaller cube.

Which one is true regarding the separation of 8 molecules in 1 ##m^{3}## cube? Is it 0.5 m or 1 m? Or both is true??

From the post it seems 1 m is true the molecules placed in the corner of the smaller cubes. But, if the molecules is in the center of the smaller cubes, the average separation is 0.5 m...
 
  • #15
agnimusayoti said:
it seems 1 m is true the molecules placed in the corner of the smaller cubes
But the size of the cubes is only 0.5 m !

##\ ##
 
  • #16
agnimusayoti said:
Well, it is more confusing because there are 2 different answer depend on how I placed the molecules in the smaller cube.
One of the placement methods is wrong!

agnimusayoti said:
Which one is true regarding the separation of 8 molecules in 1 ##m^{3}## cube? Is it 0.5 m or 1 m? Or both is true??
You need to decide for yourself! See if this helps...

Extend the volume by arranging eight of the 1m-sided cubes into a 2m x 2m x 2m cube.
You now have a 2m-sided cube (volume = 8m³) containing eight 1m-sided cubes.
And each 1m-sided cube contains eight 0.5m -sided cubes.

You have 64 molecules.

Suppose you use your (incorrect) Post #4 method and put one molecule at each corner of every 1m-sided cube.

This will produce some points where there are 8 molecules at exactly the same point (every point where eight of the small cubes meet).

Corection: This will result in 8 molecules at the centre point of the 2m-sided cube. But other points will have different numbers of molecules. For example each corner of the 2m-sided cube will only have one molecule.

You do not have an evenly-spaced arrangement of the molecules - so you can’t use it to find average spacing.

But if you put a molecule at the centre of each 0.5m-sided cube, the 64 molecules are evenly spaced which is what is needed to find the average separation.

You are told there are 64 molecules in a volume of 8m³ and you want the average separation. You would do this:
Volume per molecule = 8m³/64 = 0.125m³
Spacing = ∛V = ∛(0.125m³) = 0.5m

Sorry - I corrected some mistaskes. Hope this didn't cause confusion.
 
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  • #17
I don't know how do I get the average separation in the spherical model.
 
  • #18
Passers_by said:
I don't know how do I get the average separation in the spherical model.
This thread has been inactive for 7+ months.

Static spheres can be packed in different ways. Gas particles have random motions. The average separation depends on what system you are considering.

It might help you if you start a new thread - with a complete and clear description of what you are trying to determine.
 
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Related to Average Separation between molecules in hydrogen at STP (Alonso Finn Problem)

1. What is the average separation between molecules in hydrogen at STP?

The average separation between molecules in hydrogen at STP (standard temperature and pressure) is approximately 2.93 angstroms, or 2.93 x 10^-10 meters. This is the average distance between the centers of two adjacent molecules in a sample of hydrogen gas at a temperature of 0 degrees Celsius and a pressure of 1 atmosphere.

2. How is the average separation between molecules in hydrogen at STP calculated?

The average separation between molecules in hydrogen at STP can be calculated using the ideal gas law, which relates the pressure, volume, temperature, and number of moles of a gas. By rearranging the equation to solve for the average separation, we can use the values for the other variables at STP to calculate the average separation between molecules.

3. Does the average separation between molecules in hydrogen at STP vary with temperature and pressure?

Yes, the average separation between molecules in hydrogen at STP will vary with changes in temperature and pressure. This is because the ideal gas law states that at a constant number of moles, the volume of a gas is inversely proportional to its pressure and directly proportional to its temperature. Therefore, as temperature and pressure change, the average separation between molecules will also change.

4. What is the significance of knowing the average separation between molecules in hydrogen at STP?

Knowing the average separation between molecules in hydrogen at STP is important for understanding the behavior of gases and for various applications in chemistry and physics. It can help determine the density, compressibility, and other properties of gases, and can also be used to predict how gases will behave under different conditions.

5. How does the average separation between molecules in hydrogen at STP compare to other gases?

The average separation between molecules in hydrogen at STP is relatively small compared to other gases. For example, the average separation between molecules in oxygen at STP is approximately 3.46 angstroms, while the average separation in nitrogen is 3.66 angstroms. This is because hydrogen is the smallest and lightest element, with a smaller atomic radius and lower molecular weight compared to oxygen and nitrogen.

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