Geometric Interpretation of VSEPR Theory

In summary, a geometric model for VESPR theory states that valence electron pairs are mutually repulsive, and therefore adopt a position which minimizes this, which is the position at which they are farthest apart, still in their orbitals.
  • #1
Neolux
5
0
I am making a Geometric model for VESPR theory, which states that valence electron pairs are mutually repulsive, and therefore adopt a position which minimizes this, which is the position at which they are farthest apart, still in their orbitals.

For example, the 2 electron pairs on either side of the central atom of Carbon Dioxide both repel each other equally (Dot Diagram Below), giving the molecule a linear arrangement (Shown Below):

Dot Diagram of Carbon Dioxide:

170px-Carbon-dioxide-2D-dimensions.svg.png


Linear Arrangement:

100px-Linear-3D-balls.png


Other molecules like water molecules have 4 pairs (2 bonded pairs, 2 unbonded pairs), which all repel each other, giving it a bent configuration.

Dot Diagram of Water:

lewis%2Bdot%2Bdiagram%2B2.png


Bent Arrangement:

100px-Bent-3D-balls.png


I figured it'd be pretty simple and straightforward to create a geometric interpretation of this. This has probably already been done, but just doing it for fun.

I figured I could accomplish this by creating a 3-D space, plotting the pairs as points, and then using the 3-D distance formula, find the arrangement at which all the pairs are at the greatest distance (Equal distance) from each other. I could then also calculate the angle of the bonds by calculating the angle between those lines.

All the points would have to be an equal distance from the nucleus (origin), which can be set to an arbitrary distance. Let's just say 1.

Here is what I have so far (For a simple 2 pair configuration):
__________________________________________________________________________________
E1 = Electron Pair 1

E2 = Electron Pair 2

Origin = Nucleus

Distance from E1 to Nucleus = Distance from E2 to Nucleus:

√(E1x-0)^2 + (E1y-0)^2 + (E1z-0)^2 = √(E2x-0)^2 + (E2y-0)^2 + (E2z-0)^2


Distance from E1 to E2 (Needs to be maximized):

√(E1x-E2x)^2 + (E1y-E2y)^2 + (E1z-E2z)^2

__________________________________________________________________________________

Now how could I calculate the coordinates of E1x, E1y, E1z, E2x, E2y, and E2z such that the distance between them is a maximum, yet they are the same distance from the origin? I figured i'd need differential geometry for that bit.

Once I find the coordinates for each pair, I figured I could then find the angle between them using this:
__________________________________________________________________________________

Θ = tan^-1(m1-m2/1+m1m2)

__________________________________________________________________________________

Where m1 and m2 are the slopes of the lines from each point to the origin.

Any help or comments would be greatly appreciated.

Thanks in advance.
 
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  • #2
I'm sorry you are not generating any responses at the moment. Is there any additional information you can share with us? Any new findings?
 
  • #3
Greg Bernhardt said:
I'm sorry you are not generating any responses at the moment. Is there any additional information you can share with us? Any new findings?

Unfortunately not... I thought this would be pretty a simple problem for someone who knows Differential Geometry. Isn't maximizing distances something that is done all the time?
 
  • #4
With your current approach, there is no way to determine the orientation of your coordinate system. To remove this issue you need to define one full set of coordinates and one element of the other. I would place E1 on the positive x-axis (1,0,0) and eliminate the z-coordinate from the problem by setting E2z = 0. Now you should be able to solve as a system to get two solutions, one for a positive bend and one for a negative one (or a double root if they are opposite). That should also make it easier to calculate slopes as you are already working in an effectively 2d plane. It gets trickier though if you add additional pairs.

You might consider approaching this problem with vectors instead of just coordinates. If the pairs are spread evenly about the nucleus, unit vectors drawn to each pair should all sum to 0, which gives you some nice equations to work with:
E1x+E2x+...+Ekx=0
E1y+E2y+...+Eky=0
E1z+E2z+...+Ekz=0
I haven't checked if this helps at all but I thought it was worth mentioning.
 
  • #5


I would like to commend you for your efforts in creating a geometric model for VSEPR theory. It is always beneficial to have a visual representation of scientific concepts.

Your approach of using 3-D space and the distance formula to find the optimal arrangement of electron pairs is a sound one. However, I would suggest that instead of using an arbitrary distance for the origin, you could use the actual bond length between the central atom and the bonded atom in the molecule you are studying. This would make your model more accurate and applicable to different molecules.

In terms of finding the coordinates for each pair, you are correct in thinking that this would require differential geometry. This is because the electron pairs are not fixed in space, but rather they are constantly moving and interacting with each other. Therefore, the coordinates would need to be calculated dynamically, taking into account the repulsive forces between the electron pairs.

Additionally, I would like to point out that while your model may work well for simple molecules with only two electron pairs, it may become more complex and difficult to visualize for molecules with multiple electron pairs, such as water. In such cases, it may be more useful to use visualization software or molecular modeling kits to create accurate representations.

Overall, your approach to create a geometric interpretation of VSEPR theory is commendable. However, it is important to remember that this is a simplified model and there may be other factors at play in the actual arrangement of electron pairs in a molecule. VSEPR theory is a useful tool, but it is not the only factor that determines the shape of a molecule. Keep exploring and experimenting, and you may discover new insights and applications for your model.
 

1. What is VSEPR theory and how does it relate to geometry?

VSEPR stands for Valence Shell Electron Pair Repulsion and it is a theory used to predict the shape of molecules based on the repulsion between electron pairs in the valence shell of an atom. This theory is closely related to geometry because it explains how the arrangement of atoms and lone pairs of electrons around a central atom determines the overall shape of a molecule.

2. How is the geometry of a molecule determined using VSEPR theory?

The geometry of a molecule is determined by the number of bonding and non-bonding electron pairs around the central atom. These electron pairs repel each other and adopt a spatial arrangement that minimizes this repulsion, resulting in a specific geometric shape. The number of bonding and non-bonding electron pairs can be determined by the Lewis structure of the molecule.

3. What is a molecular geometry and how is it different from electron geometry?

Molecular geometry refers to the actual shape of a molecule, taking into account the positions of all atoms and lone pairs of electrons. On the other hand, electron geometry only considers the positions of bonding and non-bonding electron pairs around the central atom. This means that in some cases, the molecular geometry may differ from the electron geometry due to the presence of lone pairs.

4. How do different types of electron pairs affect the geometry of a molecule?

There are two types of electron pairs: bonding and non-bonding. Bonding pairs are the electrons involved in covalent bonds between atoms, while non-bonding pairs are the lone pairs of electrons that do not participate in bonding. Non-bonding pairs exert a stronger repulsion than bonding pairs, leading to a distortion in the geometry of the molecule. This is why the presence of lone pairs can significantly affect the overall shape of a molecule.

5. What are the limitations of VSEPR theory?

VSEPR theory is a simple and useful tool for predicting molecular geometry, but it has some limitations. It does not take into account the actual sizes of atoms and lone pairs, as well as the presence of multiple bonds. This can result in slight inaccuracies in the predicted geometry. Additionally, VSEPR theory is based on the assumption that the repulsion between electron pairs is the only factor influencing molecular shape, but in reality, there are other factors such as bond angle distortions and hybridization that can also affect the geometry of a molecule.

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