Permanent Dipole - Permanent Dipole Interaction Derivation

In summary, the conversation discusses the derivation of the dipole-dipole interaction equation, which involves two dipole moments (p1 and p2), a distance between them (r), and angles measured relative to the z axis (θ1 and θ2) and the y axis (ζ). The first step in the derivation involves finding the electric field (E1) of p1 in spherical coordinates. The conversation then moves on to discuss how to formulate p2 in spherical coordinates and how to find the dot product of two vectors in spherical coordinates.
  • #1
RickD

Homework Statement


I am trying to derive the dipole-dipole interaction derivation, which is:

U=(-p1p2/4πϵ_0) (1/z^3) ((2cosθ_1cosθ_2)− (sinθ_1sinθ_2cosζ))

Where p1 and p21 are the two dipole moments, r is the distance between two dipoles on the y axis, θ_1 and θ_2 are the angles between the z axis and dipoles, and ζ is the dihedral angle

I am using U = -p2 * E_1 to derive the equation, where E_1 is the electric field of p1 and * is the dot product

So far, I have deduced E-1 in spherical coordinates as:
E_1= (1/4πϵ_0)(1/r^3) (2cosθ r + sinθ θ)
Where the bolded r and θ are the unit vectors in spherical coordinates.

Now from this, I have problems:
  1. How do I formulate p2 in spherical coordinates
  2. How do I dot product two vectors of spherical coordinates

Homework Equations


See above

The Attempt at a Solution


I have tried to convert E_1 into Cartesian, but it won't solve it if I don't know how to formulate p2
 
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  • #2
RickD said:

Homework Statement


I am trying to derive the dipole-dipole interaction derivation, which is:

U=(-p1p2/4πϵ_0) (1/z^3) ((2cosθ_1cosθ_2)− (sinθ_1sinθ_2cosζ))

Where p1 and p21 are the two dipole moments, r is the distance between two dipoles on the y axis, θ_1 and θ_2 are the angles between the z axis and dipoles, and ζ is the dihedral angle.
Did you mean to say that θ1 and θ2 are angles measured relative to the y axis?

You can rotate your xyz axes about the y-axis so that p1 lies in the yz plane.
upload_2017-7-4_14-10-37.png


So far, I have deduced E-1 in spherical coordinates as:
E_1= (1/4πϵ_0)(1/r^3) (2cosθ r + sinθ θ)
Where the bolded r and θ are the unit vectors in spherical coordinates.
Can you draw the vectors r and θ at the location of p2 in the above diagram?

Then try to find expressions for the r and θ components of p2.
 

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Related to Permanent Dipole - Permanent Dipole Interaction Derivation

What is a permanent dipole - permanent dipole interaction?

A permanent dipole - permanent dipole interaction is a type of intermolecular force between two polar molecules. It occurs when the permanent dipole moments of the molecules interact with each other, causing them to attract or repel each other.

How is the permanent dipole moment of a molecule determined?

The permanent dipole moment of a molecule is determined by the distribution of its electrons and the overall shape of the molecule. It is a vector quantity, with both magnitude and direction, and is measured in units of Debye (D).

What is the significance of the angle between two permanent dipole moments?

The angle between two permanent dipole moments is important in determining the strength of the interaction between two polar molecules. The closer the angle is to 0 or 180 degrees, the stronger the interaction will be, while larger angles will result in weaker interactions.

How does temperature affect permanent dipole - permanent dipole interactions?

As temperature increases, the kinetic energy of molecules also increases, causing them to move and rotate more rapidly. This can weaken the strength of permanent dipole - permanent dipole interactions, as the molecules are less likely to remain in close proximity to each other.

What other factors can influence permanent dipole - permanent dipole interactions?

The strength of permanent dipole - permanent dipole interactions can also be influenced by the size and shape of the molecules, as well as the distance between them. Additionally, the presence of other intermolecular forces, such as hydrogen bonding, can also affect the overall strength of the interaction.

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