# Gas Of Dipoles

1. Aug 19, 2007

### Schrodu

Suppose we have a gas of randomly oriented (and in random motion) electric dipoles. Obviously the dipoles do not behave as free particles. How do we describe it's properties? Can we define potential etc. ?
I am looking for a solution without the large volume approximation. Is it possible to get an expression for the mean kinetic energy etc. of a partical given its total energy?

2. Aug 19, 2007

### pardesi

will get some answers here shreyas.
though i strictly beleive this isseroius mixture of eveidence and thermodynamic arguements

3. Aug 19, 2007

### Staff: Mentor

4. Aug 19, 2007

### quetzalcoatl9

you are talking about a truly many-body problem, where you must (usually) include the induced dipole-induced dipole interactions. if they don't matter, then why bother?

there are ways of doing this numerically, but they are difficult. you can start with an analytic expression by considering a dipole, mu=dq, interacting with a point charge, you'll then get

$$\nabla_\alpha \nabla_\beta \frac{1}{r}$$

as the induced field contribution, which needs to be solved for numerically for anything but the simplest of systems. thats why most molecular simulation techniques ignore induced dipoles.

Last edited: Aug 19, 2007
5. Aug 20, 2007

### Schrodu

Could you clarify that a bit? I am not used to the standard notations. In my original attempt, I calculated the potential energy of two interacting dipoles in terms of their spacing and orientation. I want to average this out in some way (integrating over the angle obviosly gives net potential energy 0)

Thanks for the help.

6. Aug 20, 2007

### Sojourner01

If you're looking for a simple solution, the only viable one is to treat it like a semi-ideal gas. The dipoles have translational, rotational and vibrational degrees of freedom, each of which is 1/2 kT per molecule in this simple approximation.

7. Aug 20, 2007

### quetzalcoatl9

the potential energy of interaction between a dipole and a point charge (if you draw out the two charges seperated) that is far away is $$\nabla_a (\frac{1}{r})$$

since $$E = -\nabla V$$ then the dipoles contribution to the induced field is the expression i gave.

do a scholar google search for "molecular polarization" if you are more interested. there are review articles out there that summarize the field. also, Jackson's E&M book may interest you.

8. Sep 4, 2007

### Schrodu

I found in this paper(page no. 4)

Shouldn't there be a term to take care of direction, $$U=\frac{\mu_r^2\cos\alpha}{4\pi \epsilon_0}$$?, $$\alpha$$ is angle between field and the axis

9. Sep 4, 2007

### genneth

You can also consider a mean field approximation -- that will allow you to calculate the potential energy due to the orientation of the dipoles. In fact, with that approximation, I guess you'd get a hybrid free-gas / Curie ferromagnet effect, with a phase change due to the orientation. As far as dipole-dipole forces beyond merely torque, perhaps use the Van de Waals approximation?