Gas with interacting molecules (from goldstein)

[gulsen]

Homework Statement

(from Goldstein, problem 3.12)

Suppose that there are long-range interactions between atoms in a gas in the form of central forces derivable from potential
$$U(r) = \frac{k}{r^m}$$,
where $r$ is the distance between any pair of atoms and $m$ is a positive integer. Assume further that relative to any given atom the other atoms are distributed in space such that their volume density is given by Boltzmann factor:
$$\rho(r) = \frac{N}{V} e^{-U(r) / kT}$$,
where $N$ is the total number of atoms in a volume $V$. Find the addition to the virial of Clausius resulting from these forces between pairs of atoms, and compute the resulting correction to Boyle's law. Take $N$ so large that sums may be replaced by integrals. While closed results can be found for and positive $m$, if desired, the mathematics can be simplified by taking $m = +1$.

Homework Equations

$$\bar T = \frac{1}{2} \bar{\sum_i \nabla V \cdot \mathbf r_i}$$ (average over time)
which is
$$\bar T = -\frac{1}{2} \bar{\sum_i \mathbf F_i \cdot \mathbf r_i}$$
(RHS is virial of Clausius)

The Attempt at a Solution

The force $\mathbf F_i$ acting on a particle has contributions from all other particles in the form of $\mathbf F_i = \sum_j \left(-m k/|\mathbf r_i - \mathbf r_j|^{m+1} \right) \left( \frac{\mathbf r_i - \mathbf r_j}{|\mathbf r_i - \mathbf r_j|} \right)$, which is supposed to be written as an integral...and one should consider that the particles obey a particular distribution. Then comes the second summation of $\sum_i \mathbf F_i \cdot \mathbf r_i$. But I can't even manage to compute the first sum...

I'd greatly appreciate even slightest piece of hint...

EDIT: I have some clues for $m=1$ case, which obeys the Gauss law, though. First, since the system is spherically symmteric, the force should be directed along the position vector. I'm actually considering the force acting on a piece of charge located at $\mathbf r_i$ within a charged sphere. By the Gauss law, the particles outside of $r > r_i$ do not contribute to the force, and the force is proportional to the charge within the sphere $r < r_i$ and inversely proportional with $r_i ^ 2$. So I can say

$$F_i = C \frac{ \int_0^{r_i} \rho(r) 4 \pi r^2 dr } {r_i^2}$$

and

$$\bar T = -\frac{1}{2} \int_0^R F(r) r (4 \pi r^2 dr \rho(r))$$

Does it make any sense? Though I still don't have any clue for other values of $m$...

 

[Jhero]

Hello there,

Thank you for your post and for summarizing the problem at hand. I can see that you have made some progress towards finding a solution for the m=1 case, which is a good start. However, I think it would be more helpful if I provide some general guidelines and hints rather than directly giving you the solution. This will allow you to work through the problem and understand it better.

Firstly, let's consider the force acting on a single particle i due to all other particles j in the system. As you correctly stated, this can be written as:

F_i = \sum_j \left(-m k/|\mathbf r_i - \mathbf r_j|^{m+1} \right) \left( \frac{\mathbf r_i - \mathbf r_j}{|\mathbf r_i - \mathbf r_j|} \right)

Notice that in this expression, we have a sum over all particles j in the system. This means that we can write this force as an integral over the entire volume V of the system. This is because we can think of the summation as taking into account the contributions from all particles in the system, which can be approximated by an integral over the volume V.

Next, let's consider the distribution of particles in the system. We are given that the volume density of particles is given by the Boltzmann factor:

\rho(r) = \frac{N}{V} e^{-U(r) / kT}

This tells us that the number of particles in a small volume dV at a distance r from the particle i is given by:

dN = \rho(r) dV

Using this, we can write the total force acting on particle i due to all other particles j as:

F_i = \int_V \left(-m k/|\mathbf r_i - \mathbf r|^{m+1} \right) \left( \frac{\mathbf r_i - \mathbf r}{|\mathbf r_i - \mathbf r|} \right) \rho(r) dV

Now, we can use the definition of the virial of Clausius to compute the average kinetic energy of the particle i:

\bar T = -\frac{1}{2} \bar{\sum_i \mathbf F_i \cdot \mathbf r_i}

Using the expression we found for the force F_i