# Homework Help: Gas with interacting molecules (from goldstein)

1. Nov 18, 2008

### gulsen

1. The problem statement, all variables and given/known data
(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 $m = +1$.

2. Relevant 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)

3. 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 $m$...

Last edited: Nov 18, 2008