1. May 1, 2017

### eme

1. The problem statement, all variables and given/known data
Suppose that there are long-range interactions between atoms in a gas in the form of central forces derivable from a potential. $$V(r) = \frac k r^m$$ 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 the Boltzmann favor: $$\rho(r) = \frac N V e^{\frac -U(r) kT},$$ 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 any positive $m$, if desired, the mathematics can be simplified by taking $m = +1$

2. Relevant equations
$$\overline T = -\frac 1 2 \overline{ \sum_i \mathbf{F_i \cdot r_i} }$$, where the right-hand term is the addition to the virial of Clausius.

and if $V(r) = a r^n$ then $$\overline T = -\frac 1 2 \overline V$$
3. The attempt at a solution

My idea is for the first part, to find the addition to the virial of Clausius, find the force $\mathbf F = -\nabla V$ so i can write it the first equation. For the second part i'm kind of lost, i want to use the potential energy but i'm not really sure how to find the average potential energy.

The problem is 3.12 in Goldstein third edition.

2. May 6, 2017

### PF_Help_Bot

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