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eme
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Homework Statement
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##
Homework 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 $$\overline T = -\frac 1 2 \overline V$$
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.