(adsbygoogle = window.adsbygoogle || []).push({}); 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

[tex]U(r) = \frac{k}{r^m}[/tex],

where [itex]r[/itex] is the distance between any pair of atoms and [itex]m[/itex] 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:

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

where [itex]N[/itex] is the total number of atoms in a volume [itex]V[/itex]. 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 [itex]N[/itex] so large that sums may be replaced by integrals. While closed results can be found for and positive [itex]m[/itex], if desired, the mathematics can be simplified by taking [itex]m = +1[/itex].

2. Relevant equations

[tex]\bar T = \frac{1}{2} \bar{\sum_i \nabla V \cdot \mathbf r_i}[/tex] (average over time)

which is

[tex]\bar T = -\frac{1}{2} \bar{\sum_i \mathbf F_i \cdot \mathbf r_i}[/tex]

(RHS isvirial of Clausius)

3. The attempt at a solution

The force [itex]\mathbf F_i[/itex] acting on a particle has contributions from all other particles in the form of [itex]\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)[/itex], 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 [itex]\sum_i \mathbf F_i \cdot \mathbf r_i[/itex]. 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 [itex]m=1[/itex] 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 [itex]\mathbf r_i[/itex] within a charged sphere. By the Gauss law, the particles outside of [itex]r > r_i[/itex] do not contribute to the force, and the force is proportional to the charge within the sphere [itex]r < r_i[/itex] and inversely proportional with [itex]r_i ^ 2[/itex]. So I can say

[tex]F_i = C \frac{ \int_0^{r_i} \rho(r) 4 \pi r^2 dr } {r_i^2}[/tex]

and

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

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

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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

Can you offer guidance or do you also need help?

Draft saved
Draft deleted

**Physics Forums | Science Articles, Homework Help, Discussion**