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Gas with interacting molecules (from goldstein)

  1. Nov 18, 2008 #1
    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 is virial 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]


    [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]...
    Last edited: Nov 18, 2008
  2. jcsd
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