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Question about virial theorem

  1. May 1, 2017 #1

    eme

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    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. jcsd
  3. May 6, 2017 #2
    Thanks for the thread! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post? The more details the better.
     
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