1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Isothermal compressibility and spring constant solids

  1. Jul 17, 2014 #1
    1. The problem statement, all variables and given/known data

    Consider a solid of compressibility ##\kappa##. Assume that the atoms in this solid are arranged on a regular cubic lattice, the distance between their nearest neighbors being ##a##. Assume further that a restoring force ##-k_0 \Delta a## acts on a given atom when it is displaced by a distance ##\Delta a## from its nearest neighbor. Use simple reasoning to find an approximate relation between the spring constant ##k_0## and the compressibility ##\kappa## of this solid.

    2. Relevant equations

    ##\kappa = \frac{1}{V}\frac{\partial V}{\partial p}|_{T}##.

    3. The attempt at a solution

    Consider a cubic lattice of ##N+1## atoms on each edge. Take ##N \gg 1## so that ##N+1 \approx N##. The area of any given face of the lattice is ##A = N^2 a^2##. Imagine now that the face is pushed inward by the amount ##\Delta a## where ##\frac{\Delta a}{a} \ll 1##. Then there is an increase in the total force exerted on this face due to the restoring force ##F = -k_0 \Delta a## on each of the ##(N+1)^2## atoms in this face; the increase in total force on this face will be given in magnitude by ##\Delta F = (N+1)^2 \Delta a k_0 \approx N^2 \Delta a k_0##.

    Then ##\kappa = \frac{1}{V}\frac{\partial V}{\partial p}|_T = \frac{1}{N a}\frac{\partial a}{\partial p}|_T \approx \frac{1}{a}\frac{\Delta a}{\Delta p} = \frac{1}{a}\frac{N^2 a^2 \Delta a}{N^2 \Delta a k_0} = \frac{a}{N k_0}##.

    According to the book the correct answer is ##\kappa = \frac{a}{k_0}## which is not what I got. However I am not sure of how to fix my solution. I could I suppose take a parallelipiped with sides of length ##Na## whose faces are to be pushed inwards, and sides of length ##a## which are parallel to the force applied to push the aforementioned faces inwards. However this looks to be a very contrived escape in fixing my solution on top of the assumptions I already made that make my solution above look quite non-rigorous. For example I took the solid to be a cubic lattice as opposed to any crystal lattice and I also took ##N \gg 1## to make ##(N+1)^2 \approx N^2##, neither of which are assumptions made in the problem statement. Could anyone help me in fixing my solution. Also, could any comment on whether or not I could make the calculation more rigorous? Any hints on how to do so? Thanks!
     
    Last edited: Jul 17, 2014
  2. jcsd
  3. Jul 17, 2014 #2

    TSny

    User Avatar
    Homework Helper
    Gold Member

    Can you explain how you get the second equality here?
     
  4. Jul 19, 2014 #3
    ##\kappa = \frac{1}{V}\frac{\partial V}{\partial p} = \frac{1}{A L}\frac{A \Delta a}{\Delta p} = \frac{a N^2 \Delta a}{N (N^2 k_0 \Delta a)} = \frac{a}{k_0 N}##
     
  5. Jul 19, 2014 #4

    TSny

    User Avatar
    Homework Helper
    Gold Member

    V = AL. When V changes, it appears you are assuming that only L changes and not A. Shouldn't A also change?

    That is, how do you get ##\partial V = A \Delta a##?
     
  6. Jul 22, 2014 #5
    In the calculation I was simply pushing one face of the cube inwards by some small amount so the area of the face is the same since the push is normal to the face itself.
     
  7. Jul 22, 2014 #6

    TSny

    User Avatar
    Homework Helper
    Gold Member

    I believe the usual definition of compressibility ##\kappa## involves the change in volume when the pressure is increased over the entire surface. So, for a cube, you would consider the change in volume when the pressure is increased on all six faces.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Isothermal compressibility and spring constant solids
Loading...