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Homework Help: Help - Where am I going wrong!

  1. Mar 16, 2006 #1
    Hi, I appear to have lost some mass/density.

    Q: Calculate the atom density, n, of aluminium at 600k.

    Data given (some of it not relevant to this part of the ? but i'll post it all).

    The debye temperature [tex]T_D = 428 K[/tex] . The velocity of sound at room temp, [tex] v = 5100m s^{-1}[/tex]. The interatomic spacing a = 405pm, and its relative atomic mass is 27.

    I found the atom density n to be:

    [tex] n = (KT_D / \hbar v )^3.(1 / 6\pi^2)[/tex]

    I used [tex]{\omega_m}^3 = 6\pi^2 v^3 n[/tex], and the fact [tex]{\omega_m} = K T_D / \hbar[/tex] to get my equations for n.
    I assumed the atom density was the same at 600k as at room temperature (the way the question was worded I couldn't see any other method). Plugging the numbers in I found [tex]n = 2.24x10^{28} m^{-3}[/tex].

    Then I'm asked to compare the density found with aluminiums true density of [tex]2700 kg m^{-3}[/tex] and explain any difference.
    My density is just the atom density multiplied by the atomic mass (assuming the mass is just contained within the aluminium).
    [tex]/rho = n m(amu) = n = 2.24x10^{28} * (27 / 6.022x10^{26})) = 1000 kg m^{-3}[/tex].

    There in lies my problem. I'd expect my approximations to overestimate the density if anything.
    Last edited: Mar 16, 2006
  2. jcsd
  3. Mar 16, 2006 #2
    Is there anyway my density could be right? Maybe some assumption of the debye model I'm not aware of. I've got the formulas straight out the textbook I don't see how it could be wrong and it's the right order of magnitude.
  4. Mar 16, 2006 #3
    Just had a thought is the atom density n, the number of molecules vibrating at [tex]{\omega_m}[/tex] / volume. Guess that would explain why the density is significantly lower, since many atoms won't be vibrating at [tex]{\omega_m}[/tex].
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