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Estimating the vibrational frequency of Hydrogen molecule

  1. Feb 7, 2013 #1
    1. The problem statement, all variables and given/known data
    A diatomic molecule, when excited, can vibrate in such a way that the relative distance
    between the two atoms, r(t), executes periodic oscillations. If the potential energy of the
    molecule as a function of r is given by V (r), the time dependence of r(t) is identical to that
    of a particle of reduced mass

    μ= m1 m2/ (m1 + m2)

    moving in the potential V (r). An analytic expression which is sometimes used to approximate the actual interatomic interaction is the Morse potential,
    V (r) = B[1 − exp(−β(r−r0 ))]^2 − B

    B is the depth of the well, r0 is the equilibrium separation, and β is a parameter which
    governs how rapidly the energy rises as one moves away from the equilibrium position.

    (3.a) Find the frequency of small classical vibrations about the minimum of the Morse
    potential in terms of B, r0 , and β.

    (3.b) For the molecule H2 , β = 1.93 Angstrom-1 , r0 = 0.74 angstrom, and B = 4.8 eV. The mass of a hydrogen atom is 1.67 × 10^ -24 grams. What is the frequency of small vibrations? The observed vibrational frequency is 1.32 × 10^14 Hz.



    2. Relevant equations



    3. The attempt at a solution

    I used Taylor expansion to expand the exponent in V(r) and get, to first order,
    V = Bβ^2(r-r0)^2 - B
    Compare this to the energy for harmonic oscillator, I conclude that
    ω = 2Bβ^2 / μ

    Putting in the numbers give me a frequency of 4.17 x 10^12 Hz, which is two orders off.
     
  2. jcsd
  3. Feb 7, 2013 #2

    TSny

    User Avatar
    Homework Helper
    Gold Member

    Did you take care of the mass units? g or kg?
     
  4. Feb 7, 2013 #3
    oh I got it! Thanks a lot
     
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