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Given wavenumber find spring constant harmonic oscillator

  1. Oct 30, 2016 #1
    1. The problem statement, all variables and given/known data
    The separation between energies of an oxygen molecule is 2061 cm-1 (wavenumber). Treating the molecule as a simple harmonic oscillator whose fundamental frequency is related to its spring constant and reduced mass, calculate the spring constant for an O2 molecule.

    meff = 1.33e-26 kg
    h = 4.136e-15 eV-s
    1 eV = 8065.45 cm-1

    2. Relevant equations
    ω = √(k/meff)

    k = ω2eff
    ω = 2π(v) = 2π(c/λ)
    λ = hc/ΔE

    ∴ k = 2πc/(hc/ΔE)
    = 2πΔE/h

    3. The attempt at a solution
    ΔE = 2061 cm-1 ⇒ 0.2555 eV
    k = 3.88*1014 (which just seems way too big)
    unit check: eV/(eV-s) so the above value should be even bigger?? (times the speed of light)
    k = 1.1644*1023
    Actually, I wrongly assumed that the spring constant is unitless.

    N/m (force per unit length) or N-m-1

    So instead I found this equation, but I don't know how it was derived:
    E0 = (h/2)√(k/meff)
    k = meff(2E0/h)
    k = 1.33e-26(2*0.2555/4.136e-15)^2
    k = 203 (which seems reasonable--but I don't get how the equation comes about even knowing .5mv2 = k)
     
    Last edited: Oct 30, 2016
  2. jcsd
  3. Oct 31, 2016 #2

    BvU

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    Hi,

    I gather a wavenumber of 2061 cm-1 means a wavelength of 10-2/2061 m (I'm not a spectroscopist :smile:).
    With ##\ \lambda f = c \ ## and ##\ \omega = 2\pi f \ ## I then get ##\ \omega = 2\pi {2061\over 10^{-2}}c = 3.88\ 10^{14} \ ## radian/s.
    What you call k is actually omega (the dimensions do match). From there to the spring constant gives me quite a different result ...

    ##E_0 = {1\over 2} \hbar\omega## is the energy of the ground state. But the energy difference between states is a mutltiple of ##\hbar\omega##.
     
  4. Oct 31, 2016 #3

    andrevdh

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  5. Oct 31, 2016 #4
    reduced mass ~ meff = 1.33e-26 kg
    ħ = 1.055*10-34 Js
    h = 6.626*10-34 Js
    λ = 10-2/2061 m

    E = (n+.5)ħw
    w = 2πf = 2π(c/λ)

    ω = √(k/meff)
    k = (ω)2meff
    k = (2πc/λ)2*1.33e-26
    k = 4*1.33e-26*pi*3e8*2061/10e-2
    k = 1.033e-12
     
  6. Nov 1, 2016 #5

    andrevdh

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    I get 1.11 x 10-18 kg/s2
     
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