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Prove that the Bohr hydrogen atom approaches classical conditions when [. . .]

  1. Jul 20, 2012 #1

    s3a

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    "Prove that the Bohr hydrogen atom approaches classical conditions when [. . .]"

    1. The problem statement, all variables and given/known data
    The problem and its solution are attached as ProblemSolution.jpg.

    2. Relevant equations
    E_k = chR/(n_k)^2
    E_l = chR/(n_l)^2
    ΔE = hc/λ
    hc/λ = chR[1/(n_k)^2 – 1/(n_l)^2]
    1/ λ = R[1/(n_k)^2 – 1/(n_l)^2]


    3. The attempt at a solution
    Given E_k = chR/(n_k)^2 and E_l = chR/(n_l)^2,

    ΔE = chR[1/(n_k)^2 – 1/(n_l)^2]

    Therefore, since, ΔE = hc/λ,

    hc/λ = chR[1/(n_k)^2 – 1/(n_l)^2]
    1/ λ = R[1/(n_k)^2 – 1/(n_l)^2]
    1/ λ = R[1/(n_k)^2 – 1/(n_l)^2]
    1/ λ = R[1/(n_k)^2 – 1/(n_l)^2
    ν = R[(n_l)^2 – (n_k)^2]/[(n_k)^2 * (n_l)^2]

    and, n_l – n_k = -1 which counters the negative that I had initially compared to the answer of the book so far and now the only difference is that my answer lacks the c multiplicative factor that the book has. If I did something wrong, what is it? Or is it the book?

    Also, how is the “crazier” part of equation (1.6.3) obtained?

    If more information is needed, just ask.

    Any help would be greatly appreciated!
    Thanks in advance!
     

    Attached Files:

  2. jcsd
  3. Jul 20, 2012 #2

    TSny

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    Re: "Prove that the Bohr hydrogen atom approaches classical conditions when [. . .]"

    What is the relationship between frequency, wavelength, and c?

    Just substitute the well-known Bohr model expressions for the radius of the orbit and the speed of the electron in the orbit.
     
  4. Jul 21, 2012 #3

    s3a

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    Re: "Prove that the Bohr hydrogen atom approaches classical conditions when [. . .]"

    1) The relationship between frequency, wavelength, and c is v = c/λ.

    2) I found the equation for the radius “on a silver-platter” and derived the equation for the velocity and yes, plugging them in worked. :smile:

    3) Now, I still have some work which disagrees with the solution and it is attached as MyWork.jpg. Could you please tell me if I am wrong or if it's the solution that is wrong as well as what as what to do to get the correct answer if I am wrong?
     

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  5. Jul 21, 2012 #4

    TSny

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    Re: "Prove that the Bohr hydrogen atom approaches classical conditions when [. . .]"

    S3a, The energy levels are negative: E = -chR/n2. The text that you quoted in your original post left out the minus sign for some reason (misprint?)
     
  6. Jul 21, 2012 #5

    s3a

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    Re: "Prove that the Bohr hydrogen atom approaches classical conditions when [. . .]"

    Could it be that the negative sign is to indicate that the (light) energy is being lost from the Hydrogen orbital whereas here we are talking about the energy gained by the photon?

    I deduced this based on the following quote from the screen-shot of the problem and its solution that I attached initially.: "Therefore the frequency of the emitted photon is [. . .]". The fact that that equation is the frequency of the emitted photon should imply that the frequency multiplied by h (plank's constant) is the energy of the emitted photon gained (rather than that lost from the Hydrogen orbital).

    That would explain why the book has it the way it does but not why I get the book's answer multiplied by -1. Is there something with my logic above that doesn't hold?
     
  7. Jul 21, 2012 #6

    TSny

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    Re: "Prove that the Bohr hydrogen atom approaches classical conditions when [. . .]"

    To me the natural way to think about it is that the energy of the photon equals the loss of energy of the atom as the atom goes from the higher excited state (nk) to the lower excited state (nl).

    Thus h[itex]\nu[/itex] = Ek - El where Ek = -chR/nk2 and El = -chR/nl2
     
  8. Jul 21, 2012 #7

    TSny

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    Re: "Prove that the Bohr hydrogen atom approaches classical conditions when [. . .]"

    Long before the Bohr model was developed, it was known that the frequencies of hydrogen could be calculated by taking the difference of numbers of the form cR/n2 where n is a positive integer and R was an empirically determined number. These numbers cR/n2 were called "terms".

    The Bohr model later explained this numerology by showing that the energy levels of hydrogen were just the negative of these terms multiplied by h and the model derived the value of R in terms of fundamental constants.

    Maybe your book is using these "terms" rather than energy levels of the atom.
     
    Last edited: Jul 21, 2012
  9. Jul 21, 2012 #8

    s3a

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    Re: "Prove that the Bohr hydrogen atom approaches classical conditions when [. . .]"

    What you said earlier makes sense but could it also be the case that it just doesn't matter what the sign is because what the question asks for ultimately is frequency and we just want to compare the magnitudes of the two frequencies (classical and modern) so we take the absolute value? Is that a valid thought process?
     
  10. Jul 21, 2012 #9

    TSny

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    Re: "Prove that the Bohr hydrogen atom approaches classical conditions when [. . .]"

    That sounds good to me. :smile:
     
  11. Jul 21, 2012 #10

    s3a

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    Re: "Prove that the Bohr hydrogen atom approaches classical conditions when [. . .]"

    Okay, thanks for all your help! :smile:
     
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