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Homework Help: Why does Water Vapor have 12 degrees of freedom?

  1. Sep 24, 2016 #1
    1. The problem statement, all variables and given/known data

    How many degrees of freedom does water vapor have

    2. Relevant equations
    Translational up to 3
    rotational up to 3
    Vibration up to 6

    3. The attempt at a solution

    Well I said water vapor had 3 translational. It can move along the x, y, or z axis

    I said it had 2 rotational (the answer is three because it doesn't have symmetry?) I said it had symmetry. The book said Carbon dioxide had symmetry so only 2 degrees of freedom. CO2 and H2O pretty much have the same shape don't they? so why does H2O not have symmetry?

    as far as vibration goes, I said 4. The answer is 6. I don't really understand the difference between the three.

    so: why doesn't H2O have symmetry
    and what are the three modes of vibration. (stretching, flexing, and twisting)
  2. jcsd
  3. Sep 24, 2016 #2


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    Staff: Mentor

    It can't have 12 degrees of freedom. A molecule has 3N degrees of freedom, where N is the number of atoms, since you need to specify 3 coordinates for each atom.

    You indeed have 3 degrees of freedom for translation (center of mass coordinates), and generally you have 3 for rotation (since three Euler angles need to be specified to orient a body in space). The rest, 3N - 6, correspond to vibration (relative position of the atoms relative to each other).

    There is a special case for linear molecules, since there is no rotation along the internuclear axis, so that leaves 3N - 5 dof for vibration. the extra dof appears because the bending mode is now degenerate.
  4. Sep 24, 2016 #3
    the 12 is in the soln. manual

    3 for translation
    3 for rotation
    and 6 for vibration.

    Not saying you are wrong, but that I don't understand. Can you explain it again? We are not into the quantum mechanics and so forth of it. just that
    f can equal up to 3 for translation
    f can equal up to 3 for rotation
    and f can equal up to 6 for vibration (there are three modes of vibration but each mode of vibration is equal to 2 for potential and kinetic energies
  5. Sep 24, 2016 #4


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    May I inquire the authorship of this "solutions manual?"
  6. Sep 24, 2016 #5


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    I get it now. Usually, the point is to get the number of vibrationsl modes. In your case, what is asked for is the number of quadratic degrees of freedom. Since each normal mode of vibration corresponds to two quadratic degrees of freedom, so the total is 12.

    As for your original question, the point is that for a linear molecule like CO2, rotation along the internuclear axis doesn't count (where each atom would only be turning on itself). That's why you then get only 2 rotational degrees of freedom.

    Even if water possesses symmetry, there is still three ways it can rotate.
  7. Sep 25, 2016 #6
    but why can in it rotate three ways and not carbon dioxide? I'd ask if this is something to be memorized, but the textbook asked the question right after the discussion. So I should have been able to tell based on the reading.

    and what are the difference between the 3 vibrational modes. How can I tell whether a molecule possesses one two or all three. The book said that solids have all three, but for everything else....
  8. Sep 26, 2016 #7


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    CO2 has all three atoms in a straight line (included angle between the O atoms is180o). That means one rotation axis, the line that goes thru all three atoms, does not change the orientation of the molecule.

    In H2O the two H atoms have an included angle to the O atom of 104.5o. Therefore, any rotation axis you pick wil reorient the molecule.
  9. Sep 26, 2016 #8

    I did not realize the CO2 was a straight line. So even though H2O has "symmetry" I it rotate out of the page, there is a difference in appearance. Thanks.

    Now can you explain the different modes of vibration to me?
  10. Sep 26, 2016 #9


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    Staff: Mentor

  11. Sep 26, 2016 #10
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