Rotational Energies and Spectra:

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Rotational Energies and Spectra

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Linear Molecule Spectra

• Employing the last equation twice
• ?E EJ1 EJ hB(J1)(J2) hBJ(J1)
• Or ?E 2hB(J1) for a transition from the Jth
  to the (J1)th level.
• Using ?E h? gives us
• ? 2B(J1) for rotational transitions.

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Rotational Energies Linear Molecules
J Value J(J1) Value Rotational Energy EJ1 - EJ ? (EJ1 EJ)/h
0 0 0
1 2 2hB 2hB 2B
2 6 6hB 4hB 4B
3 12 12hB 6hB 6B
4 20 20hB 8hB 8B
5 30 30hB 10hB 10B
6 42 42hB 12hB 12B
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Linear molecule Rotational Transitions

• J 4
  
• J 3
• J 2
• J 1
• J 0

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Linear molecule Rotational Spectrum

• Intensity J 4?3
• J 1?0
• 
  2B
• Absorption Frequencies ?

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Linear Molecule Spectra

• Given a molecular structure we can predict the
  appearance of a rotational (microwave) spectrum
  by calculating (a) the moment of inertia and (b)
  the value of the rotational constant. A linear
  molecule can be treated as a series of point
  masses arranged in a straight line.

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Rotational Spectra Diatomics

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Moments of Inertia Diatomics

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MomeNts of Inertia Diatomics

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Class Example Calculations

• We will calculate moments of inertia for 14N2,
  14N15N and 12C16O2 using, where possible,
  symmetry arguments to simplify the arithmetic.
• Data r(NN) 1.094 Å (109.4pm) and r(CO)
  1.163 Å
• Masses 14N (14.00307u), 15N(15.00011u),
  12C(12.00000u) and 16O(15.99492u).

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Spectra of carbon monosulfide

• In rotational spectroscopy, energy level
  separations become smaller as atomic masses
  increase and as molecular dimensions increase. On
  the next slide the effect of mass alone is
  illustrated for the 12C32S and 12C34S molecules
  which have, of course, identical bond
  distances(almost!).

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12C32S and 12C34S Rotational SpectrA
Transition 12C32S (Frequency (MHz) 12C34S Frequency (MHz)
J1?0 48990.97 48206.92
J2?1 97980.95 96412.94
J3?2 146969.03 144617.11
J4?3 195954.23 192818.46
J5?4 244935.74 241016.19
J6?5 293912.24 289209.23
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Effects of mass and molecular siZe

• The slide that follows gives B values for a
  number of diatomic molecules with different
  reduced masses and bond distances. What is the
  physical significance of the very different
  frequencies seen for H35Cl and D35Cl? All data
  are taken from the NIST site.
• http//www.nist.gov/pml/data/molspec.cfm

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Reduced Masses and Bond Distances
Molecule Bond Distance (Å) B Value (MHz)
H35Cl 1.275 312989.3
H37Cl 1.275 312519.1
D35Cl 1.275 161656.2
H79Br 1.414 250360.8
H81Br 1.414 250282.9
D79Br 1.414 127358.1
24Mg16O 1.748 17149.4
107Ag35Cl 2.281 3678.04
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Real Life Working Backwards?

• In the real world spectroscopic experiments
  provide frequency (and intensity) data. It is
  necessary to assign quantum numbers for the
  transitions before molecular (chemically useful)
  information can be determined. Sometimes all of
  the data are not available!

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Spectrum to Molecular StructurE

• Class Example A scan of the microwave
  (millimeter wave!) spectrum of 6LiF over the
  range 350 ? 550 GHz shows lines at 358856.2 MHz,
  448491.1 MHz and 538072.7 MHz. Assign rotational
  quantum numbers for these transitions. Determine
  a B value and the bond distance for 6LiF. Are the
  lines identically spaced?