ROTATIONAL ENERGIES AND SPECTRA: 

LINEAR MOLECULE SPECTRA:  Employing the last equation twice  ΔE= E J+1 – E J = hB(J+1)(J=2) – hBJ(J+1)  Or: ΔE = 2hB(J+1) for a transition from the Jth to the (J+1)th level.  Using ΔE = hν gives us  ν = 2B(J+1) for rotational transitions.

ROTATIONAL ENERGIES – LINEAR MOLECULES: J ValueJ(J+1) ValueRotational Energy E J+1 - E J ν =(E J+1 – E J )/h hB 2B 266hB4hB4B 31212hB6hB6B 42020hB8hB8B 53030hB10hB10B 64242hB12hB12B

LINEAR MOLECULE ROTATIONAL TRANSITIONS:  J = 4  J = 3  J = 2  J = 1  J = 0

LINEAR MOLECULE ROTATIONAL SPECTRUM:  Intensity J = 4←3  J = 1←0  2B  Absorption Frequencies →

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. 

ROTATIONAL SPECTRA DIATOMICS: 

MOMENTS OF INERTIA – DIATOMICS: 

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CLASS EXAMPLE CALCULATIONS:  We will calculate moments of inertia for 14 N 2, 14 N 15 N and 12 C 16 O 2 using, where possible, symmetry arguments to simplify the arithmetic.  Data: r(N≡N) = Å (109.4pm) and r(C=O) = Å  Masses: 14 N ( u), 15 N( u), 12 C( u) and 16 O( u).

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 12 C 32 S and 12 C 34 S molecules which have, of course, identical bond distances(almost!).

12 C 32 S AND 12 C 34 S ROTATIONAL SPECTRA: Transition 12 C 32 S (Frequency (MHz) 12 C 34 S Frequency (MHz) J=1← J=2← J=3← J=4← J=5← J=6←

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 H 35 Cl and D 35 Cl? All data are taken from the NIST site. 

REDUCED MASSES AND BOND DISTANCES: MoleculeBond Distance (Å) B Value (MHz) H 35 Cl H 37 Cl D 35 Cl H 79 Br H 81 Br D 79 Br Mg 16 O Ag 35 Cl

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!

SPECTRUM TO MOLECULAR STRUCTURE:  Class Example: A scan of the microwave (millimeter wave!) spectrum of 6 LiF over the range 350 → 550 GHz shows lines at MHz, MHz and MHz. Assign rotational quantum numbers for these transitions. Determine a B value and the bond distance for 6 LiF. Are the “lines” identically spaced?