Q8(a). Rotational constant; vibrational order; chelated C=O and OH

Introduction. A rigid diatomic rotor has energy levels F(J)=BJ(J+1), with the rotational constant B (in cm⁻¹) fixed by the moment of inertia. A harmonic bond vibrates at ν=(1)/(2π c)√(k/μ): a stiff bond (large k) and light atoms (small μ) give a high wavenumber.

(i) Rotational constant

Given: I = 13.97×10⁻⁴⁷ kg m²; h = 6.626×10⁻³⁴ J s; c = 2.998×10⁸ m s⁻¹.

B = (h)/(8π² I c)

B = 6.626×10⁻³⁴8×(3.1416)²×13.97×10⁻⁴⁷×2.998×10⁸ = 200.4 m⁻¹

B = 2.00 cm⁻¹ In frequency units B = h/8π² I = 6.01×10¹⁰ Hz. The lines of the pure rotational spectrum are spaced by 2B = 4.01 cm⁻¹. (A moment of inertia of this size is that of a molecule such as CO, B≈1.93 cm⁻¹.)

(ii) Order of vibrational wavenumbers

NO < CO < HI < HF NO and CO have heavy atoms (μ≈7 u), so even their strong multiple bonds vibrate at about 1900–2100 cm⁻¹; NO is lower than CO because its extra electron sits in a π^* orbital (bond order 2.5 against 3). The hydrogen halides have μ≈1 u, so even the weak H–I bond reaches 2230 cm⁻¹; HF combines the lightest μ with a strong bond and is highest.

(iii) Pentane-2,4-dione and 2'-hydroxypropiophenone

Reading of the question: “2-hydroxypropiophenone” is read as 2'-hydroxypropiophenone (OH ortho on the ring), the compound that, like the enol of pentane-2,4-dione, forms an intramolecular hydrogen bond to C=O.

Scheme: In both compounds an OH is hydrogen-bonded to C=O in a six-membered ring.

Pentane-2,4-dione is mostly enol (stabilised by conjugation and a strong intramolecular hydrogen bond). The chelated enol shows a strong broad band near 1615 cm⁻¹ (C=O, conjugated with C=C and H-bonded, coupled with the C=C stretch), and a very broad, weak O–H band spread over 2500–3200 cm⁻¹ (centred near 2750), instead of the sharp 3600 of a non-bonded OH. The minor diketo form shows two C=O bands near 1727 and 1707 cm⁻¹ (in- and out-of-phase stretching of the two C=O groups).

2'-Hydroxypropiophenone: the aryl ketone C=O (normally about 1685) is lowered to about 1640 cm⁻¹ by the chelating OH, and the O–H stretch appears as a broad band near 3000–3100 cm⁻¹. Both bands are unchanged on dilution (intramolecular H-bond), which distinguishes them from intermolecular hydrogen bonding.

Conclusion. (i) B = 200.4 m⁻¹ = 2.00 cm⁻¹. (ii) NO < CO < HI < HF. (iii) Chelation lowers C=O to about 1615 (enol of pentane-2,4-dione) and 1640 cm⁻¹ (2'-hydroxypropiophenone) and gives broad, dilution-independent O–H bands well below 3600 cm⁻¹.

Application. Rotational constants from microwave spectra give bond lengths (here r would follow from I=μ r²); chelated C=O and OH bands identify β-diketone ligands such as acac used in metal complexes and o-hydroxyaryl ketone UV absorbers in sunscreens and plastics.