Q7(a). C=O stretching wavenumber from the force constant; IR of C₅H₁₀

Introduction. (i) A bond is treated as a harmonic oscillator (Hooke's law): its wavenumber depends on the force constant k and the reduced mass μ. (ii) A band at 1380 cm⁻¹ is the symmetric (“umbrella”) bending of a methyl group; combined with the formula and the absence of alkene bands it fixes the structure.

(i) C=O stretching wavenumber

Given: k = 12.3×10⁵ dyn cm⁻¹ = 1230 N m⁻¹ (1 dyn cm⁻¹ = 10⁻³ N m⁻¹); atomic masses C = 12.000 u, O = 15.995 u.

μ=(m_Cm_O)/(m_C+m_O)=(12.000×15.995)/(27.995)=6.856 u=1.1385×10⁻²⁶ kg

ν=(1)/(2π c)√((k)/(μ))=12π×2.998×10¹⁰ cm s⁻¹1230 N m⁻¹1.1385×10⁻²⁶ kg

ν(C=O) ≈ 1745 cm⁻¹ The frequency is ν=cν≈5.23×10¹³ Hz. The value lies in the normal carbonyl region (1700–1750 cm⁻¹); with μ = 12×16/28 = 6.857 u the result is the same, 1745 cm⁻¹.

(ii) C₅H₁₀ with a band at 1380 cm⁻¹

1. Unsaturation. C₅H₁₀: degree of unsaturation =(2×5+2-10)/2=1, so one ring or one C=C.
2. 1380 cm⁻¹ = CH₃ symmetric bend: a methyl group is present. A single band (not a 1385/1370 doublet) indicates one methyl rather than an isopropyl or gem-dimethyl group.
3. No alkene bands are reported (no =C–H stretch above 3000, no C=C near 1640–1670, no strong =C–H wag at 890–990 cm⁻¹), so the one unsaturation is a ring.

Figure: methylcyclobutane

1. A cycloalkane C₅H₁₀ with one methyl group: methylcyclobutane. (Ethylcyclopropane would also fit the formula; methylcyclobutane is the simplest structure with exactly one isolated methyl on a ring.)

Conclusion. (i) ν(C=O) ≈ 1745 cm⁻¹. (ii) One degree of unsaturation, no alkene bands and a single methyl bend at 1380 cm⁻¹: methylcyclobutane.

Application. Force constants from IR wavenumbers measure bond strength: the drop of ν(C=O) on conjugation or hydrogen bonding is used to judge how much double-bond character a carbonyl group keeps.