IR Spectroscopy: Hooke's Law Approximation of Molecular Vibration

Hooke's law models atoms in a covalently bonded molecule as two vibrating masses connected by a spring. This spring-mass system allows vibrational frequencies to be determined using a derived equation. The atoms behave like masses while the bond acts like a spring, enabling prediction of how molecular vibrations occur based on bond properties and atomic masses.

Vibrational frequency is inversely proportional to the reduced mass of vibrating atoms. The reduced mass measures how the masses of both atoms combine in the system. As atomic weight increases, the reduced mass increases, causing the vibrational frequency to decrease. Therefore, bonds between heavier atoms vibrate at lower frequencies than those between lighter atoms.

Vibrational frequency is directly proportional to the force constant, which represents bond stiffness and strength. Stronger bonds have larger force constants and vibrate at higher frequencies than weaker bonds. For example, triple bonds vibrate at higher frequencies than double or single bonds between the same atoms because they are stronger and stiffer.

Bending force constants have lower values than stretching force constants for the same bond. Since vibrational frequency is directly proportional to the force constant, bending vibrations occur at lower frequencies than stretching vibrations. For instance, C–H bending occurs at a lower frequency than C–H stretching because bending requires less energy than stretching.

The force constant represents the stiffness of a bond, indicating how much energy is needed to stretch or compress it. A larger force constant reflects a stronger, stiffer bond that resists deformation more effectively. Since vibrational frequency depends directly on the force constant, bonds with higher force constants vibrate at higher frequencies and appear at different positions in infrared spectra.

C–C single bonds vibrate at approximately 1200 cm⁻¹, double bonds at 1650 cm⁻¹, and triple bonds at 2150 cm⁻¹. This progression reflects increasing bond strength and force constants as bond order increases. Triple bonds are strongest with the highest force constant, causing them to vibrate at the highest frequency, while single bonds are weakest with the lowest vibrational frequency.

Reduced mass measures how the masses of both atoms combine in a vibrating system, allowing it to be treated as a single oscillating entity. Vibrational frequency is inversely proportional to the reduced mass. A system with a larger reduced mass vibrates more slowly, while a system with a smaller reduced mass vibrates faster, making reduced mass a key factor in predicting molecular vibration behavior.