A covalently bonded heteronuclear diatomic molecule can be modeled as two vibrating masses connected by a spring. The vibrational frequency of the bond can be expressed using an equation derived from Hooke's law, which describes how the force applied to stretch or compress a spring is proportional to the displacement of the spring. In this case, the atoms behave like masses, and the bond acts like a spring.
According to Hooke's law, the vibrational frequency is directly proportional to the force constant (K) and inversely proportional to the reduced mass (μ) of the system. The reduced mass measures how the masses of both atoms are combined, allowing the system to be treated as a single oscillating entity. This means that as the atomic weight increases, the vibrational frequency decreases. Therefore, a covalent bond between heavier atoms vibrates at a lower frequency than between lighter atoms. The force constant represents the stiffness of the bond, which indicates how much energy is needed to stretch or compress it. A stronger bond (with a larger force constant) vibrates at a higher frequency than a weaker bond. Therefore, single bonds would be considered weaker than double and triple bonds, resulting in a lower frequency. For example, the single bond between two carbon atoms vibrates at 1200 cm−1. In contrast, the double bond between two carbon atoms vibrates at 1650 cm−1, and the triple bond between the same shows a maximum vibrational frequency of 2150 cm−1.
Hooke's law models atoms in a covalently bonded molecule as two vibrating masses connected by a spring, allowing their vibrational frequencies to be determined using an equation based on this principle.
The final equation is obtained by removing the Avogadro's number from the denominator of the reduced mass expression, and taking its square root.
As per this relation, the wavenumber of a vibration or its vibrational frequency is directly proportional to the force constant, K, and inversely proportional to the reduced mass of vibrating atoms, μ.
So, bonds between atoms with higher atomic masses vibrate at lower frequencies than those between atoms with lower atomic masses.
Bending force constants have a lower value than their stretching counterparts. So, C–H bending occurs at a lower frequency than C–H stretching.
As the K value depends on the bond's strength, stronger bonds with larger force constants vibrate at higher frequencies than weaker bonds.
Therefore, triple bonds vibrate at higher frequencies than double or single bonds between the same atoms.
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