Which would have the lowest frequency the C−H bond in methane, the N−H bond in ammonia (gas phase), or the O−H bond in water (gas phase)
The frequency of vibration of a bond is related to the mass of the atoms involved and the strength of the bond. This is typically described by Hooke's Law for small vibrations, and the frequency (ν) can be approximated by the equation:
ν = (1/2π) * √(k/μ)
where ν is the vibrational frequency, k is the force constant of the bond (a measure of bond strength), and μ is the reduced mass of the A-B system given by:
μ = (m_A * m_B) / (m_A + m_B)
where m_A and m_B are the masses of the two atoms.
Generally, heavier atoms result in lower vibrational frequencies, and stronger bonds result in higher frequencies.
For the C-H bond in methane, the bond is relatively strong, but carbon and hydrogen have lower masses compared to nitrogen and oxygen.
For the N-H bond in ammonia, nitrogen is heavier than carbon but lighter than oxygen, so the reduced mass will be between that of C-H and O-H.
For the O-H bond in water, oxygen is the heaviest of the three atoms being considered, and the O-H bond is also quite strong, often stronger than a C-H or N-H bond due to oxygen's higher electronegativity.
Putting these factors together, the C-H bond would generally have a higher frequency than the N-H bond because carbon is lighter than nitrogen, despite both having strong bonds with hydrogen. In turn, the N-H bond would typically have a higher frequency than the O-H bond because nitrogen is lighter than oxygen. Thus, the bond with the lowest frequency would be the O-H bond in water, due to the combination of the highest mass and a strong bond strength.