Discussion
An HCl molecule has force constant 516 N/m. Using harmonic oscillator approximation, the vibrational frequency \(\nu\) and the wave number \(\bar{\nu}\) of this molecule are :
(Given : Mass of H atom is \(1.674 \times 10^{-27}\) kg, mass of Cl atom is \(5.887 \times 10^{-26}\) kg, Planck's constant \(h = 6.626 \times 10^{-34}\) Js, speed of light \(c = 3 \times 10^{8}\) m/s.)
Answer: C
The values are \(\nu = 89.6 \times 10^{12}\) Hz and \(\bar{\nu} = 2987\ cm^{-1}\).
The vibration is that of the reduced mass on a spring of constant \(k\):
\[ \begin{aligned} \mu &= \frac{m_{H}m_{Cl}}{m_{H} + m_{Cl}} \\ &= 1.628 \times 10^{-27}\ \text{kg} \\ \nu &= \frac{1}{2\pi}\sqrt{\frac{516}{1.628 \times 10^{-27}}} \\ &= 8.96 \times 10^{13}\ \text{Hz} \\ \bar{\nu} &= \frac{\nu}{c} = \frac{8.96 \times 10^{13}}{3 \times 10^{10}\ \text{cm/s}} \\ &\approx 2987\ cm^{-1} \end{aligned} \]
Because chlorine is so much heavier, \(\mu\) is close to the mass of the hydrogen atom.
The option pairing 89.6 THz with 1890 cm−1 has an inconsistent wavenumber, and the options with 179.2 THz or 44.8 THz double or halve the frequency with wavenumbers scaled the same way, so they do not follow from \(k\) and \(\mu\).
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