?If the vibration of a diatomic A–B is modelled using a harmonic oscillator, the

Chapter 7, Problem P7E.1

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If the vibration of a diatomic A–B is modelled using a harmonic oscillator, the vibrational frequency is given by \(\omega=\left(k_{f} / \mu\right)^{1 / 2}, \text { where } \mu\) is the effective mass, \(\mu=m_{\mathrm{A}} m_{\mathrm{B}} /\left(m_{\mathrm{A}}+m_{\mathrm{B}}\right)\). If atom A is substituted by an isotope (for example \({ }^{2} \mathrm{H}\) substituted for \({ }^{1} \mathrm{H}\)), then to a good approximation the force constant remains the same. Why? (Hint: Is there any change in the number of charged species?) (a) Show that when an isotopic substitution is made for atom A, such that its mass changes from \(m_{\mathrm{A}} \text { to } m_{\mathrm{A}^{\prime}}\), the vibrational frequency of \(\mathrm{A}^{\prime}-\mathrm{B}, \omega_{\mathrm{A}^{\prime} \mathrm{B}}\), can be expressed in terms of the vibrational frequency of \(\omega_{A B} \text { as } \omega_{A^{\prime} B}=\omega_{A B}\left(\mu_{A B} / \mu_{A^{\prime} B}\right)^{1 / 2}\), where μAB and μA B′ are the effective masses of A–B and A′–B, respectively. (b) The vibrational frequency of \({ }^{1} \mathrm{H}^{35} \mathrm{Cl} \text { is } 5.63 \times 10^{14} \mathrm{s}^{-1}\). Calculate the vibrational frequency of (i) \({ }^{2} \mathrm{H}^{35} \mathrm{Cl} \text { and }(\text { ii }){ }^{1} \mathrm{H}^{37} \mathrm{Cl}\). Use integer relative atomic masses.

Text Transcription:

 ω=k_f^1/2 = μ , where μ

μ=m_A m_B m_A+ m_B

m_A to m_A′

^2H

^1H

A′–B, ω_A^B′

ω_A B, ω_A^B=ω_mu_A B mu_A^B^½

^1 H^35Cl is 5.63 × 10^14 s^ −1

^2H^35Cl

^1 H^37Cl