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Homework 5

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    Calculate the standard deviation of the bond length \(\sigma_X\) of the diatomic molecule \(\ce{^1H^{35}Cl }\) when it is in the ground state and first excited state using the quantum harmonic oscillator wavefunctions. The fundamental harmonic vibrational frequency of \(\ce{HCl}\) is 2,989.6 \(cm^{-1}\) and the equilibrium bond length is 0.091nm. How do you interpret the change in the ratio of average bond length to \(\sigma_X\) as a function of energy in the vibration?


    What are two requirements for a molecule to absorb IR radiation (through vibration)?


    (a) Using the relevant transition moment integrals, demonstrate that the probability of a vibration described by a harmonic oscillator in absorbing IR radiation form the \(v=0\) to the \(v=2\) state is forbidden.

    (b) Is the \(v=1\) to \(v=0\) transition also forbidden?


    Which of the following molecules absorb in the IR?

    1. \(F_2\)
    2. \(N_2\)
    3. \(O_3\)
    4. \(Ar\)
    5. \(Br_2\)
    6. \(HF\)
    7. \(H_2 O\)
    8. \(CD_2\)
    9. \(CO_2\)
    10. \(CH_4\)


    What do the presence of overtones in IR spectra reveal about the anharmonicity of the vibration?


    What is the energy in cm-1 of a photon of 600 nm energy that may be observed in electron (UV-VIS) spectroscopy? What is the energy of a 3-micron photon typical in IR spectroscopy? What is the energy of a photon absorbed in a typical CO rotation microwave line (\(0.11519 THz\))?



    Fill in this table.

    Spectroscopic Signature Degree of Freedom
    Type EM Range Typical Wavelength of Transition Typical Energy of Transition sensitive to electronic transition (yes/no) Sensitive to vibrational transition (yes/no) sensitive to rotational transitions (yes/no)

    If any spectroscopy is sensitive to more than one degree of freedom, explain why.


    The moment of inertia of \(\ce{^1H^{35}Cl }\) is \(2.6 \times 10^{-47} \;\textrm{kg}\times \textrm{m}^2\). What is the energy for rotation for \(\ce{^1H^{35}Cl }\) in the \(J=5\) and \(J=20\) states? For a molecule to be thermally excited, the energy of the eigenstate must be comparable to \(k_bT\), with \(k_b\) as the Boltzmann's constant and \(T\) is absolute temperature. What temperature is needed for the \(J=5\) and \(J=20\) rotational states of \(\ce{^1H^{35}Cl }\) to be thermally occupied? (Hint: assuming the term "comparable" is "equal" for this problem).


    \(\ce{^1H^{35}Cl }\) has a bond length of 0.12746 nm and fundamental stretching vibration at 2,886 cm-1. What is the temperature required for the \(v=1\) mode to be thermally excited? (Hint: assuming the term "comparable" is "equal" for this problem).


    \(\ce{^1H^{35}Cl}\) has an equilibrium bond length of 127.46 pm and a spring constant of 516 N/m. The molecule rotates freely in a three-dimensional space as a gas.

    1. What is the zero point energy associated with this rotation? Will this differ if you were considering only vibration?
    2. What is the lowest energy microwave transition observed absorbed \(\ce{^1H^{35}Cl}\) ascribed to rotational motion (assuming a rigid rotor described the rotation)?

    Homework 5 is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by LibreTexts.

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