Homework 5
- Page ID
- 143072
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Section: _____________________________
Student ID#:__________________________
Template:HideTOCQ1
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?
Q2
What are two requirements for a molecule to absorb IR radiation (through vibration)?
Q3
(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?
Q4
Which of the following molecules absorb in the IR?
- \(F_2\)
- \(N_2\)
- \(O_3\)
- \(Ar\)
- \(Br_2\)
- \(HF\)
- \(H_2 O\)
- \(CD_2\)
- \(CO_2\)
- \(CH_4\)
Q5
What do the presence of overtones in IR spectra reveal about the anharmonicity of the vibration?
Q6
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 (\(0.11519 THz\))?
Q7
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) |
| UV-Visible | |||||
| Infrared | |||||
| Microwave | |||||
If any spectroscopy is sensitive to more than one degree of freedom, explain why.
Q8
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).
Q9
\(\ce{^1H^{35}Cl }\) 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).
Q10
\(\ce{^1H^{35}Cl}\) 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.
- What is the zero point energy associated with this rotation? Will this differ if you were considering only vibration?
- What is the lowest energy microwave transition observed absorbed \(\ce{^1H^{35}Cl}\) ascribed to rotational motion (assuming a rigid rotor described the rotation)?