Homework 5
- Page ID
- 109912
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Name: ______________________________
Section: _____________________________
Student ID#:__________________________
Q1
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 2989.6 \(cm^{-1}\) and the equilibrium bond length is 0.127 nm. 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 (via its vibrations)?
Q3
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. Is the \(v=1\) to \(v=0\) transition also forbidden?
Q4
Which of the following molecule absorb in the IR?
- \(I_2\)
- \(O_2\)
- \(O_3\)
- \(HBr\)
- \(HF\)
- \(H_2 O\)
- \(CD_2\)
- \(CO_2\)
- \(CH_4\)
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 500 nm energy that may be observed in electron (UV-VIS) spectroscopy? What is the energy of a 6-micron photon typical in IR spectroscopy? What is the energy of a photon absorbed in a typical CO rotation microwave line (\(6 \times 10^{11} Hz\))?
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 \(2.6 \times 10^{-47} \;Kg\times m^2\). 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
\(\ce{^1H^{19}F}\) has an equilibrium bond length of 91.7 pm and a spring constant of 970 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 \(\ce{^1H^{19}F}\) ascribed to rotational motion (assuming a rigid rotor described the rotation)?