4.8: Problems
- Page ID
- 420501
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)- For each molecule, calculate the reduced mass (in kg) and the force constant for the bond (in N/m).
| Molecule | \(\omega_e \) (cm \({}^{-1}\) ) | \(\mu\) (kg) | k (N/m) |
|---|---|---|---|
| \(^{1} H ^{79} Br\) | 2648.975 | ||
| \(^{35} Cl _{2}\) | 559.72 | ||
| \(^{12} C ^{16} O\) | 2169.81358 | ||
| \(^{69} Ga ^{35} Cl\) | 365.3 |
- The typical carbonyl stretching frequency is on the order of 1600-1900 \(cm ^{-1}\). Why is this value smaller than the value of \(\omega_e \) for \(CO\) given in the table above?
- The first few Hermite polynomials are given below.
| v | \(H_ {v} (y)\) |
|---|---|
| 0 | 1 |
| 1 | \(2y\) |
| 2 | \(4y ^{2} – 2\) |
\(H_ {v+1}(y) = 2yH_{v} (y) – 2vH_ {v-1} (y)\)
- Use the recursion relation to generate the functions \(H_ {3}\) (y) and \(H_ {4}\) (y).
- Demonstrate that the first three Hermite polynomials (\(H_ {0}\) (y), \(H_ {1}\) (y) and \(H_ {2}\) (y)) form an orthogonal set.
- The Morse Potential function is given by
\[U\left(x\right)=D_e\left(1-e^{-\beta x}\right)\nonumber\]
where \(x = (r – r _{e})\).
- Find an expression for the force constant of a Morse Oscillator bond by evaluating
- For \(^{1} H ^{35} Cl\), \(D _{e} = 7.31 \times 10 ^{-19} J\) and \(\beta = 1.8 \times 10 ^{10} m^{-1}\). Use your above expression to evaluate k for the bond in HCl.
- On what shortcoming of the Harmonic Oscillator model does the Morse Potential improve? What shortcoming does the Morse model share with that of a Harmonic Oscillator?
- The following data are observed in the vibrational overtone spectrum in \(^{1}H ^{35} Cl\) (Meyer & Levin, 1929).
| \(v’ \leftarrow v”\) | \({\widetilde{\nu }}_{obs}\) (\(cm ^{-1}\) ) |
|---|---|
| \(1 \leftarrow 0\) | 2885.9 |
| \(2 \leftarrow 0\) | 5666.8 |
| \(3 \leftarrow 0\) | 8347.0 |
| \(4 \leftarrow 0\) | 10923.1 |
| \(5 \leftarrow 0\) | 13396.5 |
From these data, calculate a set of \(\Delta G _{v+\frac{1}{2}}\) values. Fit these results to the form
\[\mathrm{\Delta }G_{v+\frac{1}{2}}={\omega }_e-2\ {\omega }_ex_e(v+1) \nonumber\]
to determine values for \(\omega_e \) and \(\omega_ex _{e}\) for \(HCl\).
- The following wavenumber frequencies are reported for the band origins for the \(1 – v”\) bands in an electronic transition of a diatomic molecule. Using the Birge-Sponer method, determine the dissociation energy of the molecule in its ground electronic state.
| v" | Wavenumber (\(cm ^{-1}\) ) | \(\Delta G _{v+\frac{1}{2}}\) (\(cm ^{- 1}\) ) |
|---|---|---|
| 19586.9 | ||
| 19522.3 | ||
| 19504.8 | ||
| 19465.9 | ||
| 19418.3 | ||
| 19375.1 | ||
| 19323.2 | ||
| 19275.7 | ||
| 19223.8 | ||
| 19167.6 | ||
| 19111.4 | ||
| 19050.9 | ||
| 18990.4 | ||
| 18925.6 | ||
| 18860.7 | ||
| 18795.9 | ||
| 18722.4 | ||
| 18653.3 | ||
| 18579.8 | ||
| 18506.3 | ||
| 27 | 18428.5 | |
| 18342.1 | ||
| 18259.9 | ||
| 18177.8 | ||
| 18091.5 | ||
| 17996.3 | ||
| 17909.8 | ||
| 17814.8 | ||
| 17719.7 | ||
| 17624.6 |