5.10: Problems
- Page ID
- 420511
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\( \newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\)
( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\id}{\mathrm{id}}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\kernel}{\mathrm{null}\,}\)
\( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\)
\( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\)
\( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\AA}{\unicode[.8,0]{x212B}}\)
\( \newcommand{\vectorA}[1]{\vec{#1}} % arrow\)
\( \newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow\)
\( \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vectorC}[1]{\textbf{#1}} \)
\( \newcommand{\vectorD}[1]{\overrightarrow{#1}} \)
\( \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} \)
\( \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} \)
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)- Consider the data given in the table for lines found in the pure rotational spectrum of \({ }^{12} \mathrm{C}^{16} \mathrm{O}\). Determine an approximate value for \(B\) and assign the spectrum (the lower \(\rightarrow\) upper state rotational quantum numbers for each line.) Make a graph of \(\frac{\tilde{v}_{J}}{(J+1)} \) vs. \((J+1)^{2}\) and determine
| line | \(\tilde{v}\left(\mathrm{~cm}^{-1}\right)\) |
|---|---|
| 1 | \(3.84503319\) |
| 2 | \(7.68991907\) |
| 3 | \(11.5345096\) |
| 4 | \(15.378662\) |
| 5 | \(19.222223\) |
| 6 | \(23.065043\) |
the best fit line. Use these results to determine \(\mathrm{B}\) and \(\mathrm{D}\) for the molecule. Compare your results to those found in the NIST Webbook of Chemistry for the ground electronic state of CO.
-
Consider the following data for the rotation-vibration spectrum of \(\mathrm{H}^{35} \mathrm{Cl}\).
a. Using the differences in frequency, assign the location of the band origin and assign the \(\mathrm{P}\) - and R-branches accordingly.
b. Using combination differences, fir the data to find B’, D’, B" and D’.
c. Use your results to find \(B_{\mathrm{e}}, \alpha_{\mathrm{e}}\) and \(\mathrm{De}_{\mathrm{e}}\).
d. Based on your value of \(B_{e}\), find a value for \(r_{e}\) for the molecule.
e. Compare your results to those found in the NIST Webbook of Chemistry.
| line | Freq. \(\left(\mathbf{c m}^{-1}\right)\) | \(\Delta \tilde{v}\) |
|---|---|---|
| 1 | \(3085.62\) | |
| 2 | \(3072.76\) | |
| 3 | \(3059.07\) | |
| 4 | \(3044.88\) | |
| 5 | \(3029.96\) | |
| 6 | \(3014.29\) | |
| 7 | \(2997.78\) | |
| 8 | \(2980.90\) | |
| 9 | \(2963.24\) | |
| 10 | \(2944.89\) | |
| 11 | \(2925.78\) | |
| 12 | \(2906.25\) | |
| 13 | \(2865.09\) | |
| 14 | \(2843.65\) | |
| 15 | \(2821.49\) | |
| 16 | \(2798.78\) | |
| 17 | \(2775.79\) | |
| 18 | \(2752.03\) | |
| 19 | \(2727.75\) | |
| 20 | \(2703.06\) | |
| 21 | \(2677.73\) | |
| 22 | \(2651.97\) | |
| 23 | \(2625.74\) | |
| 24 | \(2599.00\) |
- A recursion formula for the Legendre Polynomials is given by
\[(l+1) P_{l+1}(x)=(2 l+1) x P_{l}(x)-l P_{l-1}(x)\nonumber\]
Based on \(P_{0}(x)=1\) and \(P_{1}(x)=x\) find expressions for \(P_{2}(x)\) and \(P_{3}(x)\).
- The function describing the \(l=1, \mathrm{~m}_{l}=0\) spherical harmonic is \(Y_{1}^{0}(\theta, \phi)=\sqrt{\frac{3}{4 \pi}} \cos (\theta)\)
- Show that this function is normalized. To do this, you must use the limits on \(\theta\) and \(\phi\) of \(0 \leq \theta \leq \pi\), and \(0 \leq \phi \leq 2 \pi\). Also, for the angular part of the Laplacian, \(d \tau=\sin (\theta) d \theta d \phi\)
- Using plane polar graph paper (or a suitable graphing program) plot the square of the function from problem 2 in the \(yz\) plane (which gives a cross-section of the probability function for the particular spherical harmonic.) Does the shape look familiar?
- Based on the given bond-length data, calculate values for the rotational constants for the following molecules:
| Molecule | Bond Length \((\mathring{\mathrm{A}})\) |
|---|---|
| \(\mathrm{H}^{35} \mathrm{Cl}\) | \(1.2746\) |
| \(\mathrm{H}^{79} \mathrm{Br}\) | \(1.4144\) |
| \(\mathrm{H}^{127} \mathrm{I}\) | \(1.6092\) |
- The spacing between lines in the pure rotational spectrum of \(\mathrm{BN}\) is \(3.31 \mathrm{~cm}^{-1}\). From this, find \(\mathrm{B}\) and calculate the bond length ( \(\mathrm{r}_{\mathrm{BN}}\) ) in the BN molecule.
- From your result in problem 6, calculate the frequencies of the first 4 lines in the pure rotational spectrum of BN.