9.4: Part II - Data Analysis
- Page ID
- 372971
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)4.7.1. Assign the fundamental bands of SO2 as: symmetric stretch, \(\widetilde{\nu}_1\); bend, \(\widetilde{\nu}_2\); asymmetric stretch, \(\widetilde{\nu}_3\) ; and, report their frequencies. Compare your value with those reported in the literature.
4.7.2. Then assign any other bands, comparing the observed frequencies with those calculated from combinations of the fundamental values \(\widetilde{\nu}_1\), \(\widetilde{\nu}_2\), \(\widetilde{\nu}_3\).
4.7.3. Calculate k1 and \(\large \frac{k_{\delta}}{\ell^2}\) from Equations (1) and (3) (on page 385 of Shoemaker, Garland and Nibler) using your values of \(\widetilde{\nu}_1\), \(\widetilde{\nu}_2\), and \(\widetilde{\nu}_3\). Check the valence-force model by calculating both sides of Eq. (2) (also on p. 385 of Shoemaker) and comparing them. Again, compare your results with literature values.
When finding literature values for the force constants of SO2, you will likely notice that different notations are used. It may be difficult to find the k1 and \(\large \frac{k_{\delta}}{\ell^2}\) notation used in Shoemaker, Garland and Nibler (and as such in this manual). For example, Kivelson8 reports these same vales as \(f_d\); and, \(\large \frac{f_a}{d^2}\) and reports the values in units of \(\large \frac{dynes}{cm}\) rather than \(\large \frac{N}{m}\). Be prepared to convert.
4.7.4 Using your values of \(\widetilde{\nu}_1\), \(\widetilde{\nu}_2\), and \(\widetilde{\nu}_1\), calculate \(\tilde{C}_{v(vib)}\) at 298 K and at 500 K from Eq. (8) on page 386 of Shoemaker, Garland and Nibler.
4.7.5 At these same temperatures, calculate \(\tilde{C}_{v}\) from the expression
\[ \tilde{C}_{v}=3R+\tilde{C}_{v(vib)} \]
and compare your results with the following \(\tilde{C}_{v}\) values (obtained from directly measured values of \(\tilde{C}_{p}\) and the expression
\[ \tilde{C}_{v} = \tilde{C}_{p} - R \]
\(\tilde{C}_{v}\) = 30.5 J K–1 mol–1 at 298 K and 37.7 J K–1 mol–1 at 500 K.
4.7.6 Calculate the uncertainty of each result you report in this experiment.