9.3: Part I - Data Analysis

Data Analysis for HCl/DCl FTIR spectra 

Note

This is one of the more laborious data analyses you will do this semester. Let's minimize the number of repetitive tasks to save you some time. The following shortcuts are recommended:

• Collaborate with a partner on steps 1-3 (allowed in this specific case only):You may collaborate with your lab partners in steps 1-3 below. One student should work up the data (steps 1-3 ONLY) for H³⁵Cl and H³⁷Cl, and the second student should do the same analysis for D³⁵Cl and D³⁷Cl. You should then share your graphs of \( \Delta \widetilde{\nu}_{(m)} \) versus \(m\) and the values determined for the slopes and y-intercepts of these plots. Each student should do one complete analysis (either HCl or DCl) alone, and the other analysis can be the work of a partner. However, you are responsible for all work that is in your ELN, so be sure your partner's work is correct. *Steps 4 and onward should be completed individually, as usual.
• Calculate errors for one isotope only: The propagation of errors that you calculate for one isotope should similar (perhaps not identical, but close enough) as for another isotope. Save time by propagating errors carefully for (\(B_e\), \(\alpha_e\), \(\widetilde{\nu}_{o}\), Ie, re, and k for one isotope, then use those errors as an approximation for the errors for analogous values for other isotopes.
• Use what you know! Do you remember that MatLab Introduction module way back in the beginning of this course? It was a long time ago, but it was a template for data analysis in this module! Go back and use what you already know, and what you already created as a template, for this data analysis.

Collaborate with a partner 

1. Label each peak in your spectra as "P" or "R" and its J-value as: P(1), P(2),..., R(O), R(1),... etc., as shown in Figure 2 on page 399 of Shoemaker, Garland and Nibler. (Note: Spectra recorded using your iS50 FTIR spectrometer are plotted with the wavenumber (abscissa) decreasing from left to right.)
2. For each of the four isotopic combinations (H³⁵Cl, H³⁷Cl, D³⁵Cl and D³⁷Cl), make a table of the m values and the corresponding peak frequencies \( \widetilde{\nu}_{(m)} \). For each isotopic combination, calculate the separation between adjacent peaks.\[ \large \Delta \widetilde{\nu}_{(m)}=\widetilde{\nu}_{(m+1)}-\widetilde{\nu}_{(m)} \]

   Note

   \( \Delta \widetilde{\nu} \)_(m) cannot be calculated for m = –1 since there is no peak for m = 0. Therefore m = –1 must be deleted from the vector of m values.
3. Plot the differences between adjacent absorption frequencies (i.e., \( \Delta \widetilde{\nu}_{(m)} \) versus \(m\). Then perform a linear least squares fit of the data with the understanding that
   \[ \large \Delta \widetilde{\nu}_{(m)}=\widetilde{\nu}_{(m+1)}-\widetilde{\nu}_{(m)}=\left ( 2B_e - 3 \alpha_e\right )-2 \alpha_e m  \label{fit}\]
   Use the standard deviations of the slope and intercept in your error analysis (Q14).

Work on your own 

4. Compute \( B_e \) and \( \alpha_e \) for each isotope from the intercept \( \left ( 2B_e - 3 \alpha_e\right ) \) and the slope \( -2 \alpha_e \) of each best fit line (Equation \ref{fit}). 
5. Calculate the frequency of the forbidden transition \(\widetilde{\nu}_{o}\) using your calculated values of \( B_e \) and \( \alpha_e \). To do this, use equation 9 in expt 37 of Shoemaker, Garland and Nibler. The equation is \(  \widetilde{\nu}_{m} = \widetilde{\nu}_{o} + (2B_e -2 \alpha _e ) m - \alpha_e m^2\). Calculate \(\widetilde{\nu}_{o}\)  from several of the lines with low \(m\) values. Calculate the average value, and standard deviation, of \( \widetilde{\nu}_{o} \).

   Note

   You will get the best results by using only the lines closest to the center (e.g., m = ±1, ±2).
6. Create a table of \( \widetilde{\nu}_{o} \), \( B_e \), and \( \alpha_e \) for the four isotopic combinations The units should be cm^–1.
7. Calculate the moment of inertia (\(I_e\)), and the internuclear distance (a.k.a. equilibrium bond length, \(r_e\)) for all four isotopic combinations.
8. Calculate the harmonic oscillator force constant, \(k\), for each isotopic combination from your calculated values of \( \widetilde{\nu}_{o} \). In doing so you will need to assume that \( \widetilde{\nu}_{o} \) and the harmonic oscillator frequency \( \widetilde{\nu}_{e} \) are the same, thus ignoring the effects of anharmonicity (see the formulas and discussion on pages 400 and 401 of Shoemaker, Garland and Nibbler).
9. Compare your values of r_e and k with those reported in the literature. Why are these values independent of isotopic substitution?
10. Determine the isotope effects: Compute the ratios \( \large \frac{\widetilde{\nu}_{o}^*}{\widetilde{\nu}_{o}} \) and \( \large \frac{B_{e}^*}{B_e} \) for ³⁵Cl and ³⁷Cl in the case of HCl, and again in the case of DCl. Compare these with the predicted values given by Eqs. (11) and (13) on page 400 of Shoemaker, Garland and Nibbler.
11.  From the results in 10, above, explain why is the ³⁵Cl /³⁷Cl isotope effect so much larger in DCl than it is in HCl? That is, when looking at your FTIR spectrum, the ³⁵Cl/³⁷Cl line separation is significantly greater in the DCl case then when compared to the HCl case. A descriptive answer is a good start, but your experimental observations should also be supported with a calculation that justifies the experimental observation.
12. Apply the Boltzmann distribution to explain observed band intensities in terms of the populations of the ground-state levels (see section 4.2). Again, a descriptive answer is a good start, but it should be supported with calculated results. You might create a spreadsheet to do these calculations. Does a calculation based on the Boltzmann population of rotational states match what you see experimentally in terms of which rotational lines show the greatest intensity?
13. From the mean line intensities for at least 10 features (for example five or more lines from the HCl spectrum and five or more from the DCl spectrum for a total of at least 10), determine the relative abundance of the two Cl isotopes. Estimate the weighted average atomic mass of Cl (that you might see on a periodic table for example) assuming that ³⁵Cl and ³⁷Cl have masses of 34.968 and 36.956 amu, respectively. Compare your result with the reported atomic mass of Cl.
14. For one isotope only, calculate the estimated uncertainty in each of the parameters determined for this experiment (\(B_e\), \(\alpha_e\), \(\widetilde{\nu}_{o}\), I_e, r_e, and k). (Another table would be helpful)
15. Why is it important to purge the sample compartment of the FTIR spectrometer for both the background and sample runs? Why is it important to purge both background and sample for the same amount of time?