The Infrared Overtone Intensity of a Simple Diatomic: Nitric Oxide Brian J. Bozlee' Whitman College, Walla Walla, WA 99362 Joseph H. Luther and Mark Buraczewski Saint Vincent College, Latrobe, PA 15650 Most laboratory exercises in molecular spectroscopy focus on the information gained from the frequencies, rather than the intensities, of spectral lines. In contrast, we describe here a laboratory exercise that illustrates the principles of calculating and measuring the relative intensities of spectral bands. Specifically, the intensity of the first infrared overtone band of nitric oxide is compared to the intensitv of the nitric oxide fundamental band. The results of the measurements are compared with a simplified theorv in which the anharmonic contributionsof the vibrationai wave functions are used to calculate the necessary transition-moment integrals, and thus to predict the relative intensities of the two hands.
Table 1. Values of Integrals for Harmonic Oscillator Wavefunctions 0
1
2
3
4
5
6
7
8
...
Background and Theory The intensity Iirof an infrared absorption from state i to state f is proportional to the square of the transition-moment integral < i I M lf > and to the frequency ( 1 ) of the transition vif as shown in eq 1. I~
ir
"i)~v~~
(1)
where M is the dipole moment of the molecule. The dipole moment is a function of the bond extension x from the equilihnum valuc and is usual1.y represented as a Taylor-series expansion.
where the subscript e indicates a quantity that is evaluated at the equilibrium bond distance. The transition-moment integral then becomes (noting that < i I f > is zero for orthogonal wave functions)
If we assume that both < i I and If > are pure harmonic-oscillator wave functions, we find that the value of the first term in eq 2 is nonzero for the fundamental transition (i = 0 and f = 1) and that it is identically zero for the first overtone (i = 0 and f = 2). This is easily verified using the integrals provided in Table 1. Thus, the overtone has no intensity unless we resort to the second term in eq 2 for which the overtone integral is small but nonzero. The contribution to the overtone intensity due to the second term is called the electronic contribution. In the case of nitric oxide, the values of the first and second dipole moment derivatives, (dMl&), and (d2M/dr2),,respectively, have been 'Author to whom correspondence should be addressed. 370
Journal of Chemical Education
0
1
2
3
4
5
6
7
8
...
0 1 =
3
Example:
< 1 1x1 2 >
=
12 element of x matrix
= @@,uo)~
note:
= 3 =W2n
shown to be nearly the same in magnitude (2). Since the < 0 lx212 z integral is much smaller than the < 0 I x I 1> integral (Table I), it is clear that the the electronic contribution to the overtone intensity is negligible in this case, and we may safely ignore it. The second way in which the overtone gains intensity is if the wave functions < i I and If > are not assumed to be pure harmonic-oscillator wave functions. In this case, only the first term of eq 2 must be used, and the resultant contribution to the overtone intensity is called the anharmonic contribution. The form of the anharmonic wave function is normally represented as a linear combination of harmonic-oscillator wave functions according to the prescription of first-order perturbation theory (3). In this approach, the internuclear potential energy function U(x) is expanded as a Taylor series in the bond extension from equilibrium.
Experimental A home-built, high-pressure IR gas cell (Fig. 1) was flushed several times with nitrogen and evacuated after each filling. The cell was then filled with 3 atm of nitrogen, and the brass valve was closed tiehtly. The cell was removed fmm the fillingasaembly and po&ioned for a backmound TI< scan, wine a Mattson Polaris FTIR. All suhsequent IR scans of s ~ m p l e scontaining nitric oxide were ratioed to this backwound. The cell was thenieevacuated in a hood and flushed several times with nitric oxide (Matheson, C.P., 99% pure). Evacuation after each flush was done with a simnle water aspirator setup in order to avoid damage to vacutm pumps and to minimize the release of toxic vapor in the hood. Finally, the cell was filled with nitric oxide to a pressure of 3 atm.
The first two terms of the expansion are zero at the bottom of the intermolecular potential well. The third term is identical to the normal harmonic-oscillator potential energy. The last two terms of the above equation are regarded as corrections or perturbations to the harmonic oscillator Hamiltonian. The perturbation Hamiltonian is then given by
Using formulas found in Levine (4) and spectral constants found in Herzberg (51, we calculated the values of the derivatives. They are listed in Table 2, along with other spectroscopic quantities of interest. The anharmonic wave functions for the ground state and the first two excited states can be evaluated by the following formula.
&
-114 inch Comer , ,
4 Ste
where N is a normalization factor, and the superscripted zero (O) indicates unperturbed harmonic-oscillator wave functions. The Ej's are harmonic-oscillatorenergies
Using eqs 3,4, and 5, we get the wave functions for the ground state and the first two excited states.
The above wave functions may be inserted into the first term of eq 2 to calculate the 0 + 1and 0 +2 transition-moment integrals, with the aid of the integrals provided in Table 1.The relative intensities for the two transitions are then found by eq 1. In this manner, the ratio of the intensities of the fundamental band to the first overtone band of nitric oxide is found to be 130. This calculated result is compared to the experimental data of students and to the literature in the discussion section.
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A survey IR scan of the gas revealed a strong fundamental band at 1876 cm-' and a weak overtone band a t 3724 cm-' (Fig. 2). In addition, a few contaminant peaks were observed. However, these were small in comparison to the nitric oxide fundamental, and they did not overlap the principal bands of interest. Both bands of the stable free radical reveal a rich P, Q, R branch structure. This is in contrast to closed-shell diatomics, such as HCI, in which the Q branch is forbidden (6). At an instrumental resolution of 0.5 em-', the P and R lines of the fundamental band of nitric oxide were seen to be doublets. These result from the overlap of rotation-vibration transitions within two electronic states, 2n12and 'ni2 hoth ofwhich are populated at mom temperature (51. s&&the molecularcon&~ntsfor both electroiic states are nearly the same, they were treated as one, single, electronic surface for our purposes. Volume 69 Number 5 May 1992
371
Table 2. Molecular Constants for Nitric Oxide (NO)a Ve
rn
(a3uia&
(a4u~a~4)e Ir
Figure 2. lnfrared spectrum of nitric oxide. Pressure is 3 atm; pathlength is 1.5cm. a. Two hundred scans forbackground andsamle; 0.5 cm-' resolution. Fundamental band is observed at 1876 cmand the first overtone band is observed at 3724 cm?. Other small bands are due to contamination or impurities in the gas sample. b. Expanded view of the overtone region.
4
As can be seen fmm Figure 2, a t 3 atm of nitric oxide, the overtone intensity (i.e. the area under the entire band) could be estimated, but the fundamental band was too intense for its area to be determined realistically. Thus, a second scan of the nitric oxide fundamental band a t lower pressure was required. Figure 3a shows the fundamental band for 0.5 a h of nitric oxide with no buffer gas. Figure 3b shows the same band when 2.5 atm of nitrogen buffer gas was added. Comparison of Figures 3a and 3b clearly demonstrates the ef-
Frequency (cm-I) Figure 3. lnfraredspectrum of the fundamental band of nitric oxide. Pathlength is 1.5 cm. a. Pure nitric oxide; 0.5 atm; 200 scans; 0.5 cm-' resolution. b. 0.5 atm nitric oxide with 2.5 atm nitrogen; 200 scans; 0.5 cm-' resolution. Note broadening of rotational lines due to higher pressure. 372
Journal of Chemical Education
5.70865 x Iol3 s-' vibrational frequency 2me = 3.58685 x IO'~S-' angular frequency =-I .25 x lot4~ l m ~ 7.7 x IoZ4~ l m ~
= 1.240 x 1o - kg ~ ~ %om tables in ref 5 and formulae in ref 4
reduced mass
fect of collisional broadening of the rotational lines at high pressure. To determine the ratio of the intensity of the fundamental band to the intensity of the overtone band for nitric oxide, the area of the fundamental band a t a total pressure of 3 atm (2.5 atm Nz and 0.5 atm NO) was compared to the area of the overtone band for pure nitric oxide at 3 atm. The total pressure was kept high in both scans to make the natural spectral line width comparable to the instrumental resolution. This minimized errors due to artificial instrumental broadening of sharp lines (2). The experimentally measured ratio of the fundamental band intensity to the overtone band intensity was then given by 101 -Ioz
larea of fundamental) (area of overtone)
where the factor of 6 corrects for the difference in the partial wessures ofthe nitric oxide in the two scans. Values of student measurements of the intensity ratio were 50-100. Discussion Chandraiah and Cho (2) report a ratio of 55.76 for the intensity of the fundamental band to the intensity of the first overtone band of nitric oxide, based on integrated band intensities a t high pressures. This is within the broad ranee of student results (70 f 25). One maior source oferror inihe student results was aircontamina;ion in the s a m ~ l ecell due to insufficient flushine or evacuation before filling. The nitric oxide immediatery formed NOz with the air, thus causing a decrease in the nitric oxide band intensities. Interference of water vapor, etc., in the overtone region was a small problem. However this was minimized using the nitrogen-filled cell for the background scan. As can be seen from Figure 2, the overtone band is still quite small at 3 atm of nitric oxide, which leads to obvious difficulties in accurate measurement of its area. However, thoughts of increasing the gas pressure or increasing the path length in order to improve the signal-to-noise ratio were dismissed on the basis of safety considerations. The use of only moderate gas pressures avoided the danger of explosive breakage of the salt windows of the IR cell. Similarly, since nitrogen oxides are toxic, it was deemed wise to work with only small quantities of nitric oxide outside of the hood. The gas cell was designed to have a short pathlength and a correspondingly low sample volume. The apparently large difference between the calculated fnndamental-to-overtone intensity ratio of 130 and the literature value of 55.76 is readily attributed to simplifications in the calculation ~rocedures.For instance. we calculated the intensity ratio for pure vibrational transitions, dlsrceardine the fan that the vibrational bands reallv encompass a number of vibration-rotation transitions. The intensity of each of these lines ideally should be calculated individually, but such a calculation would go beyond the scope of a normal laboratory exercise. Although we are, to some extent, comparing calculated apples with experimental oranges, the agreement in overall magnitude is gratify-
-
-
ing, and the calculations remain within the reach of undergraduate students in chemistry and physics.
Literature Cited I. enb berg, G.~ o b c u l o sr p d m and ~ a l o c u l o struburn; r van ~ o s t r a n d~: e ~orh w 1945VoL 2.0 261. 2. Chandr&h, O:;Cho, C. W. . IMokc Spechosc. 1913.47.134-147.
Acknowledgment
The authors would like to thank Donald O'B-on at Of the IR Whitman "'lege for assistance in cell.
3. ~ e v h *I . N . ~ u o n l v mchemistry: A I I Y ~a n d ~ a c o n~: o s t a n1974; . pp 178-182. 4. Levhe, I . N . Moleevlar Spdmaeopy; Wiley and Sona: New Yorh 1975;p 155. 5. Hemberg, G. ~ o b c v l o rS p c t m s m ~a d ~olpclrlorStrudure; van ~ o s h a n d ~: e w Y d ,1979:Vol.4, p 467. 6.Shoemaker,, . ;W m d C W.; Nibler. W,E z p r i m t s in Physiml Ckmisfry: M C G ~ ~ W - HN~~ ~ I I :ymk, 1989;pp 461-465.
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