HCl report.pb 1 Vibration-Rotation Spectrum of HCl Introduction HCl absorbs radiation in the infrared portion of the spectrum which corresponds to the molecule changing its vibrational state. A concommitant change in rotational state also occurs. In this exercise we are given a spectrum taken with a FTIR spectrometer of fairly high resolution and our objective is analyze the spectrum to obtain information about the spacing between the rotational level and the vibrational levels, from which we will ultimately extract the bond length of the molecule. Experiment The experimental spectra were given to us. Data The experimental spectra are attached. According to the procedure in SGN, each line is assigned a running index, m, and the m values and frequencies are tabulated in table I of the appendix. Theory The term values (energy levels expressed as wavenumbers, cm -1 ) for a rotating & vibrating molecule are given by SGN as T(v,J) = E(v,J) hc =ν ~(v+ ε 1 / 2 ) -ν~x ε e (v+ 1 / 2 ) 2 + B e J(J+1) -D e J 2 (J+1) 2 - α e (v+ 1 / 2 )J(J+1) In the above equation, the first term represents the energy of a harmonic oscillator, the second a correction for vibrational anharmonicity, the third is the energy of a rigid rotor, the fourth a correction for centrifugal distortion (the bond stretching a bit as the rotation is increased), and the last term represents a vibration-rotation interaction. The rotation of a rigid rotor is expressed in terms of the rotational constant B e where h B e = 8π 2 I e c where I e= µr e 2 and µ is the reduced mass, µ= m 1m 2 m 1 +m 2 The infrared transition take place from v=0 to v=1 with a concommitant change of ±1 in J. If J decreases, we have a P branch transition, and if J increases, an R branch transition. The frequency of a given v,j transition ν is thus the difference in term values or ν = T(v=1,J')-T(v=0,J") where (by convention) single prime denotes the upper state and double prime the lower. We find for the R branch ν R = T(1,J"+1) - T(0,J")=ν~ e - 2ν~ exe + (2B e -4α e )J"+(2B e -3α e )- 4D e (J+1) 3 - α e J 2 and for the P branch
HCl report.pb 2 ν P = T(1,J"-1) - T(0,J") = ~ ν ε -2ν ~ ε x e -(2B e -2α e )J" -α e J" 2 +4D e J" 3 now, let m=j"+1 for the R branch (J"=m-1) m=-j" for the P branch ν P,R = ~ ν ε -2ν ~ ε x e +(2B e -2α e )m -α e m 2-4D e m 3 ν R,P = (2B e -3α e - 4D e )- (2α e - 12D e )m- 12D e m 2 If we let D e = 0 we get ν R,P = (2B e -3α e )- 2α e m So a plot of ν R,P vs m should be a straight line with slope = -2α e and intercept (2B e -3α e ). These plots are shown in the appendix. The values of B e and α e are tabulated in Table I. The values of B e and a do not lie with the 95% CL because we have neglected D e. We may estimate D e by taking the second diferences in v and plotting them. Thus ( ν R,P ) = (2α e - 24D e )- 24D e m we plot the second difference in Fig 3 and find D e =.00048±.00024. We can then go back to the equation ν R,P = (2B e -3α e - 4D e )- (2α e - 12D e )m- 12D e m 2 but we now regard D e as a known correction. Replotting ν (corrected) vs m in Fig 4 yields a vastly improved fit and essentially the correct spectroscopic constants. TABLE I Experimental Molecular Parameters H 35 Cl D 35 Cl D 37 Cl B e (cm -1 ) (D e =0) 10.404±.068 5.37±.04 5.37±.04 D e =.00048±.00024 10.586±.024 α e (cm -1 ) 0.302±.008 0.113±.004 0.114±.004 0.307±.008 ν 0 (cm -1 ) 2886.35±0.33 2091.3±1.5 2088.5±1 I e 10 40 (g-cm 2 ) 2.6906 5.2128 5.2128 (2.6443) r e (Å) 1.2861±.004 1.2839±.005 1.2820±.005 (1.2750±.0007) Literature Values: a B e (cm -1 ) 10.59341 6 5.448794 D e 10 4 (cm -1 ) 5.3194 1.39 α e (cm -1 ) 0.30718 1 0.113291 r e (Å) 1.27455 2 1.274581 a K.P. Huber and G. Herzberg, Molecular Spectra and Molecular Structure, Vol. IV, Constants of Diatomic Molecules (Van Nostrand, New York, 1979).
HCl report.pb 3 Table II Comparison of Isotope ratios B*/B µ/µ* ν /ν µ/µ* HCl/DCl 1.9374 1.9440 1.380 1.394 D 35 Cl/D 37 Cl 1 1.003 1.0013 1.0015 Discussion Errors If D e is ignored, the slope of ν vs m is 2α, so σ slope = 2σ α. Likewise, the intercept is (2B-3α) and σ 2 B = σ int 2 ( B/ int) 2 + σ α 2 ( B/ α) 2 = σ int 2 (1/2) 2 + σ α 2 (3/2) 2. Likewise, σ 2 r = σ B 2 ( r/ B) 2 = σ B 2 (r/2b) 2. Our result with a ±95% CL means that if an infinite number of measurements were made, that there is a 95% chance that the mean of the parent distribution would lie within the stated limits. If the parent is Gaussian, then 95% of measurement will lie within ±2σ of the mean, so we want to quote 2σ as our uncertainty, and these are the numbers cited in Table I. The values of B and r e which have been extracted do notagree within the error bars with the values in Herzberg & Huber. This is because there is a systematicerror in the experiment, which results from using an incorrect or inadequate theory. We have neglected the terms in D; if these are included the fit becomes much better and the value of B and r e agree within limits to those in H&H. Questions 1. Results agree if correction for D is made. See above. 2. There is a Cl isotope effect, and D 37 Cl was analyzed. The deviations from D 35 Cl are in good agreement with theory. 3. If the gas temperature were 100K instead of 300K, we would expect the population of gas molecules in high J states to be much less, and the population would shift towards low J states. Thus, the intensities of the lines at the extremes of the spectrum would be decreased and the intensities of the lines closest to the "gap" would increase.
HCl report.pb 4 m*10-3 wavenumbers*10-3 D 35 Cl D 37 Cl HCl -0.0160 1.8940 NaN NaN -0.0150 1.9080 NaN NaN -0.0140 1.9216 1.9188 NaN -0.0130 1.9346 1.9324 2.5719-0.0120 1.9479 1.9457 2.5990-0.0110 1.9611 1.9584 2.6258-0.0100 1.9738 1.9715 2.6519-0.0090 1.9865 1.9841 2.6779-0.0080 1.9989 1.9962 2.7030-0.0070 2.0113 2.0085 2.7278-0.0060 2.0232 2.0207 2.7521-0.0050 2.0353 2.0325 2.7757-0.0040 2.0469 2.0441 2.7989-0.0030 2.0584 2.0555 2.8216-0.0020 2.0694 2.0665 2.8437-0.0010 2.0805 2.0777 2.8651 0 NaN NaN NaN 0.0010 2.1019 2.0989 2.9064 0.0020 2.1120 2.1090 2.9259 0.0030 2.1223 2.1193 2.9451 0.0040 2.1321 2.1291 2.9634 0.0050 2.1417 2.1387 2.9810 0.0060 2.1512 2.1481 2.9982 0.0070 2.1602 2.1571 3.0146 0.0080 2.1693 2.1658 3.0302 0.0090 2.1780 2.1745 3.0453 0.0100 2.1861 2.1830 3.0597 0.0110 2.1945 2.1909 3.0732 0.0120 2.2023 2.1988 3.0859 0.0130 2.2100 2.2064 3.0981 0.0140 2.2171 2.2140 NaN 0.0150 2.2244 2.2208 NaN 0.0160 2.2312 2.2276 NaN 0.0170 2.2378 NaN NaN
HCl report.pb 5 28 26 24 D=0 a=0.301693 ±0.00434851 B=10.4038 ±0.0346292 22 20 18 HCl 35 16 14 12-15 -10-5 0 5 10 15 Fig 1. Plot of ν vs running index, m, for H 35 Cl. Values of a and B are calculated from the slope and intercept according to last equation on p2. 16 14 D=0 ±0.000248733 a=0.11265 ±0.00171098 B=5.40685 ±0.0170011 12 10 DCl 35 8 6-20 -15-10 -5 0 5 10 15 20 Fig 2. Plot of ν vs running index, m, for D 35 Cl. Values of a and B are calculated from the slope and intercept according to last equation on p2. A similar plot results for D 37 Cl.
HCl report.pb 6 0.2 second differences 0-0.2-0.4 y = a + bx a=-0.602388 sa=0.045789 b=-0.0114792 sb=0.00591133-0.6-0.8-1 -1.2-15 -10-5 0 5 10 15 Fig 3. Plot of second differences vs m. The slope is -24D e. It is obviously not well determined, but it allows us to make a small correction to refine B e and α. 30 28 26 24 y = a + bx a=20.2507 sa=0.0246333 b=-0.609125 sb=0.00314968 22 20 18 16 D=0.0004783 ±0.000246305 a=0.307432 ±0.00157484 B=10.5855 ±0.0125412 14 12-15 -10-5 0 5 10 15 Fig 4. Plot of first differences vs m after correction has been made for D e.