1. Rotational-Vibrational Analysis of FTIR Spectra of the
Isotopic Variations of Hydrochloric Acid
Jiovanni Nuñez, Sanuja Mohanaraj, Rachel Dufour
Abstract
Using IR spectra of both isotopes of hydrochloric acid and deuterium chloride, the rotational
constant B, the moment of inertia and the bond length were able to be calculated. From there,
each peak in the spectra was assigned to a certain transition in the J quantum number for each
diatom. From these transitions, a plot of the transitions vs the wavenumbers they occur at were
created and used to determine the vibronic constants of the diatoms.In general, asside from the
anharmonicity constant, the constant Ae for deuterium chloride, and De, the values for the
constants calculated in this experiment are very close to those determined in other experiments
and published in the NIST Webbook.
Introduction

2. The rigid rotor is a mechanical model used to explain rotating systems. In quantum
mechanics, the linear rigid rotor is used to approximate the rotational energy of systems such as
diatomic molecules. From the rotational energy, the bond length and the reduced mass of the
diatomic molecule can also be calculated. The energy of the rotation diatom is known as the
rotational constant B, which is expressed by the formula B = ħ2/2I, where the variable I
represents the moment of inertia. From the formula for the moment of inertia, I = μR2, the bond
length R and the reduced mass μ can be determined. For this experiment, both isotopes of
hydrochloric acid and deuterium chloride were used to demonstrate the real-world applications
of the quantum mechanical rigid rotator model on diatomic molecules. The vibronic constants
will also be determined for each diatom by fitting a plot of the transitions in the J quantum
number vs their assigned wavenumbers with a third-order polynomial trendline. From the
trendline, the constants can be determined using the formula presented in the Shoemaker
handout1,
�(m) = �0 + (2B - 2αe)m - αem2 - 4Dem3
In the FTIR spectra collected, each peak represents the energy of the various transitions in the
rotational quantum number J, with the spacing between the two middle peaks representing an
energy equal to 4B. From these spectra, the goal of this experiment is to accurately calculate the
bond length, rotational constant, force constant and available rotational transitions occurring in
the diatomic molecules studied, as well as their characteristic intramolecular constants.
Experimental
All tests were run on a Nicolet 6700 FT-IR spectrophotometer, manufactured by Thermo
Scientific. A rubber capped fused quartz gas cell was flushed with nitrogen in order to produce a

3. background scan. After the background was taken, hydrochloric acid vapors were collected from
a vial and deposited into the cell using a syringe.After the HCl spectrum was taken, the cell was
again flushed with nitrogen gas and resealed. Deuterium chloride vapors were then deposited
into the cell using a clean syringe, and the DCl spectrum was taken.
Results and Discussion
From the collected infrared spectra, we were able to calculate the value of the rotational
constant B for each diatom. The space in the Q-branch in the center of both spectra, where ΔJ =
0, is assumed to equal 4B.
Table 1: Rotational Constant and Bond Length Calculations
From the rotational constant, we were then able to calculate the moment of inertia I for
each molecule using the formula B = h/B(8π2cI), where h is the Planck constant and c is the
speed of light in centimeters per second2. From the formula I = μR2, where μ is the reduced mass
of the diatom, the bond length R was then calculated. The final calculated bond lengths were
0.131 nm for both isotopes of hydrochloric acid, and 0.133 nm for both isotopes of deuterium
chloride, with margins of error of ±0.01nm for HCl and ±0.01 nm for DCl. The National Institute
of Standards and Technology lists the bond length for both HCl and DCl to be 0.12746 nm3,4,
leading to a 2.78% for hydrochloric acid, and 4.35% for deuterium chloride.
The next step in the analysis of the FTIR spectra was to assign the transitions in the
rotational quantum number J to the spectra’s peaks. In the DCl spectrum, the horizontal axis is in

4. descending order from left to right, indicating that the P-Branch, where ΔJ = -1, is on the right
side, and the R-Branch, where ΔJ = 1, is on the right side. In the HCl spectra, since the horizontal
axis is in ascending order, the orientation of the branches is reversed. In order to assign
transitions on each spectra, the empty space in the center is assigned to the ΔJ = 0 transition.
From there, the peaks from the center, outwards are assigned to consecutive positive and
negative transitions. The transitions in the rotational quantum number for the isotopes of DCl are
displayed in Tables 3 and 5, and those for HCl are displayed in Tables 4 and 6 in the
Supplementary Materials section. The margin of error for the wavenumber of the transitions in
the DCl spectra is ±0.05 cm-1, and the margin of error in the HCl spectra is ±5 cm-1.
In order to determine the characteristic intramolecular constants of each diatom, plots of
wavenumber vs transition were made, and fitted with a third-order polynomial trendline. The
equation for each trendline was then paired with the formula �(m) = �0 + (2B - 2αe)m - αem2 -
4Dem3, in order to determine the vibronic constants. From the equations �0 = �e - 2�eXe and the
isotope effect (μ/μ*) = (�eXe
*/�eXe), the anharmonicity constants for H35Cl and D35Cl can be
determined. By combining the two equations, a new formula is produced;
�eXe = (V*
0 - V0(μ/μ*)1/2 ) / (2(μ/μ*)1/2 - 2(μ/μ*))
From the constant �e, the spring force constant K or the diatomic molecular bond can also be
quantified, using the formula �e = (1/2πc)(K/μ)^(½).
Table 2: Calculated Vibronic Constants

5. Since spectra were not run for tritium chloride, we were unable to properly calculate the
anharmonicity constant, not the spring force constant for the deuterium chloride isotopes. The
values for the vibronic constants presented in Physical Chemistry: Methods, Techniques and
Experiments5 appear to generally coincide with the values calculated from the data produced in
this experiment. This is with the exception of the constants De, �eXe , and Ae for deuterium
chloride. The exact percent differences between the literature and experimental values is
displayed in Table 7.
Summary
From the spectra collected from both isotopes of hydrochloric acid and both isotopes of
deuterium chloride, the bond lengths were successfully calculated, in addition to the moment of
inertia and the rotational constant B. The peaks in each spectra were then assigned to a particular
transition in the J quantum number, and using that data, plots of the transitions vs the
wavenumbers they occur at were created. To the plots, third order polynomial trend lines were
fitted, and used to determine the vibronic constants for each diatom. Since tritium chloride was
not used in this experiment, the anharmonicity constant and the spring force constant were only
able to be calculated for the isotopes of hydrochloric acid. Based on the values collected from
the NIST Webbook and the Shoemaker handout, it appears that the constants calculated in this
experiment correlate with the accepted values.
References
1 = Shoemaker Handout, Pg. 4
2 = Barrow, G. M. Introduction To Molecular Spectroscopy; McGraw Hill: New York, NY,
1962. Pg. 52

6. 3 = Listing of experimental data for HCl. Computational Chemistry Comparison and Benchmark
Database, http://cccbdb.nist.gov/exp2.asp?casno=7647010.
4 = Listing of experimental data for DCl. Computational Chemistry Comparison and Benchmark
Database , http://cccbdb.nist.gov/exp2.asp?casno=7698057.
5 = Sime, R.J. Physical Chemistry: Methods, Techniques and Experiments, 1990, pp. 676 – 687,
Saunders College Publishing, Philadelphia, PA.
Supplementary Material

7. Figure 1: FTIR Spectra of DCl
Figure 2: FTIR Spectra of HCl

8. Table 3: J Transitions of D37Cl

9. Table 4: J Transitions of H37Cl

10. Table 5: J Transitions of D35Cl

11. Table 6: J Transitions of H35Cl

12. Figure 3: D37Cl Wavenumber vs Transition Plot
Figure 4: H37Cl Wavenumber vs Transition Plot

13. Figure 5: D35Cl Wavenumber vs Transition Plot
Figure 6: H35Cl Wavenumber vs Transition Plot

14. Table 7: Percent Error in Calculated Vibronic Constants