Lab Report on Rovibrational Spectrum of Hydrogen Chloride

The lab report below was submitted as part of the coursework for CM2101 Principles of Spectroscopy. Please do not plagiarise from it as plagiarism might land you into trouble with your university. Do note that my report is well-circulated online and many of my juniors have received soft copies of it. Hence, please exercise prudence while referring to it and, if necessary, cite this webpage.


This experiment aims to analyze the rotational fine structures of gaseous HCl, using a Fourier transform infrared (FTIR) spectrophotometer to determine the rotational constants and equilibrium internuclear distance of HCl from the rovibrational spectrum. HCl gas was prepared and a strong absorption band due to the fundamental band transition from v=0 to v=1 vibrational state. The vibrational frequency, band origin, force constant, B0, B1, α, Be and re were subsequently found. The calculated re value of 1.264 Å is very close to the literature value shows that rovibrational spectroscopy is a useful method to determine the various parameters.


Infrared spectroscopy is used to explore the structure, dynamics and concentrations of chemical compounds. FTIR spectroscopy provides simultaneous and almost immediate recording of the whole spectrum in the infra-red region. An infrared spectrum comes from transitions that arise from the vibration of molecules caused by change in their dipole moments. The energies between successive rotational levels are about 1000 times less than between vibrational levels, this result in a rovibrational spectrum. The fine structure in these absorptions observed is due to superimposed rotational energy levels.
A diatomic molecule is approximated to a harmonic oscillator by assuming that the bond obeys Hooke’s law just like a spring. In hydrogen chloride, it is a fairly reasonable to assume that during vibrations, the heavier chlorine atom stays almost still while the much lighter hydrogen moves. An elastic bond has a vibrational frequency that is dependent on the reduced mass and the force constant of the bond which approximates the strength of the bond being stretched between two atoms. By Hooke’s law, w = 1/ (2πc) (k / μ)1/2, the higher the reduced mass, the lower is the vibrational frequency.  However, in reality, molecules do not obey the laws of simple harmonic motion exactly. If the bond between atoms is stretched too much, the molecule will dissociate into atoms while compressing the bond beyond equilibrium bond length, the repulsive force increases rapidly. Thus, the graph of energy against internuclear distance, r, will become steeper at smaller r and gentler at greater r.
From the solution of the Schrodinger equation, the allowed energy levels of a vibrating diatomic molecule, G(v) = ν̃e (ν + ) – χe ν̃e (ν + )2  [cm-1], where ν is the vibrational quantum number, χe is the anharmonicity constant and ν̃e is vibrational wavenumber. The selection rules for the anharmonic oscillator are found to be Δ ν = 1, 2, ±3,…. The transition from ν = 0 to ν = 1 is known as the fundamental vibrational bandcenter. Transitions occurring between vibrational levels occur mainly in the infrared spectral region. Since each vibrational level has rotational levels associated with it, these transitions are called rovibrational transitions.
The rotational energies F of a diatomic molecule treated as a rigid rotor are F (J) = BJ(J + 1) where B is the rotational constant and J is the rotational quantum number. A rotational spectrum will be observed if the molecule possesses a dipole moment and only transitions between adjacent rotational levels are allowed (∆J 1). However, the assumption of a rigid bond is only an approximation. When the molecule is rotating rapidly, centrifugal forces stretch the bond a little resulting in an increase in bond length. From the equation B = h / (8π2cμ r2), where I is the moment of inertia and I=μ r2.The equilibrium bond length and moment of inertia are larger at higher vibrational levels, subsequently, the value of B decreases with J. In general, the rotational constant for any vibrational state is Bv = Be α(v + 1/2), where Be: and Bv are rotational constant in equilibrium and vibrational level respectively and α is the vibrational and rotational interaction constant.
By the Born-Oppenheimer approximation, the combined rotational-vibrational energy is simply the sum of the separate energies. As diatomic molecules vibrate and rotate simultaneously, the energy levels S of a molecule is the sum of rotational and vibrational energies, S(v,J) = F(J) + G(v) = ν̃e (ν + ) – χe ν̃e (ν + )2  + BvJ(J + 1), with selection rules of ∆ ν = 1, 2, ±3 etc and ∆J = 1.
The single fundamental band consists of P and R branch where P branch occurs on the low-frequency side of the band origin, ν̃0, with a decrease in rotational quantum number (∆J = -1) and R branch occurs on the high-frequency side of the band origin with an increase in rotational quantum number (∆J = +1). The wavenumber of the rotation-vibration transitions at P branch is given by ν̃ P branch = ν̃0 – (B0 + B1)J – (B0 – B1)J2 while that of R branch is ν̃ R branch = ν̃0 + (B0 + B1)(J + 1) – (B0 – B1) (J + 1)2. Since there are 2 subtraction terms in the P branch compared to 1 subtraction term in R branch, P branch has a lower wavenumber than R. In addition, the P branch lines spread out while the R branch lines cluster together as J increases.


A background spectra of a 10 cm IR cell filled with nitrogen gas was recorded using FTIR spectrophotometer with resolution 4 cm-1 over the range 600 - 4000 cm-1. Concentrated sulphuric acid was added dropwise to sodium chloride in a conical flask equipped with stopper and outlet tube. After passing through a U-tube containing CaCl2, the evolved HCl was collected into the gas cell. The IR spectrum of HCl sample was recorded. Baseline was then added and the spectrum was printed. The whole process was repeated with resolution 0.5 cm-1 over the range 2500 – 3200 cm-1.

Results and Calculation

From the low-resolution IR spectrum of gaseous HCl, there is one strong absorption band with 2 branches arise from the transition from v = 0 to v = 1. The band origin, ν̃0, is approximately at 2880 cm-1.
The allowed energy levels of vibrating diatomic molecules are expressed as term values G: 
G (v) =  , where v = 0, 1, 2, ...
For the first overtone band (v = 0 to v = 2):          = =  =  = 5668  ------ (1)
For the second overtone band (v = 0 to v = 3): = =  = = 8347  ------ (2)
2 × (1) – (2):  = 2989 cm-1
Substitute = 2989 cm-1 into (1):  2(2989) - 6 = 5668      
                    = 51.67 cm-1        and       = 0.01729
Band origin, ν̃0 = ve (1-2ce) = 2989 (1- 2 x 0.01729) = 2886 cm-1
Reduced mass of HCl, µ =  =  = 1.628 × 10-27 kg
 Force constant, k =  =   = 515.9 Nm-1
High-resolution spectrum
Compared to the vibrational band center, P branch has lower wavenumbers (2571.99 cm-1 to 2864.92 cm-1) while R branch has higher wave numbers (2904.22 cm-1 to 3097.82 cm-1).
The presence of the double peaks is due to the isotopic effect in chlorine, 35Cl and 37Cl.
0(J) = R(J) – P(J + 2) = 2B0 (2J + 3) 
1(J) = R(J) – P(J) = 2B1(2J + 1) 
Sample calculations for J = 1:
0(J) = R(0) – P(2) = 2906.39 - 2844.19 = 62.20 cm-1
1(J) = R(1) – P(1) = 2926.16 - 2864.92 = 61.24 cm-1
Table 1. Values of P(J), R(J), ∆0(J) and ∆1(J) based on high-resolution spectrum
P(J) / cm-1
R(J) / cm-1
0(J) / cm-1
1(J) / cm-1

Graph 1: Graph of ∆0(J) / cm-1 against J to get B0

0(J) = 2B0 (2J + 3) = 4B0J + 6B0

The gradient of graph of ∆0(J) / cm-1 against J is 4B0 = 42.04.

Therefore, B0 = 10.51 cm-1

Graph 2: Graph of 1(J) / cm-1 against J to get B1

1(J) = 2B1(2J + 1) = 4B1J +2B1

The gradient of graph of ∆1(J) / cm-1 against J is 4B1 = 39.95.

Therefore, B1 = 9.988 cm-1

Bv = Be  - a(v +1/2)
v=0, 10.51 = Be - a(1/2)                       ----- (3)                           v=1, 9.988 = Be - a(3/2)                    ----- (4)
Solving simultaneous equations (3) and (4), a = 0.5225 (4 s.f)       Be  = 10.77 cm-1 (4 s.f)
B = h / (8p2cm r2)               =>              re2 = [h / (8p2cmBe)]
re = [6.626 x 10-34 / {8p2 x 2.998 x 108 x 1.628 × 10-27 x 10.77}]1/2   = 1.264 Å (4 s.f)


In this experiment, Fourier transform infrared (FTIR) spectrophotometer was used to obtain a rovibrational spectrum of hydrogen chloride. The advantage of the FTIR instrument is that it acquires the interferogram in less than a second because the whole spectrum is obtained across the entire frequency range at once. This allows many samples to be collected and averaged together producing a greater sensitivity. The FTIR spectrophotometer operates in a single beam, it cannot remove the background absorptions at the same time the sample spectrum is obtained. Thus, to obtain the spectrum of hydrogen chloride, an interferogram of the background which is nitrogen gas was first subjected to a Fourier transform and recorded. Then, hydrogen chloride gas was placed into the beam and the spectrum obtained contains absorption bands for both the compound and the background. The computer software automatically subtracts the spectrum of the background from the sample spectrum, yielding the spectrum of hydrogen chloride being analyzed.


From the low resolution spectrum, one strong absorption band with two branches was observed. This is the fundamental band which corresponds to the vibrational transition of HCl from v = 0 to v = 1, where v is the vibrational quantum number. The band origin was found at approximately 2880 cm-1. In addition, a peak near 2350 cm-1 was observed. This corresponds to the presence of carbon dioxide impurities, which may be due to carbonate impurities in the CaCl2 in the U-tube or incomplete removal of carbon dioxide after flushing with nitrogen gas.
The high intensity of fundamental band is because of the large population of the ground state. The relative intensity of the absorption lines are largely governed by the populations of molecules occupying the various rotational levels in the ground v=0 vibrational state. IJ α NJ’’ / Ntotal = (2J + 1) e-Erot’’/kT, where Erot’’ = hc BeJ’’ (J’’ + 1) and k is the Boltzmann’s constant. The most intense spectral line arises from the level which initially has the greatest population. Overtones bands which are transitions differing by Δ v>1 were not observed in the spectrum due to the low intensity. However, if the HCl gas is prepared at higher temperatures, more molecules will be excited and the intensity of hot band will increase and may be observed. There is a gap between P(1) and R(0) as ΔJ=0 is not allowed by the selection rules. This band origin at the midpoint of P(1) and R(0) can be calculated and found to be 2886 cm-1.
In reality, the bond does not obey Hooke’s Law as it is non-rigid. The Morse Function is a better approximate to the bond when stretched. The internuclear distance between the atoms in a diatomic molecule is proportional to the change in the vibrational quantum number. An increase in the vibrational quantum number will increase the internuclear distance between the atoms. This translates to the fact that the rotational constant, B, would be further decreased. This is consistent with our results since the value of B0 determined (10.51cm-1) was greater than B1 (9.988cm-1). As we also observe that decreasing B as J increase means that the R-branches are progressively closer to one another while the P-branches get progressively furthur as J increases.
The theoretical equilibrium internuclear distance is 1.274 Å while that calculated from the spectrum results is 1.264 Å. The different value obtained from experimental calculations was due to the effect of centrifugal distortion which increases the equilibrium internuclear distance but was not included in the calculations. However, the percentage error of 0.785% [(1.274 – 1.264)/(1.274) x 100% ]  , thus the experiment is accurate to a large extent. The results prove that vibrations of molecules do not follow simple harmonic model, instead they are anharmonic and the bond in diatomic molecule is not rigid. The R2 values are equal to 1 both in graph 1 and 2 indicating that there is a linear relationship between ∆0 or ∆1 and J values.

Double Peaks

From the high resolution spectrum, double peaks for both the P and R branches were observed. This is due to the presence of chlorine isotopes, 35Cl and 37Cl, which occurs in a 3:1 abundance. When an isotopic substitution is made in a diatomic molecule, the equilibrium bond length re and the force constant k remain unchanged, since they depend only on the bonding electrons. However, the reduced mass changes and this will affect the vibration and rotation of the molecule. Since H35Cl has a smaller reduced mass, by Hooke’s law, w = 1/ (2πc) (k / μ)1/2, it will vibrate at a higher  wavenumber compared to H37Cl.  From the ratio of the reduced masses, the frequency of H35Cl is 1.00075 times that of the H37Cl, this means that the shift in frequency will be very small. As 35Cl has a greater abundance than 37Cl, the intensity of H35Cl peak is stronger than that of H37Cl. Thus, the spectrum obtained is a superposition of the H35Cl and the H37Cl spectra.


Care has to be taken to avoid touching the sides of IR  gas cell as fingerprint markings will scatter the light passing through, thus affecting the transmittance readings.

Gloves must be worn during the preparation of the HCl gas as concentrated sulphuric acid which is highly corrosive was used. In addition, blue litmus paper was used at the mouth of the cell to determine that the cell has been entirely filled with gaseous HCl.

N2 gas can be used to flush out unwanted gas as it is IR inactive because N2 is a homonuclear diatomic molecule that does not have permanent dipole moment and in addition, it will not react with HCl. CaCl2 was used in the preparation of HCl gas as a drying agent to remove the IR active water.


We had obtained and analysed the rovibrational spectrum of gaseous HCl, and found the following values based on the experiment, ωe = 2989 cm-1, χeωe = 51.67 cm-1, χe = 0.01729, ν̃0 = 2886 cm-1, force constant k = 516.1 Nm-1, B0 = 10.51 cm-1, B1 = 9.988 cm-1, α = 0.5225, Be = 10.77 cm-1 and re = 1.264 Å. The close correspondence of the calculated re value, with that of the literature value shows rovibrational spectrum is a very useful way to determine the various parameters and the calculation of equilibrium bond length.


Literature values taken from:
[1] Banwell and McCash, 2007. Fundamentals of Molecular Spectroscopy. 4th ed. New Delhi: Tata McGraw-Hill Publishing Company Limited.
Experimental Methods:
Born-Oppenheimer/Morse Function:
[4] Banwell and McCash, 2007. Fundamentals of Molecular Spectroscopy. 4th ed. New Delhi: Tata McGraw-Hill Publishing Company Limited