Abstract
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, B_{0},
B_{1}, Î±, B_{e} and r_{e} were subsequently found. The calculated
r_{e} value of 1.264 Ã… is very close to the literature value shows that
rovibrational spectroscopy is a useful method to determine the various
parameters.
Introduction
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 infrared 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/ (2Ï€c) (k / Î¼)^{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)
= Î½̃_{e} (Î½ +
)_{ }– Ï‡_{e} Î½̃_{e} (Î½ +
)^{2 } [cm^{1}], where Î½ is the vibrational
quantum number, Ï‡_{e }is the anharmonicity
constant and Î½̃_{e }is vibrational wavenumber. The selection rules for the anharmonic
oscillator are found to be Î” Î½ =
1,
2, ±3,…. The transition from Î½ = 0 to Î½ = 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 / (8Ï€^{2}cÎ¼
r^{2}), where I is the moment of inertia and I=Î¼ r^{2}.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 B_{v} = B_{e }– Î±(v + 1/2), where B_{e: }and
B_{v} are rotational constant in equilibrium and vibrational level
respectively and Î± is the vibrational and rotational interaction constant.
By the BornOppenheimer approximation, the combined
rotationalvibrational 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) = Î½̃_{e} (Î½ +
)_{ }– Ï‡_{e} Î½̃_{e} (Î½ +
)^{2 } + B_{v}J(J + 1), with selection rules
of ∆
Î½ =
1,
2, ±3 etc and ∆J =
1.
The single fundamental band consists of P and R
branch where P branch occurs on the lowfrequency side of the band origin, Î½̃_{0}, with a decrease in
rotational quantum number (∆J = 1) and R branch occurs on the highfrequency
side of the band origin with an increase in rotational quantum number (∆J =
+1). The wavenumber of the rotationvibration
transitions at P branch is given by Î½̃ _{P branch }= Î½̃_{0} – (B_{0} +
B_{1})J – (B_{0} – B_{1})J^{2} while that of R
branch is Î½̃_{ R branch }= Î½̃_{0} + (B_{0} +
B_{1})(J + 1) – (B_{0} – B_{1}) (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.
Experimental
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 Utube containing CaCl_{2}, 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 lowresolution 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, Î½̃_{0, }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, Î½̃_{0} = v_{e}
(12c_{e}) =
2989 (1 2 x 0.01729) = 2886 cm^{1}
Reduced mass of HCl, Âµ =
=
= 1.628 × 10^{27}
kg
Force constant, k =
=
= 515.9 Nm^{1}^{ }
Highresolution 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, ^{35}Cl and ^{37}Cl.
∆_{0}(J) = R(J) – P(J + 2) = 2B_{0}
(2J + 3)
∆_{1}(J) = R(J) – P(J) = 2B_{1}(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 highresolution spectrum
J

P(J) / cm^{1}

R(J) / cm^{1}

∆_{0}(J) / cm^{1}

∆_{1}(J) / cm^{1}

0



2906.39

62.20



1

2864.92

2926.16

87.03

61.24

2

2844.19

2943.28

144.90

99.09

3

2819.36

2962.57

187.09

143.21

4

2798.38

2980.65

228.56

182.27

5

2775.48

2998.01

270.03

222.53

6

2752.09

3014.40

311.25

262.31

7

2727.98

3030.31

352.48

302.33

8

2703.15

3045.26

393.22

342.11

9

2677.83

3059.49

433.49

381.66

10

2652.04

3072.99



420.95

11

2626.00

3085.77



459.77

Graph 1: Graph of ∆_{0}(J) / cm^{1} against J to get B_{0}

∆_{0}(J) = 2B_{0} (2J + 3) = 4B_{0}J
+ 6B_{0}
The gradient of graph of ∆_{0}(J) / cm^{1}
against J is 4B_{0} = 42.04.
Therefore, B_{0} = 10.51 cm^{1}

Graph 2: Graph of ∆_{1}(J) / cm^{1} against J to get B_{1}

∆_{1}(J) = 2B_{1}(2J + 1) = 4B_{1}J
+2B_{1}
The gradient of graph of ∆_{1}(J) / cm^{1}
against J is 4B_{1} = 39.95.
Therefore, B_{1} = 9.988 cm^{1}

B_{v} = B_{e }  a(v +1/2)
v=0, 10.51 = B_{e } a(1/2) 
(3) v=1, 9.988 = B_{e } a(3/2) 
(4)
Solving simultaneous equations (3) and (4),
a = 0.5225 (4 s.f) B_{e
} = 10.77 cm^{1 }(4 s.f)
B = h / (8p^{2}cm r^{2})
=> r_{e}^{2}
= [h / (8p^{2}cmB_{e})]
r_{e} = [6.626 x 10^{34}
/ {8p^{2}
x 2.998 x 10^{8} x 1.628
× 10^{27} x 10.77}]^{1/2} = 1.264 Ã… (4 s.f)^{}
Discussion
Experimental
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.
Spectrum
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 CaCl_{2} in the Utube 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. I_{J }Î± N_{J’’}
/ N_{total} = (2J + 1) e^{Erot’’/kT}, where E_{rot’’}
= hc B_{e}J’’ (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 Î” 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 Î”J=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 nonrigid.
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 B_{0}
determined (10.51cm^{1}) was greater than B_{1} (9.988cm^{1}).
As we also observe that decreasing B as J increase means that the Rbranches
are progressively closer to one another while the Pbranches get progressively
furthur as J increases.
The theoretical equilibrium internuclear
distance is 1.274 Ã… while that calculated from the
spectrum results is 1.264 Ã…. 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 R^{2} 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, ^{35}Cl
and ^{37}Cl, which occurs in a 3:1 abundance. When an isotopic
substitution is made in a diatomic molecule, the equilibrium bond length r_{e}
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 H^{35}Cl has a smaller
reduced mass, by Hooke’s law, w = 1/ (2Ï€c) (k / Î¼)^{1/2}, it will
vibrate at a higher wavenumber compared
to H^{37}Cl. From the ratio of
the reduced masses, the frequency of H^{35}Cl is 1.00075 times that of
the H^{37}Cl, this means that the shift in frequency will be very
small. As ^{35}Cl has a greater abundance than ^{37}Cl, the
intensity of H^{35}Cl peak is stronger than that of H^{37}Cl.
Thus, the spectrum obtained is a superposition of the H^{35}Cl and the H^{37}Cl spectra.
Precautions
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.
N_{2} gas can be used to flush out unwanted gas as it is IR
inactive because N_{2} is a homonuclear diatomic molecule that does not
have permanent dipole moment and in addition, it will not react with HCl. CaCl_{2}
was used in the preparation of HCl gas as a drying agent to remove the IR
active water.
Conclusion
We had obtained and analysed the rovibrational spectrum of gaseous
HCl, and found the following values based on the experiment, Ï‰_{e} =
2989 cm^{1}, Ï‡_{e}Ï‰_{e} = 51.67 cm^{1}, Ï‡_{e
}= 0.01729, Î½̃_{0} = 2886 cm^{1}, force constant k =
516.1 Nm^{1}, B_{0} = 10.51 cm^{1}, B_{1} =
9.988 cm^{1}, Î± = 0.5225, B_{e} = 10.77 cm^{1} and r_{e}
= 1.264 Ã…. The close correspondence of the calculated r_{e} 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.
References
Literature
values taken from:
[1] Banwell and
McCash, 2007. Fundamentals of Molecular
Spectroscopy. 4^{th} ed. New Delhi: Tata McGrawHill Publishing
Company Limited.
Experimental
Methods:
[2] http://hyperphysics.phyastr.gsu.edu/HBASE/molecule/vibrot.html
[accessed 15/09/09]
BornOppenheimer/Morse
Function:
[4] Banwell and
McCash, 2007. Fundamentals of Molecular
Spectroscopy. 4^{th} ed. New Delhi: Tata McGrawHill Publishing
Company Limited
FTIR:
[5] http://www.wcaslab.com/TECH/tbftir.htm
[accessed 12/09/09]
[6] http://mmrc.caltech.edu/FTIR/FTIRintro.pdf
[accessed 12/09/09]
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