Download The Vibrational-Rotational Spectrum of HCl

CHEM 332L
Physical Chemistry Lab
Revision 2.2
The Vibrational-Rotational Spectrum of HCl
In this experiment we will examine the fine structure of the vibrational fundamental line for
H35Cl in order to determine several spectroscopic constants; the Rotational Constants B0 and B1,
the Centrifugal Distortion Constant De, and the Fundamental Vibrational Band Origin
.
Inclusion of a few literature values for Rotational Constants Bv for higher vibrational states
v=2,3,4,… will allow us to determine the Equilibrium Rotational Constant Be. Inclusion of
literature values for the 1st thru 4th vibrational overtones will allow us to determine further
spectroscopic constants; the Equilibrium Fundamental Vibrational Frequency and the
Anharmonicity Constant
. Finally, we will assume the HCl molecule is behaving
approximately as two masses connected by a stiff spring of constant k rotating with internuclear
distance r. We will then extract these molecular parameters from the appropriate spectroscopic
constants.
First, we must recognize that Hydrogen Chloride is a diatomic gaseous substance. Because it is a
diatomic molecule, it will have one vibrational mode and one rotational mode of
motion. Thus, we can model HCl molecules as vibrating rotors.
We begin by examining the vibrational motion of the molecule and treat the
atoms of the molecule as being connected by a Hookean spring of constant k. In
this case, the Hamiltonian operator for the system is written as:
=
Solving the Schrodinger wave equation
Ev = h (v + ½)
(Eq. 1)
 = E for the energy yields:
v = 0, 1, 2, …
(Eq. 2)
where  = (1/2)
is the Fundamental Vibrational Frequency of the system and  is its
reduced mass. v is the quantum number associated with the vibrational energy state Ev. In
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anticipation of discussing spectral lines arising from with energy transitions, we can define the
vibrational spectral "Term" as:
G(v) =
=
(v + ½)
(Eq. 3)
where = /c. h and c are Planck's Constant and the speed of light, respectively. If we allow
for non-Hookean behavior, then:
G(v) =
(v + ½) -
(v + ½)2
(Eq. 4)
where
is an anharmonicity constant. This anharmonicity correction lowers the energy of
each vibrational state because the parabolic potential of Hookean behavior tends to broaden and
flatten and become more Morse like. The anharmonicity correction becomes more severe at
higher vibrational states where the deviation from Hookean behavior is more pronounced.
A couple of points must be kept in mind when
considering equation 4. First, anharmonicity
will cause the fundamental frequency of the
oscillator  to depend on which vibrational
state the system is in. Thus, we define an
equilibrium fundamental frequency ; this
equals  at the bottom of the potential well
where anharmonicity vanishes. Second, and
of equation 4 are spectroscopic constants
that can be measured regardless of the model
used for representing the molecule. On the
other hand, the spring constant k of equation 2
depends on the nature of the model used to
represent the molecule. This must be kept in mind when we extract values of k from the
measured spectroscopic constant .
Now, vibrational spectral lines will occur when a photon is absorbed causing an energy transition
from Ev'' to Ev', where v'' is the lower quantum state and v' is the upper. In terms of spectral
Terms, this transition is represented as:

= G(v') - G(v'')
The fundamental line vibrational spectral line
will be given by 
= G(1) - G(0).
Overtones   , etc. are quantum
forbidden by the selection rule v = ±1 and so
are very weak. (Hot Bands, 1 2, 2 3, etc. are
also very weak because higher vibrational states
will not be significantly populated due to
(Eq. 5)
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insufficient thermal energy in the system.) Schematically, the spectrum will appear as:
Using high resolution spectrometers, it is observed each of these lines exhibits more detailed
structure.
For instance, at fairly low resolution, the Fundamental Line appears as a doublet. Using a higher
resolution spectrometer, it is observed this line exhibits fine structure. At higher resolution yet,
each fine structure line is observed to split into a pair of lines.
The fine structure can be explained by considering the rotational motion of the HCl molecule.
The splitting of each fine structure line can be explained by considering the sample to be made
up of a mixture H35Cl and H37Cl molecules.
So, turning to the rotational motion of the HCl molecule, we treat the molecule as a rigid rotor.
Again, we write the system's Hamiltonian operator:
=
(Eq. 6)
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Solving the Schrodinger wave equation
El =
 = E for the energy yields:
l = 0, 1, 2, …
l (l + 1)
(Eq. 7)
where I = r2 is the moment of inertia of the molecule. We define the rotational spectral "Term"
as:
F(J) =
= B J (J + 1)
(Eq. 8)
where B = h/2c is the system's Rotational Constant. (Note: In spectroscopic notation, the
quantum number l is denoted by the symbol J.) If we allow for centrifugal distortion as the
molecule begins to "spin" faster and faster, then the Term becomes:
F(J) = Be J (J + 1) - De J2 (J + 1)2
(Eq. 9)
where De is the Centrifugal Distortion Constant. Again, Be and De are spectroscopically
determinable constants that are independent of molelcular model.
Expected values of "I" suggest these terms will lie very close together and that the spectra lines:

= F(J') - F(J'')
(Eq. 10)
will lie in the microwave region. Here the selection rule for allowed transitions is J = ± 1.
"Hot" bands are not a problem because the energy levels lie close enough
together that thermal population of upper rotational states is
considerable. Therefore, in microwave spectroscopy, energy transitions,
as pictured to the left, will occur.
These rotational states will lie on top the vibrational states, giving us
spectral Terms T(v,J) = G(v) + F(J). This allows us to describe the
fine structure lines as being due to transitions of the form:

= T(v',J') - T(v'',J'')
(Eq. 11)
Inserting our expressions for G(v) and F(J) into this equation, we obtain
the following general result for the spectral lines:

= (v' - v'')
-
[(v' - v'')(v' + v'' + 1)]
+ Bv' J'(J' + 1) - Bv'' J''(J'' + 1) - De J'2(J' + 1)2 + De J''2(J'' + 1)2
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(Eq. 12)
Note that because the molecule "stretches" slightly as it moves into higher vibrational states, due
to anharmonicity, the internuclear distance r will depend on vibrational quantum number v
(denoted as rv), hence "I" will depend on the vibrational quantum number and therefore "B" will
also depend on the vibrational quantum number (denoted as Bv). We define a vibration-rotation
Coupling Constant as e such that the nature of B's dependence on v is:
Bv = Be - e (v + ½)
(Eq. 13)
A similar problem occurs for the distortion constant D, except the coupling is weak enough we
can neglect it.
With all the overlayed rotational fine-structure on the vibrational band, how do we define the
"position" of the vibrational spectral line? Well, formally, we define the line's Band Origin
as:
= (v' - v'')
-
[(v' - v'')(v' + v'' + 1)]
(Eq. 14)
If we are dealing with the fundamental line, then v'' = 0 and v' = 1. Then:


- 2
+ B1 J'(J' + 1) - B0 J''(J'' + 1) - De J'2(J' + 1)2 + De J''2(J'' + 1)2
(Eq. 15)
Now, one of two situations can occur; J = J'-J'' = +1 or J = J'-J'' = -1. If J = +1, we are said
to be in the R-branch of the line; which occurs at higher wavenumbers. If J = -1, we are in the
P-branch; at lower wavenumbers.
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Thus, for the R-branch, we have:
R(J'') = 
J=+1
- 2
+ (B1 + B0) (J''+1)
+ (B1 - B0) (J''+1)2 - 4De (J''+1)3
(Eq. 1)
and, for the P-branch, we have:
P(J'') =  J=-1
- 2
- (B1 + B0) J'' + (B1 - B0) J''2 + 4De J''3
(Eq. 17)
We now turn to the question of how to extract the spectroscopic constants from the measured
positions of the fine-structure lines; R(0), R(1), R(2), …., P(1), P(2), P(3), ….
First, how do we extract the rotational constants B0 and B1 and the distortion constant De?
Equations 16 and 17 can be re-arranged to obtain the following forms:
= B1 - 2 De (J''2 + J'' + 1)
= B0 - 2 De (J''2 + J'' + 1)
(Eq. 18)
(Eq. 19)
Thus, measurements of R(J'') and P(J''), when plotted according to equations 18 and 19, and the
results subjected to a linear least square analysis, will yield the desired constants.
Next, to determine the band origin we use another manipulation of equations 16 and 17:
=
+ (B1 - B0) (J'' + 1)2
(Eq. 20)
Another linear least square analysis is needed to obtain this constant.
To obtain and
we will need additional information concerning the overtone lines for the
case where v'' = 0. We can rearrange equation 14 to yield the following.
=
-
(v' + 1)
(Eq. 21)
Again, a linear least square analysis will give us the desired constants. Overtone data for H35Cl
is:
Overtone
0-2
0-3
0-4
0-5
[cm-1]
5667.9841
8346.782
10922.81
13396.19
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Next, we can use our measured values of B0 and B1, along with additional data, and a linear least
square analysis according to equation 13, to determine Be. Needed data for H35Cl is:
v
2
3
4
5
B [cm-1]
9.834663
9.534909
9.2360
8.942
Finally, once the spectroscopic constants and Be are determined, we can use our model for the
HCl molecule to determine the molecular parameters k and re.
e =
c = (1/2)
(Eq. 22)
Be = h / 2cre2
(Eq. 23)
(See development above.) Here re represents the equilibrium internuclear distance when the
molecule is not vibrating. When the molecule moves into vibrational states v = 0,1,2,… it begins
to stretch, due to the anharmonicity of the potential, and r increases. Hence, the internuclear
distance (really root-mean-square internuclear distance
) will depend on the vibrational state
v. From the rotational constants B0 and B1, the root-mean-square internuclear distance between
the H and Cl atoms in vibrational states v=0 and v=1,
and
respectively, can be
determined.
Bv = h /2c 

(Eq. 24)
Thus, we will experimentally determine the positions of all the fine-structures lines R(J'') and
P(J'') of the fundamental band for H35Cl. We will use this information to extract the
spectroscopic constants B0, B1, De and
. We will supplement the fundamental line data with
additional spectroscopic data to determine ,
and Be. Finally, we will invoke our
molecular model for HCl and determine the constants k and re,
, and
.
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Procedure
Obtain a high resolution Infrared spectrum of HCl gas. You will use a 10 cm gas cell as pictured
below.
The cell will be filled with HCl gas via one of the methods listed:
i) Use a "lecture" gas bottle, as pictured on page one of this laboratory, to fill the cell
directly.
ii) Treat Sodium Chloride with concentrated Sulfuric Acid to produce the Hydrogen
Chloride gas:
NaCl(s) + H2SO4(l)
HCl(g) + NaHSO4(s)
You can do this by dropping, very slowly, ~30 mL of conc. H2SO4 into about 5g of the
solid NaCl as pictured below.
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Your laboratory instructor will indicate which method of filling the gas cell will be used.
Once your gas cell is filled with HCl, your laboratory instructor will demonstrate the use of the
IR spectrometer. Take the spectrum with an appropriate resolution to see the fine structure in the
fundamental line. Measure the positions of R(J'') and P(J'') spectral lines for H35Cl.
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Data Analysis
The following calculations will require linear least squares analyses that are of fairly high
precision. Track your significant figures closely and make sure you perform an error
analysis for each directly measured quantity.
1.
From your data, determine the spectroscopic constants:
B0, B1 and
2.
Use the supplemental overtone data to determine the spectroscopic constants:
and
3.
Use the supplemental data for rotational constants to determine the spectroscopic constant
Be.
4.
Determine the molecular parameters k, re,
5.
Use the following data to determine k for alternate isotopic substitutions of HCl.
Molecule
D35Cl
D37Cl
6.
and
.
e [cm-1]
2145.1630
2141.82
Use the following data to determine re for alternate isotopic substitutions of HCl.
Molecule
H37Cl
D35Cl
D37Cl
Be [cm-1]
10.578
5.448794
5.432
P a g e | 11
References
Arnaiz, Francisco J. "A Convenient Way to Generate Hydrogen Chloride in the Freshman Lab"
J. Chem. Ed. 72 (1995) 1139.
Herzberg, G. Spectra of Diatomic Molecules Van Nostrand, Princeton, New Jersey, 1950.
Levine, Ira N. Physical Chemistry McGraw-Hill, Boston, 2009.
Levine, Ira N. Quantum Chemistry Prentice Hall, Englewood Cliffs, New Jersey, 1991.
Meyer, Charles F. and Levin, Aaron A. "On the Absorption Spectrum of Hydrogen Chloride"
Phys. Rev. 34 (1929) 44.
Pickworth, J. and Thompson, H.W. "The Fundamental Vibration-Rotation Band of Deuterium
Chloride" Proc. R. Soc. London Ser. A 218 (1953) 218.
Rank, D.H.; Eastman, D.P; Rao, B.S.; and Wiggins, T.A. "Rotational and Vibrational Constants
of the HCl35 and DCl35 Molecules" J. Opt. Soc. Am. 52 (1961) 1.
Sime, Rodney J. Physical Chemistry: Methods, Techniques, and Experiments Saunders College
Publishing, Philadelphia, 1990.
Tipler, Paul A. Physics Worth Publishers, New York, 1976.
Van Horne, B.H. and House, C.D. "Near Infrared Spectrum of DCl" J. Chem. Phys. 25 (1956)
56.