Physical Chemistry Laboratory Experiment II-4

Physical Chemistry Laboratory
Experiment II-4
See relevant sections in undergraduate text
Learn from your instructor how to use the spectrometer.
Know definitions of the following and their interrelations:
Reduced mass, µ
Frequency, ν (sec-1)
Wave number, "˜ (cm-1)
Rotational constant, Be (cm-1)
internuclear separation," re
Anharmonicity constant, xe
Vibration-rotation interaction constant, αe
Quantum numbers υ and J
P and R branches
Moment of inertia, I
Measurement of:
Vibration-rotation spectrum of HCl gas
Frequency of rotational lines of HCl gas
Computation of:
Vibration-rotation interaction constant, αe
Rotational constant, Be
Fundamental vibration frequency, "˜ 0 (cm-1)
Moment of inertia of HCl
Bond length of HCl, re
Force constant for the H-Cl bond
Experiment II-4
Physical Chemistry Laboratory
Conc. H2SO4, NaCl and anhydrous CaCl2
IR gas cell and filling system
Research grade IR spectrometer
Theory and Spectral Analysis
The spectra of molecular systems can be quite complicated
and the "sorting out" of the profusion of lines and their
assignment to the appropriate transitions may be a formidable
task. However a proper molecular model may be used towards this
objective as well as the deduction of structural parameters of
the molecule of interest.
The vibrational-rotational band:
An infrared, high resolution
spectrum of a heteronuclear diatomic molecule, such as HCl gas,
consists of a vibrational-rotational band due to transitions in
which both vibrational and rotational energies change.
Such a
band can be conveniently considered to have three portions as
shown in Figure (1).
The zero gap is the central portion and is signified
by no absorption.
The P- branch is the low frequency (cm-1) portion to
the right of the zero gap.
It consists of several
lines which diverge towards low frequency.
The R- branch is the high frequency (cm-1) portion to
the left of the zero gap.
It consists of several
lines which converge towards high frequency.
The vibrotor model:
The spectral features of the vibrationalrotational band of HCl (gas) may be explained in terms of a
model for a diatomic molecule simultaneously executing both
vibrational and rotational motions.
The simplest model is an
approximate one in which the vibrational and rotational motions
are treated independently.
For the vibrational motion the
molecule is assumed to vibrate as a simple harmonic oscillator.
The energy of the allowed vibrational levels is given by:
Physical Chemistry Laboratory
Experiment II-4
E(υ) = hν(υ + 1/2)
The vibrational quantum number υ is of integral values 0, 1,
The rotational motion of the molecule is that of a rigid
rotor with the energy of the allowed levels given by:
E( J ) = 2 J ( J + 1)
8! I
The rotational quantum number J is of integral values 0, 1, 2,
The moment of inertia I is related to the internuclear
distance r, and the reduced mass by:
I = µr2
As a first approximation the energy of a vibrating rotating
molecule (vibrotor) is the sum of expressions (1) and (2):
E(!, J ) = h"$ ! + %& + 2 J ( J + 1)
8' I
Each vibrational energy
rotational energy levels.
A more complete expression, but not necessarily the most
exact one, should include:
The effect of anharmonicity which is accounted for by
a term involving the anharmonicity constant, xe.
A term involving the distortion constant, Be, which
accounts for the centrifugal stretching. This results
from the stretching of the non-ridged chemical bond
during rotation and is important at high rotational
energies (high J).
A term involving the vibration-rotation interaction
constant, αe, which accounts for changes in r during
The vibrational-rotation
may be represented by:
Experiment II-4
Physical Chemistry Laboratory
T (!, J ) =
E( !, J )
1 2
T (!, J ) = ˜"e $ ! + %& ' xe "˜ e $ ! + %& + Be J ( J + 1) + De J ( J + 1) ' ( e $ ! + %& J ( J + 1)
where T (υ, J) is in cm-1.
The frequency "˜ e (cm-1) refers to the frequency of the molecule
vibrating about its equilibrium internuclear separation re. The
rotational constant Be is defined by (c is the speed of light,
Be = 2
8! Ic
Selection Rules:
The fine structure of the P- and R- branches
represents transitions from a particular rotational level (J =
J") in a given vibrational state (υ = υ") to a different
rotational level (J = J') in an excited vibrational state (υ =
The superscripts (') and (") refer to high and low
vibrational states, respectively.
A common pitfall for students first exposed to spectroscopy
is to confuse energy levels T (υ, J) with transitions between
energy levels Δ[T (υ, J)] where
Δ [T(υ, J)] = T (υ', J') - T (υ", J")
It is the latter which is directly observed.
For transitions to be allowed the selection rules for a
diatomic molecule to be infrared active are:
Δυ = ± 1
ΔJ = ± 1
Change to dipole moment ≠ 0 during vibration.
A homonuclear diatomic molecule, being nonpolar, has a
dipole moment equal to zero, and invariant, unless the molecule
is in a unique chemical environment. Thus, homonuclear diatomic
molecules are, in general, infrared inactive.
In the present experiment the transitions take place from
Physical Chemistry Laboratory
Experiment II-4
various J" - levels of the vibrational ground state (υ" = 0) to
J' - levels of the first excited vibrational state (υ' = 1).
Frequencies of the P- and R- lines:
From the selection rules, and application of equation (7)
with proper substitution from eq (5) the frequencies (cm-1) of
the P- and R- lines are:
P- branch:
ΔJ = -1 ; which is J' - J"
where J" = 1, 2, 3
˜!P = ˜!0 " 2(Be " # e ) J " # e J 2 ;
R- branch:
ΔJ = + 1 ; J' = J" + 1
˜! R = ˜!0 " ( 2Be " 3# e ) + (2Be " 2# e ) J " ae J 2
˜!0 = ˜!e " 2x e ˜!e
where "˜ 0 is the frequency of the forbidden transition (ΔJ = 0).
It corresponds to the missing line, somewhere near the midpoint
in the zero gap.
In the derivation of equations (8) and (9)
from equation (7) the term in De is neglected.
The frequency of the P- and R- branches, in the observed
spectrum can be represented by the empirical equation:
"˜ = c + dm + em2
where m is a running number which is:
+1, +2, +3, …!
for the R- branch
-1, -2, -3, …
for the P- branch.
A single equation similar to equation (11) may be obtained from
equations (8) and (7) by substituting for J as follows:
for the P- branch (equation 8):
J = -m
for the R- branch (equation 9):
J = m - 1
This leads to:
˜! = ˜!0 " 2( Be " # e )m " # em 2
Experiment II-4
Physical Chemistry Laboratory
The separation between adjacent lines in each branch is
given by:
! ˜"(m ) = ˜"(m + 1) # ˜"(m) = (2Be # 3$ e ) # 2$ e m
If we plot values of "#˜ ( m) as ordinate and m as abscissa, a
fit of the best straight line through the points has its slope
equal to -2αe, yielding the vibrotor interaction constant.
the points are from a sufficiently highly resolved spectrum and
are plotted on a sufficiently
large scale, the best fit will be
a slightly curved line. The slope of the tangent to the curve
at m = 0 should be used to obtain αe. The intercept of the curve
with the "#˜ ( m) axis gives a value from which Be, the rotational
constant, is obtained. The value of "˜ 0 is obtained from equation
(12) by using αe, Be and with data from an m that has a "#˜ that
falls on the least squares line.
The moment of inertia, and
the bond length, are obtained from the definition for Be.
Specific instructions for the operation of the infrared
spectrometer to be used will be given in the laboratory.
is a precision research instrument -- use it carefully.
If in
doubt about anything after being instructed in its use, ask the
Don't take chances with expensive equipment when
you are unsure.
The gas cell is constructed from a 10 cm length of large
diameter PTFE tubing with stopcocks attached.
Infraredtransparent windows are attached to the ends by an adhesive or
by mechanical clamps or both.
The windows are cut from large
crystals of NaCl and are easily fogged by moisture from
atmosphere or skin. Never touch the windows and store the cell
in a desiccator when not in use.
Connect the dropping funnel and flask (properly supported
by a ring stand) to the drying tube and cell, with stopcocks
open. This setup must be in a fume hood. To about 25 g NaCl,
slowly add about 25 mL concentrated sulfuric acid to produce a
steady evolution of gas, probably not all the acid will be
Check the open end of the cell for HCl evolution as
evidenced by white fumes and/or blue litmus turning pink. Allow
the gas to continue passing through the cell for a few minutes
then disconnect and close the stopcocks. This usually provides
Physical Chemistry Laboratory
Experiment II-4
a reasonable partial pressure of HCl in the cell.
Record the spectrum on a fresh piece of paper using the
highest expansion of wavelength scale available with the
instrument. This spectrum should run between values of 2700 cm-1
and 3000 cm-1 the frequencies as recorded by the FTIR are then
used for the calculations.
Results and Calculations:
In a single table, list the following information for
each identifiable
in your high
J, ˜!0 , P or R branch, m, ! ˜"(m )
Using the method given above, i.e. the plot of Δν
~ (m)
vs. m, from the tabulated data compute:
Using the value of Be computed above, obtain a value
for I, the moment of inertia of the HCl molecule.
Using 1.67379 x 10-27 kg and 5.80752 x 10-26 kg for the
masses of individual atoms of hydrogen and chlorine,
respectively, compute the reduced mass, µ, and bond
length re (in Angstroms and nm) for HCl from I = µ re2.
Calculate the force constant for the HCl bond.
Repeat steps 1-5 for the DCl spectrum given by the