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Infrared Spectroscopy Part I: IR of HCl and DCl (REQUIRED)

Please refer to M&S Ch. 13 for important background material.

Experiment

Record the IR spectrum of a mixture of HCl(g) and DCl(g) at 0.4 cm-1 resoluGon

Calculate all constants discussed in the experimental procedure

Note that the pre-lab is preJy extensive This writeup will have an extended deadline due to the large amount of data analysis required.

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Harmonic Oscillator Allowed energies for the HO are

Ev = v+ 12( )h v=0, 1, 2, ...

E=Eu El = hwhere is the fundamental vibrational frequency

given by:

= 12k

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VibraGonal TransiGon SelecGon rules:

v = 1 (this rule is only hard and fast for a purely harmonic oscillator, which molecules arent)

Dipole moment of the molecule must change as a result of the vibraGon; highly symmetric modes (such as the symmetric stretch of CO2 are IR-inacGve

For simplicity, we will assume that we are looking at v=0 to v=1 transiGon (called the fundamental) For HCl and DCl its also possible to see v=0 to v=2 transiGon

(called the rst overtone); this allows us to determine the anharmonicity constant. Overtones are signicantly weaker than fundamentals

v=0 to v=1 is the transiGon shown on IR spectra of organic molecules (occasionally you might see an overtone, parGcularly for aromaGc compounds)

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VibraGonal Energy in cm-1

In M&S the Glde over a symbol indicates the quanGty is expressed in wavenumbers

G(v) = v+ 12( )!where

! = 12ck

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Rigid Rotor SelecGon rule for rigid rotor:

J = 1

This rule is strict

EJ =2

2I J J +1( ) J = 0,1,2...Frequently expressed in wavenumbers as

F(J ) = BJ J +1( )where

B = h8 2cIand I is the moment of inertia, Re2

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VibraGonal-RotaGonal Energy Diagram When a molecule absorbs IR radiaGon, the vibraGonal and rotaGonal energies change simultaneously.

IR Spectroscopy The HO and RR models can be combined to represent the simultaneous change in vibraGonal and rotaGonal energies (in units of cm-1):

E! v ,J = G(v)+ F(J ) = v + 12( )" + B!J J +1( )with v=0,1,2,and J=0,1,2,and selection rules v= 1, J= 1

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IR Spectroscopy In an IR spectrum of organic molecules in the liquid or solid state (typically v=0 to v=1), the rotaGonal transiGons are not resolved, leading to a spectrum with relaGvely broad bands.

With gases, rotaGonal transiGons can be seen. The group of vibraGonal transiGons with

J=+1 is called an R-branch J=-1 is called a P-branch

If present, the group of transiGons with J=0 is called a Q-branch. These transiGons are not allowed in HCl J=0 is observed in bending mode of CO2 and similar molecules

with a doubly-degenerate bending mode (more on this later)

IR Spectroscopy Lets consider the energies of just the R-branch transiGons (J=+1):

! obs =E" v+1,J+1 - E" v,J

! obs = !(v +1+ 12 )+ B"(J +1)(J + 2)

!(v + 12 )+ B"(J )(J +1)

= ! + 2B"(J +1) J=0, 1, 2, ...

Remember the ~ means a quantity is expressed in wavenumbers

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IR Spectroscopy Similarly, the it can be shown that the P branch transiGons are given by:

! obs =E" v+1,J1 - E" v,J= ! 2B"J J=1, 2, 3, ...

Summary

! + 2B"(J +1) for the R branch

! 2B"J for the P branch

Thus, the vibration-rotation spectrum is centered about a frequency of with the first lines appearing 2 B above and below with a gap in the center (corresponding to the position where J = 0 bands would be, if present.

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Resolution 2.0 cm-1

P branch

R branch

R-Branch P-Branch HCl IR spectrum

40

50

60

70

80

90

100

260027002800290030003100

Wavenumber

Inte

nsity

Actual IR spectrum of HCl (fundamental) recorded with our OLD spectrometer

Note that the scale is opposite of that on the previous slide

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ObservaGons on HCl Spectrum

Most IR spectrometers plot data as %T rather than absorbance (as shown in M&S)

Wavenumber values are usually in the opposite order from the spectra in the text

Gap in the middle - corresponds to Bands are not equally spaced Peaks are all split into doublets

! vib

Spacing The previous equaGons assume that the value of B (the rotaGonal constant) is the same for v=0 and v=1.

In fact, this is not the case; bonds stretch as they vibrate; the larger v, the longer the bond length becomes. So R1 > R0

B! = h8 2cRe2

Since R1 > R0, then B!1 < B! 0

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Spacing

Since B0 > B1, the P branch will diverge slightly and the R branch will converge. This is readily apparent in the HCl/DCl spectrum

P or R branch prole The overall shape of either the P branch or the R branch is due to the distribuGon of molecules among the rotaGonal states.

Thus, the prole of either branch can be used to esGmate the temperature of the gas sample using the Boltzmann equaGon (see quesGon 4 in the pre-lab)

To do this properly, you really need absorbance rather than %T. Our spectrometer can do this easily (absorbance is proporGonal to populaGon; transmiJance is not)

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VibraGon-RotaGon InteracGon Since B is known to depend on v, we can replace B in the equaGons by BvindicaGng its dependence on v.

The dependence of B on v is called vibra8on-rota8on interac8on.

The dependence of B on v can be expressed as:

E! v,J = " v+ 12( ) + B! vJ J +1( )

B!V = B! e ! e v+ 12( )! e values are pretty small for HCl/DCl

Dont confuse v and !

VibraGon-RotaGon InteracGon Consider v=0 v=1 R-branch:

P-branch:

! R = E1,J+1 E0,J= 32!V + B"1(J +1)(J + 2) 12!V B" 0J(J +1)= !V + 2B"1 + (3B"1 B" 0 )J + (B"1 B" 0 )J 2 J=0, 1, 2, ...

! P = E1,J1 E0,J= !V (B"1 + B" 0 )J + (B"1 B" 0 )J 2 J=1, 2, 3, ...

In both equations, J corresponds to the initial rotational quantum number

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Analyzing the Data Using Trendline Look at the equaGons for the R and P branches

Both are equaGons in J2 If we plot frequency vs. J and t a quadraGc trendline, we can nd all the B values from the coecients of the various terms in J

! R = !V + 2B"1 + (3B"1 B" 0 )J + (B"1 B" 0 )J 2

! P = !V (B"1 + B" 0 )J + (B"1 B" 0 )J 2

Analyzing the Data Using Regression Trendline analysis gives all the coecients but doesnt give any info about the uncertainty of the data (just plot against J and do quadraGc t).

For regression, create columns of J and J2 and do regression as usual, selecGng both J columns as your x variables in LINEST

AlternaGvely, you can do the following: LINEST(y-values, x-values^{1, 2},TRUE,TRUE)

Also works with cubic t with {1,2,3} instead

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Centrifugal DistorGon Bonds are not truly rigid but instead tend to stretch as the molecule rotates, due to centrifugal forces on the atoms.

This is corrected by applying a centrifugal distorGon term called De

In general, De values are most important at high J values. You should assess how important De seems in your experiment by evalua

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Anharmonicity The harmonic oscillator model is only an approximaGon To improve on it, we can expand Ev as a power series in v+1/2 (keeping up through the squared term)

The coecient ee is called the anharmonicity constant. In order to evaluate the anharmonicity, it is necessary to have overtone spectra; this should be preJy straighqorward for DCl; less so for HCl

Ev = e(v +12 )ee(v + 12 )

2 + ...

Anharmonicity By observing fundamental (v=+1) and overtone (v=+2) transiGons for an anharmonic oscillator, then the vibraGonal energy spacings can be used to determine e and ee. (the anharmonicity constant)

E! V = " e v+ 12( )" ee v+ 12( )2thus:

E! 0 = 12"

e 14"

eeE!1 = 32

"e 94

"ee

E! 2 = 52"

e 254 "

ee

E! 01 = " e 2" e ! eandE! 02 = 2" e 6" e ! ethus:2E! 01 E! 02 = 2" e ! e

Therefore:

This is based on the fact that the overtone would be exactly 2X the fundamental if the molecule is a harmonic oscillator. The difference between the actual overtone and 2X the fundamental is due to the anharmonicity

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DissociaGon Energy Knowing the anharmonicity constant allows us to calculate the

dissociaGon energy, using the Morse potenGal (a beJer approximaGon than the harmonic oscillator)

The dissociaGon energy of a Morse oscillator can be expressed in terms of e and ee, as:

This will likely give a value that is too high, because we are only using one correcGon term for the anharmonicity. AddiGonal terms are needed to bring it into beJer agreement (but one term is OK for our purposes!

De =

e

2

4ee

Dont confuse this with the centrifugal distortion constant! Unfortunately they have the same symbol but they are vastly different quantities!

Isotopic Splitng

Remember that the peaks in the HCl/DCl spectra are split due to the presence of two naturally occurring chlorine isotopes, 35Cl and 37Cl.

If we assume that the molecule is a HO, we can calculate the degree of splitng using the following:

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Isotopic Splitng For HCl, this gives ~ 2 cm-1 difference between the two isotopic forms. This is easily resolved on the spectrometer (0.4 cm-1 resolution). The splitting for DCl is slightly larger, due to the larger mass of D.

= 12k

(H 35Cl)(H 37Cl) =

(H 37Cl)(H 35Cl)

0.97370.9722 = 1.00075