Exp4 Iodine

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CHEM 379, Fall 2011 Page 1 of 6

Physical Chemistry Structure Laboratory

Experiment 4: The Absorption Spectrum of Iodine

Background Reading:In addition to the theory presented below, read the article by R. B. Snadden 1. For further

details on the theory of electronic-vibrational spectra of diatomic molecules (I2 in particular),read the Theory section (pp. 425-428) and the Experimental section (Absorption spectrum,pp. 428-429) of experiment 40 in Shoemaker 6th edition (This experiment is not in the 7th or 8theditionsthe relevant text is posted on the course web page). Refer to your Physical Chemistrytextbook for additional background on electronic, vibrational, and rotational spectra ofdiatomic molecules.

Total Internal Energy and the Harmonic Oscillator:

A diatomic molecule such as I2 in the gas phase has an overall internal energy,Eint.To a degree of approximation, this energy can be seen as the sum of the energies arisingfrom the molecules electronic state energy,Ee, itsvibrational state energy,Ev, and itsrotational state energy,Er; i.e.,

Eint =Ee +Ev +Er (1)

The electronic, vibrational, and rotational energies are individually quantized (Figure 1). Theground state value ofEint occurs when all three components are in their ground states(which arises when the quantum number associated with the type of energy has is lowestvalue). However, in a real sample all combinations of allowed values ofEe,Ev, andEr mayoccur for various fractions of the total population of molecules.

Figure 1. Quantized electronic, vibrational, and rotational levels for the internal energy of a

molecule, as Eint =Eelec +Evib + Erot. The quantum numbers v and J are associated with Evib and

Erot, respectively.


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As shown in Fig 1, the energy separations between the various electronic states are relativelylarge, the separations between vibrational states in any electronic state are smaller, and theseparations between various rotational states in any vibrational state are the smallest.Furthermore, note that the separation between vibrational states gets smaller as the quantum

numbervgets bigger, but the separation between rotational states gets larger as the quantumnumberJgets larger.

A photon of the appropriate energy could induce a transition between any of thesestates. In this experiment we will be looking at the visible spectrum due to transitions from

the electronic ground state to the first electronic excited state; i.e.,E0E*. This transitioncan occur from any of the vibrational states in the electronic ground state to any of thevibrational states in the electronic excited state. Transitions from any of the availablerotational states associated with each vibrational state are also possible, but the energydifferences are too small to result in observable bands at the resolution available on ourspectrophotometers. Thus, we will be looking at bands arising from the combination ofelectronic-vibrational transitions.

In either electronic state, the vibrations of the molecule behave approximatelylike aharmonic oscillator with energies

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

where, =2 is the vibrational frequency. The energy levels of a harmonic oscillator,compared to the parabolic energy variation of a classical (non-quantum) oscillator, are shownin Fig. 2. Note that the lowest energy, at the bottom of the parabolic potential well, is notpossible. At that point, the molecule would be static with its atoms at their equilibriuminternuclear separation. The difference between this energy and the first vibrational state (v= 0) is the zero-point energy.

Figure 2. The energy levels of a harmonic oscillator


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The vibrational frequency, , in any state is the same and is related to the bondsforce constant, k, by the harmonic oscillator expression:

k (3)

From this it follows:

k = = 4c2 (4)

where =/2c=/cis the vibrational frequency in wavenumbers (cm-1).Note that the harmonic oscillator model predicts equal spacing between vibrational

levels. Real molecules have potential energy curves with some degree of anharmonicity,which causes the spacing between states to decrease as v increases. The potential is oftenapproximated by a Morse curve, as shown in Fig. 3.

Figure 3. Energy levels of an anharmonic oscillator (Morse potential curve). Note the successively

smaller separations between energy states as v increases (bottom to top).

From the Schrdinger equation for the vibrational levels of a diatomic molecule, theenergy in any state is given by

Ev =(v+ )(v+ )2 + ... (5)

where is the anharmonicity constant. Higher terms in Eq. (5) are usually neglected.At room temperature, most molecules in a sample of I2 exist in the electronic ground state.In that state, most are also in the vibrational ground state (v= 0). However, significantportions of the population of molecules exist in higher vibrational energy states (v= 1, 2, ...).

Thus, the electronic transitionE0E* may be from any vibrational statevof the groundelectronic state to any vibrational state v of the excited electronic state, as shown in Fig. 4.Note that the spacing of the vibrational energy levels in the excited electronic state aredifferent from those in the ground state because of changes in the force constant k(a


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measure of bond strength).

Figure 4. Electronic-vibrational transitions for the fundamental (left), first hot band (center) and

second hot band (right). The transition notations n,m refer to the vibrational states in the excited andground electronic states, respectively.

Following Snadden, we will designate the change in vibrational states with theelectronic transition by the index notation n,m, where nis the value ofv' in the electronicexcited state, and mis the value ofvin electronic ground state. All transitions fromv= 0(designated n,0) are part of the fundamental band. All transitions fromv= 1 (designatedn,1) are part of the first hot band, and all transitions fromv= 2 (designated n,2) are part ofthe second hot band. In the visible spectrum, instead of a single band for the electronictransition, we observe three broad bands (the fundamental and two hot bands) with saw-tooth fine structure corresponding to the various vibrational transitions from a particular

vibrational statevin the electronic ground state to all possible vibrational statesv' in theelectronic excited state. The appearance of the band structures can be seen in Snadden's Fig.2. The fundamental band involves the largest energy separations between states, and so itoccurs at the highest energy (lowest wavelengths) of the three bands.

We will use data from the frequencies (measured as wavelengths) of the vibrationalfine structure of the electronic transitions arising from the fundamental band and the twohot bands. Within the electronic ground state, we define the energy difference between thevibrational statesv= 0 andv= 1 asX, and the difference between the vibrational statesv=1 andv= 2 as Y, as indicated in Fig. 5.

ThusX=E0 (v= 1)E0 (v= 0) and Y=E0(v= 2)E0 (v= 1) (6)Using the anharmonic vibrational energy given the Morse potential function (Eq. 5),

forXwe haveX =E0(v=1) - E0(v=0) = (3/2) (9/4) (1/2) (1/4)

= 2(1 2) hc(12) (7)that, expressed in units of wavenumbers, gives

X = (12) (8)


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Figure 5. Pairs of electronic-vibrational transitions that define X as the energy separation between v

= 0 and v = 1 of the electronic ground state, and Y as the energy separation between v = 1 and v = 2

of the electronic ground state.

Similarly, for Y we have

Y=E0 (v= 2)E0 (v= 1) = (5/2) (25/4) (3/2) (9/4)

= 4(1 4) hc(14) (9)

that expressed in units of wavenumbers (cm-1)

Y= (1 4) (10)

As can be deduced from Fig. 5, X= 0,00,1 = 1,0 1,1 = 2,02,1 = ... (11)

Y= 0,10,2 = 1,11,2 = 2,12,2 = ... (12)

In other words,Xis the frequency difference (in cm-1) between all peaks in thefundamental band (v= 0) and first hot band (v= 1) that correspond to transitions to thesame vibrational statev' in the electronic excited state, and Yis the frequency difference (incm-1) between all peaks in the first hot band (v= 1) and second hot band (v= 2) thatcorrespond to transitions to the same vibrational statev' in the electronic excited state.

By matching all peak pairs that defineXand all peak pairs that define Y, averagevalues for both parameters can be obtained. The method for matching the pairs is describedin Snadden's paper. Basically, you are looking for sequential pairs in two bands that give arelatively constant difference for all pairs. UsingXand Yfrom these data, Eqs. 8 and 10 can

be solved simultaneously to obtain and , the vibrational frequency and anharmonicity

constant for the ground electronic state. Knowing, the force constant k can be obtainedfrom Eq. (4). Furthermore, the dissociation energy of I2, Do, can be estimated as

Do = hc(1 2)/4hc/4(13)

The relative populations of molecules in any two states separated by an energyE =EbEa, expressed as a fraction, is given by the Boltzmann distribution expression


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)exp(/Tk

ENN

b

ab

(14)

Here kb is the Boltzmann constant (kb = 1.38066 x 10-23

JK-1) and Tis the temperature in

Kelvin. Thus, the relative populations of molecules atv= 1 tov= 0 andv= 2 tov= 1 can

be estimated as

)exp(/01

Tk

XhcNN

b

and )exp(/

12Tk

YhcNN

b

(15)

Note that in these expressions, kb is the Boltzmann constant, not the force constant of I2.Also, note that in Snadden's sample calculation, N0:N1:N2 = 1:0.35:0.12, because N1/N0N2/N1 = 0.35 and (0.35)(0.35) = 0.12.

Laboratory Procedures:

Recording the Spectrum

This experiment will be conducted using the Cary-14 UV-Vis spectrometer. Obtainthe medium-resolution visible absorption spectrum of gaseous diatomic iodine, whichconsists of overlapping bands from electronic transitions, each having pronouncedvibrational fine structure. Under the conditions of the experiment, rotational fine structurecannot be resolved.

Glass sample cells filled with iodine vapor in equilibrium with solid I2 will beprovided by the instructor. A wavelength scan from 500 nm to 630 nm covers the region ofinterest. Each group will generate their own spectra and assign peaks. If additional data isneeded from other groups, then it will be necessary to average all of the relevant peak values.

Calculations1. Carry out an analysis of the spectrum following the procedure and notation described in

Snaddens paper. Tabulate your data and calculateX, Y, , , k and Do.2. Values ofXand Yshould be quoted with standard deviations, and the consequences of

those confidence limits on calculated values of, , k and Do should be assessed anddiscussed.

3. Compare your calculated values of, , k and Do with literature values.4. Estimate the relative populations of the vibrational states v = 0, 1, 2 of I 2 in its groundelectronic state [In carrying out this calculation, note that k is the Boltzmann constant].

1 R. B. Snadden, J. Chem. Educ. 1987, 64, 919-921.

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Exp4 Iodine