The Vibrational-Rotational Spectrum of HCl

CHEM 332LPhysical Chemistry LabRevision 2.2The Vibrational-Rotational Spectrum of HClIn this experiment we will examine the fine structure of the vibrational fundamental line forH 35 Cl in order to determine several spectroscopic constants; the Rotational Constants B 0 and B 1 ,the Centrifugal Distortion Constant D e , and the Fundamental Vibrational Band Origin .Inclusion of a few literature values for Rotational Constants B v for higher vibrational statesv=2,3,4,… will allow us to determine the Equilibrium Rotational Constant B e . Inclusion ofliterature values for the 1 st thru 4 th vibrational overtones will allow us to determine furtherspectroscopic constants; the Equilibrium Fundamental Vibrational Frequency and theAnharmonicity Constant . Finally, we will assume the HCl molecule is behavingapproximately as two masses connected by a stiff spring of constant k rotating with internucleardistance r. We will then extract these molecular parameters from the appropriate spectroscopicconstants.First, we must recognize that Hydrogen Chloride is a diatomic gaseous substance. Because it is adiatomic molecule, it will have one vibrational mode and one rotational mode ofmotion. Thus, we can model HCl molecules as vibrating rotors.We begin by examining the vibrational motion of the molecule and treat theatoms of the molecule as being connected by a Hookean spring of constant k. Inthis case, the Hamiltonian operator for the system is written as:= (Eq. 1)Solving the Schrodinger wave equation = E for the energy yields:E v = h (v + ½) v = 0, 1, 2, … (Eq. 2)where = (1/2) is the Fundamental Vibrational Frequency of the system and is itsreduced mass. v is the quantum number associated with the vibrational energy state E v . In

P a g e | 2anticipation of discussing spectral lines arising from with energy transitions, we can define thevibrational 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 allowfor non-Hookean behavior, then:G(v) = (v + ½) - (v + ½) 2 (Eq. 4)where is an anharmonicity constant. This anharmonicity correction lowers the energy ofeach vibrational state because the parabolic potential of Hookean behavior tends to broaden andflatten and become more Morse like. The anharmonicity correction becomes more severe athigher vibrational states where the deviation from Hookean behavior is more pronounced.A couple of points must be kept in mind whenconsidering equation 4. First, anharmonicitywill cause the fundamental frequency of theoscillator to depend on which vibrationalstate the system is in. Thus, we define anequilibrium fundamental frequency ; thisequals at the bottom of the potential wellwhere anharmonicity vanishes. Second, andof equation 4 are spectroscopic constantsthat can be measured regardless of the modelused for representing the molecule. On theother hand, the spring constant k of equation 2depends on the nature of the model used torepresent the molecule. This must be kept in mind when we extract values of k from themeasured spectroscopic constant .Now, vibrational spectral lines will occur when a photon is absorbed causing an energy transitionfrom E v'' to E v' , where v'' is the lower quantum state and v' is the upper. In terms of spectralTerms, this transition is represented as: = G(v') - G(v'') (Eq. 5)The fundamental line vibrational spectral linewill be given by = G(1) - G(0).Overtones , etc. are quantumforbidden by the selection rule v = ±1 and soare very weak. (Hot Bands, 1 2, 2 3, etc. arealso very weak because higher vibrational stateswill not be significantly populated due to

P a g e | 3insufficient thermal energy in the system.) Schematically, the spectrum will appear as:Using high resolution spectrometers, it is observed each of these lines exhibits more detailedstructure.For instance, at fairly low resolution, the Fundamental Line appears as a doublet. Using a higherresolution 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 madeup of a mixture H 35 Cl and H 37 Cl 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)

P a g e | 4Solving the Schrodinger wave equation = E for the energy yields:E l = l (l + 1) l = 0, 1, 2, … (Eq. 7)where I = r 2 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/ 2 c is the system's Rotational Constant. (Note: In spectroscopic notation, thequantum number l is denoted by the symbol J.) If we allow for centrifugal distortion as themolecule begins to "spin" faster and faster, then the Term becomes:F(J) = B e J (J + 1) - D e J 2 (J + 1) 2 (Eq. 9)where D e is the Centrifugal Distortion Constant. Again, B e and D e are spectroscopicallydeterminable 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 enoughtogether that thermal population of upper rotational states isconsiderable. Therefore, in microwave spectroscopy, energy transitions,as pictured to the left, will occur.These rotational states will lie on top the vibrational states, giving usspectral Terms T(v,J) = G(v) + F(J). This allows us to describe thefine 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 obtainthe following general result for the spectral lines: = (v' - v'') - [(v' - v'')(v' + v'' + 1)]+ B v' J'(J' + 1) - B v'' J''(J'' + 1) - D e J' 2 (J' + 1) 2 + D e J'' 2 (J'' + 1) 2

P a g e | 5(Eq. 12)Note that because the molecule "stretches" slightly as it moves into higher vibrational states, dueto anharmonicity, the internuclear distance r will depend on vibrational quantum number v(denoted as r v ), hence "I" will depend on the vibrational quantum number and therefore "B" willalso depend on the vibrational quantum number (denoted as B v ). We define a vibration-rotationCoupling Constant as e such that the nature of B's dependence on v is:B v = B e - e (v + ½) (Eq. 13)A similar problem occurs for the distortion constant D, except the coupling is weak enough wecan 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 Originas:= (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+ B 1 J'(J' + 1) - B 0 J''(J'' + 1) - D e J' 2 (J' + 1) 2 + D e 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 saidto be in the R-branch of the line; which occurs at higher wavenumbers. If J = -1, we are in theP-branch; at lower wavenumbers.

P a g e | 6Thus, for the R-branch, we have:R(J'') = J=+1 - 2 + (B 1 + B 0 ) (J''+1)+ (B 1 - B 0 ) (J''+1) 2 - 4D e (J''+1) 3and, for the P-branch, we have:(Eq. 1)P(J'') = J=-1 - 2 - (B 1 + B 0 ) J'' + (B 1 - B 0 ) J'' 2 + 4D e J'' 3(Eq. 17)We now turn to the question of how to extract the spectroscopic constants from the measuredpositions 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 B 0 and B 1 and the distortion constant D e ?Equations 16 and 17 can be re-arranged to obtain the following forms:= B 1 - 2 D e (J'' 2 + J'' + 1) (Eq. 18)= B 0 - 2 D e (J'' 2 + J'' + 1) (Eq. 19)Thus, measurements of R(J'') and P(J''), when plotted according to equations 18 and 19, and theresults 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:= + (B 1 - B 0 ) (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 thecase 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 H 35 Clis:Overtone [cm -1 ]0 - 2 5667.98410 - 3 8346.7820 - 4 10922.810 - 5 13396.19

P a g e | 7Next, we can use our measured values of B 0 and B 1 , along with additional data, and a linear leastsquare analysis according to equation 13, to determine B e . Needed data for H 35 Cl is:v B [cm -1 ]2 9.8346633 9.5349094 9.23605 8.942Finally, once the spectroscopic constants and B e are determined, we can use our model for theHCl molecule to determine the molecular parameters k and r e . e = c = (1/2) (Eq. 22)B e = h / 2 cr e2(Eq. 23)(See development above.) Here r e represents the equilibrium internuclear distance when themolecule is not vibrating. When the molecule moves into vibrational states v = 0,1,2,… it beginsto stretch, due to the anharmonicity of the potential, and r increases. Hence, the internucleardistance (really root-mean-square internuclear distance ) will depend on the vibrational statev. From the rotational constants B 0 and B 1 , the root-mean-square internuclear distance betweenthe H and Cl atoms in vibrational states v=0 and v=1, and respectively, can bedetermined.B v = h / 2 c (Eq. 24)Thus, we will experimentally determine the positions of all the fine-structures lines R(J'') andP(J'') of the fundamental band for H 35 Cl. We will use this information to extract thespectroscopic constants B 0 , B 1 , D e and . We will supplement the fundamental line data withadditional spectroscopic data to determine , and B e . Finally, we will invoke ourmolecular model for HCl and determine the constants k and r e , , and .

P a g e | 8ProcedureObtain a high resolution Infrared spectrum of HCl gas. You will use a 10 cm gas cell as picturedbelow.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 celldirectly.ii) Treat Sodium Chloride with concentrated Sulfuric Acid to produce the HydrogenChloride gas:NaCl(s) + H 2 SO 4 (l)HCl(g) + NaHSO 4 (s)You can do this by dropping, very slowly, ~30 mL of conc. H 2 SO 4 into about 5g of thesolid NaCl as pictured below.

P a g e | 9Your 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 theIR spectrometer. Take the spectrum with an appropriate resolution to see the fine structure in thefundamental line. Measure the positions of R(J'') and P(J'') spectral lines for H 35 Cl.

P a g e | 10Data AnalysisThe following calculations will require linear least squares analyses that are of fairly highprecision. Track your significant figures closely and make sure you perform an erroranalysis for each directly measured quantity.1. From your data, determine the spectroscopic constants:B 0 , B 1 and2. Use the supplemental overtone data to determine the spectroscopic constants:and3. Use the supplemental data for rotational constants to determine the spectroscopic constantB e .4. Determine the molecular parameters k, r e , and .5. Use the following data to determine k for alternate isotopic substitutions of HCl.Molecule e [cm -1 ]D 35 Cl 2145.1630D 37 Cl 2141.826. Use the following data to determine r e for alternate isotopic substitutions of HCl.Molecule B e [cm -1 ]H 37 Cl 10.578D 35 Cl 5.448794D 37 Cl 5.432

P a g e | 11ReferencesArnaiz, 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 DeuteriumChloride" 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 Constantsof the HCl 35 and DCl 35 Molecules" J. Opt. Soc. Am. 52 (1961) 1.Sime, Rodney J. Physical Chemistry: Methods, Techniques, and Experiments Saunders CollegePublishing, 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.