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 H 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 states v=2,3,4, will allow us to determine the Equilibrium Rotational Constant B e. Inclusion of literature values for the 1 st thru 4 th 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: = (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 its reduced mass. v is the quantum number associated with the vibrational energy state E v. In
P a g e 2 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 E v'' to E v', 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'') (Eq. 5) 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
P a g e 3 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 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 4 Solving 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, 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) = 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 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)] + 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, due to 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" will also depend on the vibrational quantum number (denoted as B v ). We define a vibration-rotation Coupling 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 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 + 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 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.
P a g e 6 Thus, 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) 3 and, 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 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 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 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: = + (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 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 H 35 Cl is: Overtone [cm -1 ] 0-2 5667.9841 0-3 8346.782 0-4 10922.81 0-5 13396.19
P a g e 7 Next, we can use our measured values of B 0 and B 1, along with additional data, and a linear least square analysis according to equation 13, to determine B e. Needed data for H 35 Cl is: v B [cm -1 ] 2 9.834663 3 9.534909 4 9.2360 5 8.942 Finally, once the spectroscopic constants and B e are determined, we can use our model for the HCl molecule to determine the molecular parameters k and r e. e = c = (1/2 ) (Eq. 22) B e = h / 2 c r e 2 (Eq. 23) (See development above.) Here r e 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 B 0 and B 1, 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. B v = h / 2 c (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 H 35 Cl. We will use this information to extract the spectroscopic constants B 0, B 1, D e and. We will supplement the fundamental line data with additional spectroscopic data to determine, and B e. Finally, we will invoke our molecular model for HCl and determine the constants k and r e,, and.
P a g e 8 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) + 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 the solid NaCl as pictured below.
P a g e 9 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 H 35 Cl.
P a g e 10 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: B 0, B 1 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 B 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 2145.1630 D 37 Cl 2141.82 6. Use the following data to determine r e for alternate isotopic substitutions of HCl. Molecule B e [cm -1 ] H 37 Cl 10.578 D 35 Cl 5.448794 D 37 Cl 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 HCl 35 and DCl 35 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.