Fall 2020 CHE 4480 Palmer This wor k by Alycia Palme r is licensed under CC BY 4.0 1 Rotationalvibrational Spectroscopy Determining rovibrational constants for HCl Objective Students will use the IR spectrum of HCl to determine the values of e, e, e, and for H35Cl and H37Cl. Logistics The duration is two weeks: one week for analysis of the spectrum and peak picking and one week for calculations and spreadsheet work . Introduction In Experiment #2, â€œVibronic spectra of I2 and Br2â€, we used visible light to investigate electronic transitions within the molecules. With this energy of light we were able to approximate the energy to break the bond. In this experiment, we will be using infrared (IR) light that is lower energy than visible light, and is too low to cause electronic transitions. IR photons interact with the vibrational and rotational wavefunctions of a molecule. These â€œrovibrationalâ€ transitions will be investigated in this experiment. Vibrational Energy Levels â€“ Harmonic and Anharmonic Oscillators It is convenient to treat the molecule's vibrations as those of a harmonic oscillator, which assumes a chemical bond acts like a spring . The energy of a vibration is quantized in discrete levels and given by Equation 1. Equation 1 Where the vibrational quantum number, v (pronounced vee), can have integer values 0, 1, 2..., etc. And the frequency of the vibration, (the Greek letter nu) is given by Equation 2. Equation 2
Fall 2020 CHE 4480 Palmer This wor k by Alycia Palme r is licensed under CC BY 4.0 2 Where k is the force constant and is the reduced mass of a diatomic molecule with atom masses m1 and m2, given by Equation 3. Equation 3 Equation 1 gives energy in units of Joules , which is inconvenient because of the extremely small energies involved with molecules. It is more convenient to work with cm1 (called â€œwavenumbersâ€ and equal to reciprocal wavelength). Dividing both sides of Equation 1 by Planckâ€™s constant and the speed of light gives the vibrational term value, G( v ). Equation 4 Where e is the harmonic vibrati onal frequency in wavenumbers. For small values of the vibrational quantum number, v , the harmonic oscillator gives a good approximation of the real energy, but at higher v , the bond no longer acts like a spring because it is near it breaking point. Thus real molecules are better modeled as an anharmonic oscillator. The difference is highlighted in Figure 1. Note that in an anharmonic oscillator, the vibrational energy levels get closer together as v increases. CC-SA
Fall 2020 CHE 4480 Palmer This wor k by Alycia Palme r is licensed under CC BY 4.0 3 A nharmonicity is better represented by a power series , as in Equation 5. Equation 5 Where e is the harmonic vibrational frequency and ee is the first anharmonic correction constant, both in wavenumbers. The second anharmonicity constant ee is usually small, and so the third term in the power series is usually neglected. The vibrational term value modeled by Equation 5 is only part of the total energy of a molecule. The rotational term value will be derived in the next section. Rotational Energy Levels â€“ Rigid and Non rigid Rotors We treat the molecule's rotations as those of a rigid rotor . The energy of a rotation is also quantized in discrete levels given by Equation 6. Equation 6 In which is the moment of inertia, given by Equation 7. Equation 7 where is the reduced mass (Equation 3) and r is the equilibrium bond length. Again, the energy in Equation 6 is in units of Joules (if the moment of inertia is in . Dividing by gives the rotational term value F(J) in units of wavenumbers. All constant s in Equation 8 are set equal to , which is the rotational constant. Equation 8 Real molecules are not accurately modeled by Equation 8 due to nonrigid behavior. Atoms in real molecules behave like masses connected by a spring that is subject to centrifugal force. As the rotational energy increases, the internuclear distance increases and the moment of inertia also increases. This causes higher energy rotational levels to become more closely spaced than the rigid rotor model predicts. Equation 9
Fall 2020 CHE 4480 Palmer This wor k by Alycia Palme r is licensed under CC BY 4.0 4 Equation 9 provides a close approximation of the energy due to the rotational term , where J is the rotational quantum number and the constants are as follows: is the rotational constant in wavenumbers D is the centrifugal distortional constant in wavenumbers It may be noted that value of D is very small compared to B with the result that centrifugal distortion is significant only for very large J values. Rotational Vibrational Transitions The total energy of an anharmonic, nonrigid molecule is a combination of its vibrational and rotational energy, as in Equation 10. Equation 10 In real molecules, the rotational and vibrational motions are coupled together. As a result, Equation 10 contains the rotational e, which has units of wavenumbers. Equation 11 In this experiment, when HCl absorbs IR light, the molecule will undergo a transition from the ground state to an excited state vibrational level . Because 99.999% of molecules in an ordinary sample of HCl gas at room temperature are in their lowest possible vibrational state ( v = 0) , only transi tions from the lowest vibrational state are possible. These are called fundamental transitions. However, many rotational levels can be populated at room temperature, and we will see many combinations within the fundamental band, as shown in Figure 2. T rans itions in which branch , and transitions in 1) belong to the P branch. The Q A convention in spectroscopy is to label lower energy quantum states with a double prime (â€) and higher energy states with a single prime (â€™).
Fall 2020 CHE 4480 Palmer This wor k by Alycia Palme r is licensed under CC BY 4.0 5 Cartoon depiction of rotational energy levels, J, imposed on vibrational energy levels, v . The transitions between levels that would result in the P and R branches are depicted in purple and red, respectively, in addition to the theoretical Q -branch line in blue.All transitions in the R branch will be of the form ( v â€=0, Jâ€=J) ( vâ€™ =1, Jâ€™=J+1). T ransitions in the P branch will be of the form ( v â€=0, Jâ€=J) ( vâ€™ =1, Jâ€™=J 1). Peaks in the R and P branches can be labeled using shorthand notation. For the P and R branch, the convention is to use the ground state rotational level for the number in parenthesis , i.e. P(Jâ€). Some examples are noted below. Label Transition R( 1) ( v â€=0, Jâ€=1 ) vâ€™ =1, Jâ€™=2 ) P(2) ( v â€=0, Jâ€=2) vâ€™ =1, Jâ€™=1) Q(0) ( v â€=0, Jâ€=0) ( vâ€™ =1, Jâ€™=0) The peaks v = 1) for HCl will be observed at the energies calculated by Equation 12. Equation 12 To solve for the freque ncy of a transition, Equation 12 can be used with the vibrational term from Equation 5 and the rotational term from Equation 9 for the appropriate rotational quantum number. The result after simplification and defining N in terms of the J value is Equat ion 13. Equation 13
Fall 2020 CHE 4480 Palmer This wor k by Alycia Palme r is licensed under CC BY 4.0 6 In Equation 13, the value of N is defined as follows : P Branch N P 0, e, , and D can be determined for each 1H35Cl and 1H37Cl. Procedure Download the Excel template and HCl spectrum from Blackboard. Spreadsheet Assignment Plot the IR spectrum of HCl and label each peak with the wavenumber and name (i.e. P(1)) of the transition. Construct a plot of wavenumber vs. N value, and fit the data with a 3rd order polynomial using the LINEST function. Use the data generated using LINEST and Equation 13 to find the con stants 0e, , and D for each 1H35Cl and 1H37Cl. References Halpern, A. M.; McBane, G. C. , 3rd Ed.; W. H. Freeman and Company: New York, 2006. LibreText https://chem.libretexts.org/Courses/University_of_California_Davis/UCD_Chem_110B%3 A_Physical_Chemistry_II/Text/13%3A_Molecular_Spectroscopy/1305._Overtones_Are_Observed_in_Vibrational_Spectra (accessed Feb 3, 2020)