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SpectroscopicTransitions in Molecules. Rotational-Vibrational Transitions. The harmonic oscillator has already been treated and the model may be applied to molecules in their lower vibrational states. The allowed energies of the quantum harmonic oscillator are. Where:.
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Rotational-Vibrational Transitions The harmonic oscillator has already been treated and the model may be applied to molecules in their lower vibrational states. The allowed energies of the quantum harmonic oscillator are Where: and k is the force constant and μ is the reduced mass.
In spectroscopy, functions for the energy are called terms. In the case of the vibrational energy of a molecule, G(ν) is called the vibrational term and the energy is usually expressed in wavenumbers (cm-1). The vibrational energy then becomes: Where: And the energies of vibrations range from hundreds to thousands of wavenumbers.
The quantum mechanics of the diatomic rigid rotor were also previously discussed and the energy was expressed as: where I is the moment of inertial of the molecule. The degeneracy associated with the J-th state is In similar fashion to the vibrational energy, the rotational term, F(ν), is in units of wavenumbers. It is customarily written as:
~ B is called the rotational constant and is specific to each molecule: The tilde means the energy is expressed in wavenumbers and Typical values range from 1 to 10s of cm-1. Using the model of a vibrating but rigidly rotating molecule gives the total rotational-vibrational energy as:
E N E R G Y R Each vibrational state has its own set of associated rotational energy levels.
Every spectroscopic transition must obey at least one or Possibly a set of selection rules. These are usually, although not necessarily, limitations on how much the quantum numbers of the energy levels associated with the transition can change. For absorption of electromagnetic radiation, the rigid rotor- harmonic oscillator selection rules are: The change of the vibrational quantum number can also be -1 in which case the process is one of emission of radiation. Note that Δ J = 0 is a forbidden transition.
For the case of absorption, there are two possibilities. The first is for Δ J = +1. This means that Ju = Jl + 1 = J + 1 And the second is for Δ J = -1 :
The spectrum that is produced will be centered about the vibrational frequency which, because of the Δ J = 0 selection rule being forbidden, cannot appear. There will be rotational absorptions to higher and lower energies than the pure vibrational frequency, The lower energy branch, the P-branch arises from the Δ J = -1 transitions and the higher energy branch, the R-branch , arises from the Δ J = +1 transitions.
Each of the lines arises because of the absorption of energy from light so that the molecule goes from a lower energy state to a higher one. Examination shows the lines to be apparently evenly spaced. What is the significance of the energy differences between an adjacent pair of absorption lines? Clearly, these will be the differences in energies between two transitions. Both equations show this spacing will be the value 2B. Close examination of the spectrum of HBr shows that the Spacing is not constant and differs between the two branches and differs within a branch as the rotational quantum number increases. ~
Vibrational-Rotational Coupling The simple rigidly rotating harmonic oscillator has energies: The rotational constant is inversely proportional to the moment of inertia: B ~ ( μRe2 )-1 ~ So as the molecule vibrates Re should increase with υ thereby leading to a decrease in the rotational constant of the molecule.
This is corrected for by using a rotational constant that depends upon the vibrational quantum number. This couples the rotational levels to the vibrations and this coupling is called the vibration-rotation interaction or ro-vib coupling. For a υ = 0 1 vibrational transition: for the R-branch.
And similarly for the P-branch: Were there no ro-vib coupling then B1 = B2 and the equations reduce to those for the system with no interaction. The Re will increase with increasing vibrational energy, so the moment of inertia will decrease, thus B1 < B2 thereby leading to a decrease in the spacing between the lines of the R-branch. This dependence is described by:
The Non-Rigid Rotor If the shape of the molecule is allowed to distort upon rotation, Then the restriction of the rigid rotor is lifted. It may be expected that as the rotational energy increases the molecule will have it’s bond lengthened because of the centrifugal distortion. This will lead to an effective decrease in the rotational energy since the longer bond will lead to a decrease in the rotational energy.
~ Here, D is the centrifugal distortion constant. For the JJ + 1 transitions, the absorption frequency is given by
When both anharmonicity and non-rigidity ( D = αe ) are accounted for, the P and R branches are: Where The P and R branches can be expressed in a single equation that is in terms of a new “quantum number”, m which is equal to J” + 1 for the R branch and –J” for the P branch: I And the frequency difference between two adjacent lines as:
Isotope Effect Recall that for the harmonic oscillator: Writing the equation for two molecules with different isotopes And ratioing the two equations to each other gives”
Similarly, for rotational spectra: Which gives a ratio of rotational constants:
Isotopic Exchange Statistical thermodynamics gives the equilibrium constant for isotopic exchange for ideal gases through the molecular partition functions: The first term arises from translational freedom, the second represents rotational contributions, with the last term clearly associated with the vibrational contributions.
Experimental • Gases ( DCl and HBr) must be in a gas spectrophotometery cell with a spectrometer capable of resolution of ~2.0cm-1. • Gases must be dry before filling the cell. • Cell windows (NaCl) must be relatively clear. • Gases are in lecture bottles. • Cells are filled on a vacuum line. • Rapid scans at different pressures should be made to obtain best transmission and resolution
Calculations and Analysis • Select best spectra, assign the spectra for each isotope and index the lines with appropriate m values. • Express frequencies to 0.01 cm-1. • Make a table of frequency differences between adjacent lines. • Plot these frequency differences against m. • Compute Be and αfrom the intercept and slope.
Use the Be and αvalues to calculate the vibrational frequencies using low value m’s. • Least-squares calculation should be used for the straight line analyses. • Calculate the moment of inertia and bond distances for the isotopomer diatomics. • Compute the ratios vo*/v and Be*/Be and compare with theoretical values. • Calculate splittings due to isotopes. • Calculate the K for isotopic exchange.
Calculate the force constants for the isotopomer diatomic molecules. • Calculate the vibrational partition function. • Calculate the vibrational contribution to the heat capacity. • Compare all calculated values with theory and their measurement using other techniques.
