![]() ![]() When derived, the energy for the J th rotational state is found by This is given by the equation found by solving for eigenvalues of a rotational system Schrodinger equation. However, because these rotational energy levels are not a continuum but rather discrete, quantized levels, a different definition must be used. Where I is the moment of inertia, ω is the angular velocity, and it is then put into terms of L, the angular momentum, μ, the reduced mass, and r, the bond length. To describe the rotation of an object, the classical mechanics form would describe the energy as The ability to extrapolate out from the spectral data into molecular information depends on some mathematical relations that begin with classical mechanics and are brought into the quantum mechanical field through some manipulation. This difference allows a very small but significant amount of transitions where Δν=+2, despite that being against the typical “selection rules” for quantum mechanic systems. The overtone’s appearance is due to the potential of the molecule being more similar to a Morse potential rather than a true harmonic oscillator. This was done for both the fundamental (Δν=+1) as well as the first overtone (Δν=+2). Once a spectrum like this was obtained for HCl and DCl, peak values were assigned to the P and R branches, for both 35Cl and 37Cl. The peaks to the left correspond to what is called the “P-Branch,” where ΔJ = -1, and the peaks to the right correspond to the “R-Branch,” where ΔJ = +1. ![]() This change in vibrational energy (ν=0 to ν=1) with several different rotational energy changes (J=☑) leads to the splitting in a typical ro-vibrational spectrum shown in Figure 2.įigure 2 – Typical appearance of a ro-vibrational FTIR spectrum 3 Figure one depicts this overlay of both rotational and vibrational energies:įigure 1 – Rotational energy levels, J, superimposed upon the vibrational energy levels, v, and the separation thereof into P & R branches. With sufficiently high resolving power, a vibration spectrum can be obtained that not only shows the vibration excitation, but also the rotation excitation within it. In essence, as the light from the source passes through the sample cell, the sample is energetically excited both rotationally and vibrationally. In this experiment, FTIR spectroscopy was used to analyze rotational-vibrational transitions in gas-state HCl and DCl and their isotopomers (due to 35Cl and 37Cl) to determine molecular characteristics. ![]() FTIR spectroscopy uses gratings for diffraction and then processes the raw data from an interferogram into the actual spectrum via the mathematical process known as a Fourier transform. 1 Thus, a higher resolving power is required. ~2900 cm^-1, respectively, simple dispersive IR spectroscopy is simply not viable. Because the rotational energy levels are much more closely spaced than the vibrational energy levels, ~20 cm^-1 vs. More simple spectrometers using prisms to diffract light do not have the resolving power necessary to separate the rotational effects in the vibrational regime of energy transitions. These constants were then used to determine the force constants, H 35Cl=527667(error) dynes/cm, H 37Cl=527781(error) dynes/cm, D 35Cl=522901(error) dynes/cm, and D 37Cl=521422(error) dynes/cm and equilibrium bond length, H 35Cl=1.274(error)Å, H 37Cl=1.275(error)Å, D 35Cl=1.201(error)Å, D 37Cl=1.202(error)Å.įourier transform infrared spectroscopy (FTIR) is a form of spectroscopy used when high resolution spectroscopy is required. frequency, from which several physical constants were determined. Energy transitions from the spectra were plotted vs. This is the focus of most of the rest of this section.Spectra and Molecular Structure – HCl & DClįTIR spectroscopy was used to analyze rotational-vibrational transitions in gas-state HCl and DCl and their isotopomers (due to 35Cl and 37Cl) to determine molecular characteristics. It is best to work out specific examples in detail to get a feel for how to calculate the moment of inertia for specific shapes. This, in fact, is the form we need to generalize the equation for complex shapes. ![]()
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