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This graph shows the absorbance of different wavenumbers of infrared radiation causing different forms of vibrations of the water molecule. I would like to know why the graph turns out the way it is.
How do we decide which kind of vibration would take in the most energy? How do we know their respective absorbances?

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Molecules' bonds vibrate at all temperatures, which manifests e.g., by change of bond lengths, or (intermolecular) bond angles. At $T = \pu{0 K}$, molecules still vibrate, too.

The energy in electromagnetic radiation is discontinous, discrete (you recall $\hbar$). For some energies in electromagnetic radiation, the electric field vector brings to the molecule in question a packet of energy which matches in frequency the frequency of an already existing vibration. In an analogy, think of a pendulum; its swing is amplified if you tap with the matching frequency. Similar with molecules (as if consisting of multiple swings) and IR radiation (the pusher); on certain energies, the molecule's vibrations get amplified. The electric field vector gets attenuated.

A match in energy (here: in frequency) is not the only criterion to consider. IR spectroscopy is selective to show only these excited vibrations which change the dipole moment of the bond/the molecule. (Somewhat complementary to this is Raman spectroscopy, reporting about changes in the ease to polarize the electron cloud in a bond.)

Now for the shape of the spectrum you called graph. On the ordinate, the energy is plot; wavenumbers is one representation of the different frequencies probed during the IR spectroscopy experiment. The abscissa relates how efficient the transfer of energy from the electric field vector of the IR radiation to the molecule is: great absorbance equals to exciting these vibrations further is efficient, low absorption equals to there is not much change in intensity of the electric field vector passing the sample. (Similar to the Beer-Lambert law typically introduced with UV-Vis spectroscopy, the number, the concentration of molecules in the optical path length contribute to these readings.) Typically IR spectroscopy reports vibrations with high resonance frequency on the left, and vibrations with low resonance frequency on the right side. Instead of absorbance, the ordinate may scale transmission.

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(IR spectrum of ethanol as liquid film (neat), recorded by SDBS, entry 1300)

In your example, $\nu_1$ and $\nu_3$ are shown under one envelope. It is one of the characteristics of IR spectra for liquid/solid samples that the quantitized transitions are less easy to discern, because close molecules interact with each other. Equally, the more complex the molecule, the greater the chance that signatures of similar vibrations overlap. In addition, you may observe the vibrations as fundamentals, their harmonics, and combinations of them. This one reason why often only a few characteristic absorption bands in the spectrum are assigned to vibrations in the molecule.


With the large body of recorded IR spectra, it is possible to assign typical bonds in molecules a force constant. Like in a spring, how strong their connection is. For one, this relates the position of the associated vibrations in the spectrum in energy (the frequency scale, ordinate). For two, it became possible to predict from scratch for still unknown molecules drawn where vibrations are likely to occur.

For the example of water, paste the SMILES string O into MolCalc, hit once «optimize» to get bond lengths reasonable, then click «calculate properties». On the results, select the tab «vibrational frequencies» to display the three vibrations of water:

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(animation of the IR-active vibrations of water, molcalc)

Note: If you carefully compare the spectrum you got with the one with the one of MolCal's prediction, you recognize a energetic different sequence of the vibrations shown and different frequencies. This is where the how the computation is performed becomes important. Does it consider an ensemble of molecules in condensed state (like the one one records in the lab with a spectrometer), where the molecules of the matrix interact with each other? A neat spectrum (only the compound itself, e.g. in ATR-IR spectroscopy) may very well show the bands at a different position than one recorded for the same compound in a suspension of nujol, or a pill of $\ce{KBr}$. On the other hand, to obtain the result faster, computations often assume the molecule as isolated in vacuum. And there are different levels of theory of computation; for teaching purpose (up to ten non-H atoms), the affordable one by MolCalc often suffices.

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