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# What does infrared spectroscopy measure?

Mar 1, 2018

I like to think of it measuring the shadow of a molecule.

Certain bonds in a molecule vibrate at certain rates/conformations when irradiated by infrared radiation. It is mainly used coupled with nuclear magnetic resonance or mass spectrometry to identify unknown compounds in analytical organic or inorganic chemistry.

Mar 1, 2018

Infrared (IR) spectroscopy measures the change in dipole moment of molecules due to irradiating them with light at frequencies that trigger transitions between vibrational energy levels.

The peaks that show up on an IR spectrum lie in the range $400 - {\text{4000 cm}}^{- 1}$ or so, and are given by the transition from one vibrational energy level to another.

When we land on a resonant frequency as we scan the range of frequencies, that corresponds to the frequency for a vibrational mode of the molecule. Some examples are shown below of methane:

Typically we measure absorptions. For simplicity, a non-rotating linear anharmonic oscillator has energy levels given to second order by:

$t i l \mathrm{de} {E}_{\upsilon} = t i l \mathrm{de} {\omega}_{e} \left(\upsilon + \frac{1}{2}\right) - t i l \mathrm{de} {\omega}_{e} {\chi}_{e} {\left(\upsilon + \frac{1}{2}\right)}^{2}$

where:

• $t i l \mathrm{de} {\omega}_{e}$ is the fundamental vibrational frequency of the molecule in its equilibrium position (no displacement).
• $t i l \mathrm{de} {\omega}_{e} {\chi}_{e}$ is the anharmonicity constant of the molecule in its equilibrium position (no displacement).

Those absorption transitions are given by:

$\textcolor{b l u e}{t i l \mathrm{de} {\nu}_{\upsilon \to \upsilon + 1}} = \left[t i l \mathrm{de} {\omega}_{e} \left(\upsilon + 1 + \frac{1}{2}\right) - t i l \mathrm{de} {\omega}_{e} {\chi}_{e} {\left(\upsilon + 1 + \frac{1}{2}\right)}^{2}\right] - \left[t i l \mathrm{de} {\omega}_{e} \left(\upsilon + \frac{1}{2}\right) - t i l \mathrm{de} {\omega}_{e} {\chi}_{e} {\left(\upsilon + \frac{1}{2}\right)}^{2}\right]$

$= t i l \mathrm{de} {\omega}_{e} \left(\upsilon + \frac{3}{2} - \upsilon - \frac{1}{2}\right) + t i l \mathrm{de} {\omega}_{e} {\chi}_{e} {\left(\upsilon + \frac{1}{2}\right)}^{2} - t i l \mathrm{de} {\omega}_{e} {\chi}_{e} {\left(\upsilon + \frac{3}{2}\right)}^{2}$

$= t i l \mathrm{de} {\omega}_{e} + t i l \mathrm{de} {\omega}_{e} {\chi}_{e} \left[{\left(\upsilon + \frac{1}{2}\right)}^{2} - {\left(\upsilon + \frac{3}{2}\right)}^{2}\right]$

$= t i l \mathrm{de} {\omega}_{e} + t i l \mathrm{de} {\omega}_{e} {\chi}_{e} \left[{\upsilon}^{2} + \upsilon + \frac{1}{4} - {\upsilon}^{2} - 3 \upsilon - \frac{9}{4}\right]$

= color(blue)(tildeomega_e -2tildeomega_echi_e(upsilon + 1)