Covalent bonds in a molecule absorb radiation in the IR region and vibrate at characteristic frequencies. (a) The \(\mathrm{C}-\mathrm{O}\) bond absorbs radiation of wavelength \(9.6 \mu \mathrm{m}\). What frequency (in \(\mathrm{s}^{-1}\) ) corresponds to that wavelength? (b) The \(\mathrm{H}-\mathrm{Cl}\) bond has a frequency of vibration of \(8.652 \times 10^{13} \mathrm{~Hz}\). What wavelength (in \(\mu \mathrm{m}\) ) corresponds to that frequency?

Infrared (IR) spectroscopy is a powerful technique used to study molecules. It works by shining infrared light on molecules causing them to absorb specific frequencies of radiation and vibrate. The unique absorption patterns can help identify different types of bonds and molecules. For instance, a carbon-oxygen (C-O) bond might absorb IR light at a different frequency than a hydrogen-chlorine (H-Cl) bond. These differences in absorption are due to the bonds vibrating at characteristic frequencies. When IR radiation is absorbed, it matches the energy level difference between vibrational states of the bond.
This is why IR spectroscopy is essential for identifying molecules and understanding their structure.

Wavelength (\text{λ}) and frequency (u) are important properties of waves, including the waves in IR spectroscopy. They are related by the equation:
\[ u = \frac{c}{\text{λ}} \] where \[ c \] is the speed of light (approximately \[ 3.0 \times 10^{8} \text{ m/s} \]). This relationship implies that as the wavelength of a wave increases, its frequency decreases, and vice versa. To solve problems involving these properties, it's crucial to first convert all units to be consistent. For example, in the given exercise, the wavelength from micrometers must be converted to meters before calculating the frequency. Using this relation, we can determine the frequency of an IR absorption given the wavelength and vice versa.

The speed of light (\text{c}) is one of the fundamental constants in physics and equals approximately \[ 3.0 \times 10^{8} \text{ m/s} \]. This constant is crucial when working with the relationship between frequency and wavelength. Light travels incredibly fast, which is why changes in wavelength lead to significant changes in frequency. In the context of the exercise:
- If we are given a wavelength, we can find the corresponding frequency by dividing the speed of light by the wavelength.
- Conversely, if we know the frequency, we can find the wavelength by dividing the speed of light by the frequency.
This relationship is key to understanding how different types of light (including infrared) interact with matter and why certain bonds absorb specific frequencies of IR light.