Question

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?

Short answer

(a) 3.125 \times 10^{13} \text{ s}^{-1}, (b) 3.468 \mu \text{m}

Step-by-step solution

Understand the relationship between frequency and wavelength

To solve these problems, use the equation that relates frequency (u), wavelength (\text{λ}), and the speed of light (c): \[ u = \frac{c}{\text{λ}} \] where \( c = 3.0 \times 10^8 \text{ m/s} \).

Convert wavelength to meters for part (a)

The given wavelength is \( 9.6 \mu \text{m} \). Convert this to meters: \[ 9.6 \mu \text{m} = 9.6 \times 10^{-6} \text{ m} \].

Calculate the frequency for part (a)

Use the equation from step 1 to find the frequency: \[ u = \frac{3.0 \times 10^8 \text{ m/s}}{9.6 \times 10^{-6} \text{ m}} \] Simplifying this gives: \[ u = \frac{3.0 \times 10^8}{9.6 \times 10^{-6}} \text{ s}^{-1} \] \[ u = 3.125 \times 10^{13} \text{ s}^{-1} \]

Use the same equation for part (b)

For part (b), the frequency is given as \( 8.652 \times 10^{13} \text{ Hz} \). Use the same equation, but solve for wavelength (\text{λ}): \[ \text{λ} = \frac{c}{u} \]

Calculate the wavelength for part (b)

Substitute the given frequency into the equation: \[ \text{λ} = \frac{3.0 \times 10^8 \text{ m/s}}{8.652 \times 10^{13} \text{ Hz}} \] Simplifying this gives: \[ \text{λ} = 3.468 \times 10^{-6} \text{ m} \] Converting to micrometers: \[ 3.468 \times 10^{-6} \text{ m} = 3.468 \mu \text{m} \]

Key concepts

IR Spectroscopy

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.

Frequency and Wavelength Relationship

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.

Speed of Light

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.