Question

The wavelength of the fundamental $O-H$stretching vibration is about $3.0$µm. What is the approximate wavenumber and wavelength of the first overtone band for the $O-H$stretch?

Short answer

Approximate wave number and wavelength is$6.6\times {10}^3{\mathrm{cm}}^{-1}$and$1.5\mu m$respectively.

Step-by-step solution

Step 1. Given information

Stretching vibration is given as$3.0\mu m$

Step 2. Formula used

The distance between the peak and trough of a wave is specified as its wavelength, and that wave might be a sound wave, an electromagnetic wave, or any other wave. A wave has two points: a high point, also known as the crest, and a low point, also known as the trough. The wavelength sign is lambda $\lambda$, and the wavelength unit is metres.

The wavelength formula is as follows:

$\lambda =\frac{\mathrm{v}}{\mathrm{f}}$

Where vis velocity and fis frequency.

Wave number is defined as the ratio of unit distance to number of wavelength. The symbol of Wave number is $\overline{V}$.

$\overline{v}=\frac{1}{\lambda }.....\left(1\right)$

rearranging the above equation,

$\lambda =\frac{1}{\overline{v}}......\left(2\right)$

Step 3. Unit conversion

Convert the units as,

$$\begin{gathered} 3.0\mu \mathrm{m}=(3.0\mu \mathrm{m})\left(\frac{1\times {10}^{-6}\mathrm{m}}{1\mu \mathrm{m}}\right)\left(\frac{1\times {10}^{2}c\mathrm{m}}{1m}\right) \\ =3.0\times {10}^{-4}c\mathrm{m} \\ \overline{v}=\left(1\right)/\left(3.0\times {10}^{-4}c\mathrm{m}\right) \\ =3.3\times {10}^3{\mathrm{cm}}^{-1} \end{gathered}$$

Step 4. calculating wavelength

The first over tune is at $2\times \overline{v}$,

$$\begin{gathered} =\left(2\right)\left(3.3\times {10}^3{\mathrm{cm}}^{-1}\right) \\ =6.6\times {10}^3{\mathrm{cm}}^{-1} \end{gathered}$$

Therefore approximate wavenumber for

$O-H$is $6.6\times {10}^3{\mathrm{cm}}^{-1}$

Now substitute the value in equation $\left(2\right)$,

$$\begin{gathered} \lambda =1/6.6\times {10}^3{\mathrm{cm}}^{-1} \\ =1.5\times {10}^{-4}\mathrm{cm} \\ =1.5\mathrm{\mu m} \end{gathered}$$

Therefore the approximate wavelength of $O-H$is$1.5\mu m$