Question

Complete the following conversion table:

$\overline{\mathbf{v}}\mathbf{\left(}{\mathbf{cm}}^{\mathbf{-}\mathbf{1}}\mathbf{\right)}$	4000				1800	1670	1620	400
$\mathbf{\lambda }\mathbf{\left(}\mathbf{\mu m}\mathbf{\right)}$	2.50	3.40	3.13	4.87				25.0

Short answer

$\overline{\mathbf{v}}\mathbf{\left(}{\mathbf{cm}}^{\mathbf{-}\mathbf{1}}\mathbf{\right)}$	4000	2941.17	3095.97	2053.38	1800	1670	1620	400
$\mathbf{\lambda }\mathbf{\left(}\mathbf{\mu m}\mathbf{\right)}$	2.50	3.40	3.13	4.87	5.55	5.98	6.17	25.0

Step-by-step solution

Relation between wavelength and wavenumber

Wavenumber is calculated as,

$\overline{\mathbf{v}}\mathbf{\left(}{\mathbf{cm}}^{\mathbf{-}\mathbf{1}}\mathbf{\right)}\mathbf{=}\frac{\mathbf{1}}{\mathbf{wavelength}\mathbf{\left(}\mathbf{\lambda }\mathbf{\right)}\mathbf{incm}}$

The units of wavenumber are cm^-1.

The units of wavelength are µm .

Conversion of  wavenumber to wavelength and vice-versa

Since, 1 cm =10,000$\mu m$

Wavenumber can be calculated by dividing 10,000 by the wavelength in microns.

Conversion of 3.40 µm wavelength to wavenumber

An absorption of wavelength 3.40 µm corresponds to a wavenumber that is calculated as:

$\begin{array}{rcl}\overline{v}\left({\text{cm}}^{\text{- 1}}\right)& =& \frac{10,000\mu m/cm}{3.40\mu m}\\ & =& 2941.17c{m}^{-1}\end{array}$

Conversion of 3.13 µm wavelength to wavenumber

An absorption of wavelength 3.13 µm corresponds to a wavenumber that is calculated as:

$\begin{array}{rcl}\overline{v}& =& \frac{10,000\mu m/cm}{3.13\mu m}\\ & =& 3095.97c{m}^{-1}\end{array}$

Conversion of 4.87 µm wavelength to wavenumber

An absorption of wavelength 4.87 µm corresponds to a wavenumber that is calculated as:

$\begin{array}{rcl}\overline{v}& =& \frac{10,000\mu m/cm}{4.87\mu m}\\ & =& 2053.38c{m}^{-1}\end{array}$

Conversion of 1800 cm-1 wavenumber to wavelength

A wavenumber of 1800cm^-1 absorbs a wavelength that is calculated as:

$\begin{array}{rcl}\lambda & =& \frac{10,000\mu m/cm}{1800c{m}^{-1}}\\ & =& 5.55\mu m\end{array}$

Conversion of 1670 cm-1  wavenumber to wavelength

A wavenumber of 1670 cm^-1 absorbs a wavelength that is calculated as:

$\begin{array}{rcl}\lambda & =& \frac{10,000\mu m/cm}{1670c{m}^{-1}}\\ & =& 5.98\mu m\end{array}$

Conversion of 1620 cm-1 wavenumber to wavelength

A wavenumber of 1620cm^-1 absorbs the wavelength that is calculated as:

$\begin{array}{rcl}\mathrm{\lambda }& =& \frac{10,000\mathrm{\mu m}/\mathrm{cm}}{1620{\mathrm{cm}}^{-1}}\\ & =& 6.17\mu m\end{array}$