Question

Characteristic orange light produced by sodium in a flame is due to an intense emission called the sodium D line, which actually a doublet, with wavelength (measured in vacuum) of 589.157 88 and 589.755 37 nm. The index of refraction of air at a wavelength near 589 nm is 1.000 292 6. Calculate the frequency, wavelength, and wavenumber of each component of the D line, measured in air.

Short answer

Given data,

The frequency in vacuum is$\mathrm{v}=5.0869.{10}^{14}\mathrm{Hz}$

The wavelength of air ${\mathrm{\lambda }}_{\mathrm{air}}=589.58285\mathrm{nm}$

The wavenumber is$\tilde{v}=1.6961145.{10}^4{\mathrm{cm}}^{-1}$

Step-by-step solution

Calculate frequency:

$$\begin{gathered} \mathrm{v}=\frac{\mathrm{c}}{{\mathrm{\lambda }}_{\mathrm{va}\mathrm{ccum}}} \\ =\frac{3.{10}^8\mathrm{m}/{\mathrm{s}}^2}{589.15788.{10}^{-9}\mathrm{m}} \\ \mathrm{v}=5.0920.{10}^{14}\mathrm{Hz} \\ \mathrm{v}=\frac{\mathrm{c}}{{\mathrm{\lambda }}_{\mathrm{va}\mathrm{ccum}}} \\ =\frac{3.{10}^8\mathrm{m}/{\mathrm{s}}^2}{589.75537.{10}^{-9}\mathrm{m}} \\ \mathrm{v}=5.0869.{10}^{14}\mathrm{Hz} \end{gathered}$$

Calculate wavelength of air:

$$\begin{gathered} {\mathrm{\lambda }}_{\mathrm{air}}=\frac{589.15788\mathrm{nm}}{1.0002926} \\ {\mathrm{\lambda }}_{\mathrm{air}}=588.98554\mathrm{nm} \\ {\mathrm{\lambda }}_{\mathrm{air}}=\frac{589.75537\mathrm{nm}}{1.0002926} \\ {\mathrm{\lambda }}_{\mathrm{air}}=589.58285\mathrm{nm} \end{gathered}$$

Calculate wavenumber:

$$\begin{gathered} \tilde{\mathrm{v}}=\frac{1}{{\mathrm{\lambda }}_{\mathrm{air}}} \\ =\frac{1}{588.98554\mathrm{nm}} \\ \tilde{\mathrm{v}}=1.6978345.{10}^{-3}{\mathrm{nm}}^{-1} \\ \tilde{\mathrm{v}}=1.6978345.{10}^4{\mathrm{cm}}^{-1} \\ \tilde{\mathrm{v}}=\frac{1}{{\mathrm{\lambda }}_{\mathrm{air}}} \\ =\frac{1}{589.58285\mathrm{nm}} \\ \tilde{\mathrm{v}}=1.6961145.{10}^{-3}{\mathrm{nm}}^{-1} \\ \tilde{\mathrm{v}}=1.6961145.{10}^4{\mathrm{cm}}^{-1} \end{gathered}$$