Q.Yellow light emitted from a sodium lamp has a wavelength ($\lambda$) of 580 nm. Calculate the frequency ($\nu$) and wavenumber ($\bar{\nu}$) of the yellow light.

Concept: Electromagnetic Wave Wavelength

Yellow light from sodium is an electromagnetic wave. Three quantities describe it: wavelength $\lambda$, frequency $\nu$, and wavenumber $\bar{\nu}$. They are related through the speed of light $c = 3.0 \times 10^8 \, \text{m s}^{-1}$.

Step 1: Convert wavelength to meters.

Step 2: Calculate frequency using $c = \nu \lambda$.

$\nu = \frac{c}{\lambda} = \frac{3.0 \times 10^8}{5.80 \times 10^{-7}} = 5.17 \times 10^{14} \, \text{Hz}$ …

Why these quantities matter

Electromagnetic radiation can be described in three equivalent ways: wavelength $\lambda$ (the spatial extent of one cycle), frequency $\nu$ (how many cycles pass a point per second), and wavenumber $\bar{\nu}$ (how many wavelengths fit in a unit distance). All three are connected through the speed of light $c$, which is constant in vacuum at $3.00 \times 10^8 \, \text{m/s}$.

The wavelength tells us the color we perceive. The frequency tells us the energy per photon (through $E = h\nu$). The wavenumber is particularly useful in spectroscopy because it's directly proportional to energy and conveniently expressed in $\text{cm}^{-1}$ for infrared and Raman work.

Step-by-step calculation

1. Convert wavelength to SI units

The given wavelength is $\lambda = 580 \, \text{nm}$. Since all our constants are in SI, we need meters:

2. Calculate frequency using the wave equation

The fundamental relationship for any wave is:

Rearranging for frequency:

This enormous number makes sense: visible light oscillates hundreds of trillions of times per second.

3. Calculate wavenumber

Wavenumber is simply the reciprocal of wavelength:

$\bar{\nu} = 1.72 \times 10^{6} \, \text{m}^{-1}$ …