Question

What will be the wave number of yellow radiation having wavelength${\text{240}}\,{\text{nm}}$?
A.$1.724\, \times {10^4}\,{\text{c}}{{\text{m}}^{ - 1}}$
B. $4.16\, \times {10^6}\,{\text{c}}{{\text{m}}^{ - 1}}$
C.$4\, \times {10^{14}}\,{\text{Hz}}$
D. $2.193\, \times {10^3}\,{\text{c}}{{\text{m}}^{ - 1}}$

Answer 1

Hint:We have to determine the wavenumber form the given wavelength, so, we should know the relation between wavenumber and wavelength. Wavenumber and wavelength have a simple relation. Wavenumber is inversely proportional to the wavelength. First we will convert the wavelength from nm to cm then by substituting the value of wavelength we can determine the wavenumber.


Complete solution:
The wavelength is defined as the distance between two successive crests and troughs is of a wave. The wavenumber is defined as the number of wavelengths per unit distance.
The relation between wavenumber and wavelength is as follows:
$\mathop \nu \limits^ - = \dfrac{1}{\lambda }$
Where,
\[\mathop \nu \limits^ - \]is the wavenumber.
$\lambda $ is the wavelength
First we will convert the wavelength from nm to cm because we have to answer in cm so,
We know that $1$ nm is equal to ${10^{ - 7}}$ cm so, $240$nm will be equal to,
$1$ nm = ${10^{ - 7}}$ cm
$240$nm = $2.40\, \times {10^{ - 5}}$ cm
So, $200$nm will be equal to $2.40\, \times {10^{ - 5}}$ cm.
On substituting $2.40\, \times {10^{ - 5}}$ cm for wavelength in wavenumber and wavelength relation,
$\mathop \nu \limits^ - = \dfrac{1}{\lambda }$
$\mathop \nu \limits^ - = \dfrac{1}{{2.40\, \times {{10}^{ - 5}}}}$
$\mathop \nu \limits^ - = 4.16 \times {10^6}\,{\text{c}}{{\text{m}}^{ - 1}}$
So, the wave number of yellow radiation having wavelength${\text{240}}\,{\text{nm}}$is $4.16 \times {10^6}\,{\text{c}}{{\text{m}}^{ - 1}}$.

Therefore, option (B) $4.16 \times {10^6}\,{\text{c}}{{\text{m}}^{ - 1}}$ is correct.


Note: As the wavenumber and wavelength are inversely related so, they units also has inverse relation. The unit of wavelength is cm whereas the unit of wavenumber is per cm. The wavenumber is also defined as reciprocal of wavelength. The wavenumber and wavelength are slo related with frequency as, $\nu = \dfrac{c}{\lambda }$
Where,
${\text{c}}$is the speed of light and \[\nu \] is the frequency.
As we know, $\mathop \nu \limits^ - = \dfrac{1}{\lambda }$ so, we can also write $\nu = c\mathop \nu \limits^ - $
The wavenumber also related with energy as, \[\delta E\, = h\nu \]
$\delta E\,$is the energy change.
$h$is the Plank constant.
Or $\delta E\, = hc\mathop \nu \limits^ - $