Question

A near ultraviolet photon of $ 300nm $ is absorbed by a gas and then remitted as two photons. One photon is red with a wavelength of $ 760nm $. What would be the wave number of the second photon?

Answer 1

Hint: We know that radiation which travels perpendicular to each other and perpendicular to electric and magnetic fields which are electromagnetic radiations. Wavelength is the distance between two crests and two successive troughs.

Complete answer:
As we know that radiation which travels perpendicular to each other and perpendicular to electric and magnetic fields are electromagnetic radiations. Wavelength is the distance between two crests and two successive troughs and the number of wavelengths per unit length is called wavenumber.
We know that,
 $ E = \dfrac{{hc}}{\lambda } $
 The wavenumber is given as:
 $ \mathop \upsilon \limits^ - = \dfrac{1}{\lambda } $
So we are given a wavelength of energy absorbed by the gas so the energy absorbed will be equal to the sum of energies of both the quanta. It will be calculated as:
 $ \Rightarrow \dfrac{{hc}}{{300 \times {{10}^{ - 9}}}} = \dfrac{{hc}}{{760 \times {{10}^{ - 9}}}} + \Rightarrow \dfrac{{hc}}{{\lambda \times {{10}^{ - 9}}}} $
 $
\Rightarrow \dfrac{1}{\lambda } = \dfrac{1}{{300}} - \dfrac{1}{{760}} \\
\Rightarrow \lambda = 496m \\
 $
 $
\Rightarrow \mathop \upsilon \limits^ - = \dfrac{1}{\lambda } = \dfrac{1}{{498}} \\
\Rightarrow \mathop \upsilon \limits^ - = 2.02 \times {10^{ - 3}}{m^{ - 1}} \\
 $

Therefore the correct answer is $ 2.02 \times {10^{ - 3}}{m^{ - 1}} $.

Additional information:
Wave number is usually specified for an electromagnetic field in a vacuum, also called free space. In other than free space media, the wave number for a given electromagnetic field may increase. For light in a medium, the wavenumber is the vacuum wavenumber times the refractive index. The wavenumber is related to the phase change per unit length of a plane wave in a homogenous medium. $ 5.8 \times {10^{ - 5}}cm $ is the wavelength for a typical spectral line in the visible region of the spectrum.

Note:
The electromagnetic waves do not travel in a straight line and do not require any medium for their propagation. When such radiations are arranged in decreasing order of frequency and increasing order of wavelength then the spectrum so obtained is called electromagnetic spectrum.
The wavenumber for an electromagnetic field is equal to $ \dfrac{{2\pi }}{\lambda } $ in meters. As the wavelength decreases, the wave number increases.