Question

Describe the relationship of the electronic transition of spectral lines with which the wavelength is inversely proportional?
A. Number of electrons under-going transition.
B. The nuclear charge of the atom.
C. Undergoing transition due to the electron’s velocity.
D. The difference in the energy involved in the transition.

Answer 1

Hint: According to Plank’s, Energy is proportional to the frequency of the light.
According to De-Broglie, Wavelength is inversely related to the mass of the radiation and velocity of the radiation.

Formula Used: According to Plank’s, \[E = hv\]
Frequency, \[v = \dfrac{c}{\lambda }\]
c = light’s speed, λ = wavelength

Complete Step by Step Solution:
Electromagnetic radiation may be defined as the radiation which is produced by the electric field and magnetic field which are perpendicular to each other and propagate through the empty space also. Electromagnetic radiation shows dual character- particle nature and wave nature. The wave nature is calculated in terms of wavelength, which shows the distance travelled by the wave/electromagnetic radiation.
According to Plank’s, he states that the difference in the energy of different states of the electromagnetic radiation is directly proportional to the frequency of the electromagnetic radiation.
\[
  {E_2} - {E_1} \propto v \\
   \Rightarrow \Delta E = hv \\
 \]
Formula for the speed of particle
\[Speed{\text{ }} = {\text{ }}frequency \times wavelength\]
In this equation frequency is termed in place of time and wavelength is used in terms of distance.
Now,
\[
  Wavelength{\text{ }} = {\text{ }}\dfrac{{speed}}{{frequency}} \\
   \Rightarrow \lambda = \dfrac{c}{v} \\
 \]
Here, c = speed of light and v = frequency
Frequency can be defined as the periodic repetition of wave through a given point in 1 second.
Now according to above equations,
\[\begin{array}{*{20}{l}}
  {\Delta E = {\text{ }}\dfrac{{hc}}{\lambda }} \\
  { \Rightarrow \lambda \propto \dfrac{1}{{\Delta E}}}
\end{array}\]
So, the wavelength of the electromagnetic transition of the spectral lines is inversely proportional to the difference in the energy of the involved transition.
The correct option is D.

Note: The wavelength is used in terms of length which can be measured by measuring the distance between the two trough and two crust. Here, electromagnetic radiation is also a form of energy.