Question

In which of the following transitions will the wavelength be minimum?
A) n = 5 to n = 4
B) n = 4 to n = 3
C) n = 3 to n = 2
D) n = 2 to n = 1

Answer 1

Hint: The distance between the two successive crests or troughs of a wave is called Wavelength. It is inversely proportional to the frequency. Hence the lower the frequency, the larger will be the wavelength and vice-versa.

Complete step by step solution:
The wavelength of the photon is given by,
\[\dfrac{1}{\lambda } = R\left( {\dfrac{1}{{n_f^2}} - \dfrac{1}{{n_i^2}}} \right)\]
Hence for the wavelength to be minimum, \[\left( {\dfrac{1}{{n_f^2}} - \dfrac{1}{{n_i^2}}} \right)\] should be maximum.
1. For n = 5 to n=4,
\[\Rightarrow \left( {\dfrac{1}{{n_f^2}} - \dfrac{1}{{n_i^2}}} \right) = \dfrac{1}{{{4^2}}} - \dfrac{1}{{{5^2}}} = 0.0225\]
2. For n = 4 to n=3,
\[\Rightarrow \left( {\dfrac{1}{{n_f^2}} - \dfrac{1}{{n_i^2}}} \right) = \dfrac{1}{{{3^2}}} - \dfrac{1}{{{4^2}}} = 0.0486\]
3. For n = 3 to n=2,
\[\Rightarrow \left( {\dfrac{1}{{n_f^2}} - \dfrac{1}{{n_i^2}}} \right) = \dfrac{1}{{{2^2}}} - \dfrac{1}{{{3^2}}} = 0.14\]
4. For n = 2 to n=1,
\[\Rightarrow \left( {\dfrac{1}{{n_f^2}} - \dfrac{1}{{n_i^2}}} \right) = \dfrac{1}{{{1^2}}} - \dfrac{1}{{{2^2}}} = 0.75\]
Hence \[\left( {\dfrac{1}{{n_f^2}} - \dfrac{1}{{n_i^2}}} \right)\] is maximum for n = 2 to n = 1 and thus the corresponding wavelength will be minimum.

Hence the correct option is D.

Note: 1. Transition is defined as the change of state from one to another. Similarly, the electronic transition is defined as the transfer of the electron from one energy to the other energy level.
2. Wavelength is sometimes called a waveform. Wavelength is denoted by $\lambda $. It is measured as the distance from crest to crest or the distance from trough to trough. The length of one complete wave cycle is called the wavelength of the wave. The energy is directly proportional to frequency and inversely proportional to the wavelength.
3. Radio Waves have the longest wavelength and lowest frequencies. Gamma rays have the highest frequencies and shortest wavelengths.
4. Wavelength depends on the medium, the frequency, and, the speed of the wave. Thus the wavelength is different for different mediums. The frequency will not change when the wave travels from one medium to another medium.