Question

In a hydrogen atom, if an electron makes a transition from the fourth orbit to the second orbit, then the wavelength of the emitted radiation is: (R = Rydberg constant)

• A. $ \frac{16}{3R} $
• B. $ \frac{8}{3R} $
• C. $ \frac{3}{16R} $
• D. $ \frac{3}{8R} $

Hint:

For transitions in the hydrogen atom, use the formula $ \Delta E = -R_h \left( \frac{1}{n_2^2} - \frac{1}{n_1^2} \right) $ to find the energy difference, and relate it to wavelength using $ \Delta E = \frac{hc}{\lambda} $.

Answer 1

The Correct Option is A

Step 1: Understanding the Bohr Model and Energy Levels
In the Bohr model of the hydrogen atom, the energy of an electron in the $ n $-th orbit is given by: $$ E_n = -\frac{R_h}{n^2} $$ where:
$ R_h $ is the Rydberg constant,
$ n $ is the principal quantum number.
The energy difference between two orbits ($ n_1 $ and $ n_2 $) when an electron transitions is: $$ \Delta E = E_{n_2} - E_{n_1} = -R_h \left( \frac{1}{n_2^2} - \frac{1}{n_1^2} \right) $$ For the transition from the fourth orbit ($ n_1 = 4 $) to the second orbit ($ n_2 = 2 $): $$ \Delta E = -R_h \left( \frac{1}{2^2} - \frac{1}{4^2} \right) $$ $$ \Delta E = -R_h \left( \frac{1}{4} - \frac{1}{16} \right) $$ $$ \Delta E = -R_h \left( \frac{4}{16} - \frac{1}{16} \right) = -R_h \left( \frac{3}{16} \right) $$ $$ \Delta E = -\frac{3R_h}{16} $$ The negative sign indicates that energy is released during the transition. Step 2: Relating Energy to Wavelength
The energy of a photon is related to its wavelength by: $$ E = \frac{hc}{\lambda} $$ where:
$ h $ is Planck's constant,
$ c $ is the speed of light,
$ \lambda $ is the wavelength.
For the emitted radiation: $$ \Delta E = \frac{hc}{\lambda} $$ Substitute $ \Delta E = \frac{3R_h}{16} $: $$ \frac{3R_h}{16} = \frac{hc}{\lambda} $$ Solve for $ \lambda $: $$ \lambda = \frac{16hc}{3R_h} $$ Using the relationship $ R_h = \frac{hc}{R} $, where $ R $ is the Rydberg constant: $$ \lambda = \frac{16hc}{3 \cdot \frac{hc}{R}} = \frac{16R}{3} $$ Thus, the wavelength of the emitted radiation is: $$ \boxed{\frac{16}{3R}} $$ Step 3: Analyzing Options Option (1): $ \frac{16}{3R} $
Correct — matches the calculated value. Option (2): $ \frac{8}{3R} $
Incorrect — does not match the calculation. Option (3): $ \frac{3}{16R} $
Incorrect — incorrect numerator and denominator. Option (4): $ \frac{3}{8R} $
Incorrect — incorrect numerator and denominator.