Question

The ratio of energies of photons produced due to transition of electron of hydrogen atom from its (i) second to first energy level and (ii) highest energy level to 2\(^\text{nd}\) level is respectively

• A. 4:1
• B. 2:1
• C. 2.5:1
• D. 3:1

Hint:

The energy of a photon in hydrogen atom transitions depends on the difference in the reciprocals of the square of the energy levels involved.

Answer 1

The Correct Option is D

Step 1: Energy of photons.
The energy of a photon emitted during a transition between two energy levels in a hydrogen atom is given by the formula: \[ E = -13.6 \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right) \, \text{eV} \] where \( n_1 \) and \( n_2 \) are the initial and final energy levels, respectively.
Step 2: Calculate energies for the two transitions.
For the first transition (second to first energy level): \[ E_1 = -13.6 \left( \frac{1}{1^2} - \frac{1}{2^2} \right) = -13.6 \left( 1 - \frac{1}{4} \right) = -13.6 \times \frac{3}{4} = -10.2 \, \text{eV} \] For the second transition (highest to second energy level, \( n_2 \to \infty \)): \[ E_2 = -13.6 \left( \frac{1}{2^2} - \frac{1}{\infty^2} \right) = -13.6 \times \frac{1}{4} = -3.4 \, \text{eV} \]
Step 3: Conclusion.
The ratio of energies is: \[ \frac{E_1}{E_2} = \frac{10.2}{3.4} = 3:1 \] Thus, the correct answer is (D).