Atomic Physics

Given the angular momentum quantization condition:

The de Broglie wavelength is calculated as:

For a Li²⁺ ion, the radius of the electron orbit is given by:

Substituting the radius into the wavelength equation:

Comparing with the given form , we find:

Given:

• Energy of electron in 2nd orbit of hydrogen atom:
• Find energy of electron in 3rd orbit of He ion

Step 1: Recall energy level formula for hydrogen-like atoms

The energy of an electron in nth orbit is given by:

where:

• = atomic number
• = principal quantum number

Step 2: Express given energy for hydrogen (Z=1)

For hydrogen's 2nd orbit:

Step 3: Calculate energy for He (Z=2) in 3rd orbit

For He 's 3rd orbit:

Step 4: Find ratio of energies

We need to express in terms of :

Thus:

Step 1: Identify the given energy levels and transitions clearly:
Three energy levels (E₁ < E₂ < E₃) of a hydrogen atom are given. Wavelengths of radiation emitted due to transitions between these levels are λ₁, λ₂, and λ₃:

• Transition from level E₃ → E₂ emits radiation λ₁.
• Transition from level E₂ → E₁ emits radiation λ₃.
• Transition from level E₃ → E₁ emits radiation λ₂.

Step 2: Use the energy-wavelength relationship clearly:
The energy difference (ΔE) between two levels is related to the wavelength (λ) of emitted radiation by:

where = Planck’s constant, = speed of light.

Step 3: Express the energy differences for each transition clearly:

• For transition E₃ → E₂:
• For transition E₂ → E₁:
• For transition E₃ → E₁:

Step 4: Relate the above energy differences:
Clearly, energy levels satisfy:

Substitute wavelength terms:

Canceling out , we get clearly:

Step 5: Solve this clearly to find λ₂ explicitly:

Final Conclusion:
Thus, the correct relation is clearly: