Question

Three energy levels of hydrogen atom and the corresponding wavelength of the emitted radiation due to different electron transition are as shown. Then

• A. \(λ_1=\frac{λ_2λ_3}{λ_2+λ_3}\)
• B. λ₂ = λ₁ + λ₃
• C. \(λ_2=\frac{λ_1λ_3}{λ_1+λ_3}\)
• D. \(λ_3=\frac{λ_1λ_2}{λ_1+λ_2}\)

Answer 1

The Correct Option is C

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:

\[ ΔE = \frac{hc}{λ} \]

where \(h\) = Planck’s constant, \(c\) = speed of light.

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

• For transition E₃ → E₂: \[ E_3 - E_2 = \frac{hc}{λ_1} \]
• For transition E₂ → E₁: \[ E_2 - E_1 = \frac{hc}{λ_3} \]
• For transition E₃ → E₁: \[ E_3 - E_1 = \frac{hc}{λ_2} \]

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

\[ (E_3 - E_1) = (E_3 - E_2) + (E_2 - E_1) \]

Substitute wavelength terms:

\[ \frac{hc}{λ_2} = \frac{hc}{λ_1} + \frac{hc}{λ_3} \]

Canceling out \(hc\), we get clearly:

\[ \frac{1}{λ_2} = \frac{1}{λ_1} + \frac{1}{λ_3} \]

Step 5: Solve this clearly to find λ₂ explicitly:

\[ λ_2 = \frac{λ_1λ_3}{λ_1 + λ_3} \]

Final Conclusion:
Thus, the correct relation is clearly:

\[ λ_2 = \frac{λ_1λ_3}{λ_1 + λ_3} \]

Answer 2

In a hydrogen atom, the energy levels are quantized, and the transition between these energy levels corresponds to the emission or absorption of electromagnetic radiation. The energy difference between two levels is related to the wavelength of the emitted radiation by the equation: \[ E_1 - E_2 = \dfrac{hc}{\lambda} \] where \(E_1\) and \(E_2\) are the energies of the two levels, \(h\) is Planck's constant, \(c\) is the speed of light, and \(\lambda\) is the wavelength. From the given energy level diagram, the wavelengths \(\lambda_1\), \(\lambda_2\), and \(\lambda_3\) correspond to different electron transitions between these levels. To find the relationship between the wavelengths, we use the energy difference formula and the fact that energy is inversely proportional to wavelength. Using the fact that: \[ E_1 - E_2 = \dfrac{hc}{\lambda_1}, \quad E_2 - E_3 = \dfrac{hc}{\lambda_2}, \quad E_3 - E_1 = \dfrac{hc}{\lambda_3} \] By combining these relations, we obtain: \[ \lambda_2 = \dfrac{\lambda_1 \lambda_3}{\lambda_1 + \lambda_3} \] Thus, the correct answer is (3).