Physics Q43 – JEE-MAIN 2025

For a hydrogen atom, the ratio of the largest wavelength of Lyman series to that of the Balmer series is

Clue 1 of 3: Understand Spectral Series

The Lyman series corresponds to electron transitions to the $n = 1$ energy level, while the Balmer series corresponds to transitions to the $n = 2$ energy level.

🧠 Full Solution Path

Step 1: Identify Transitions for Largest Wavelength✦ Active

The largest wavelength in a spectral series corresponds to the transition with the smallest energy difference. This occurs when the electron transitions from the lowest possible initial energy level ( $n_{i}$ ) to the final energy level ( $n_{f}$ ) of that series.

For the Lyman series, the final state is $n_{f} = 1$ . The smallest possible initial state is $n_{i} = 2$ .

For the Balmer series, the final state is $n_{f} = 2$ . The smallest possible initial state is $n_{i} = 3$ .

Step 2: Apply Rydberg Formula for Each Series○ Expand

Using the Rydberg formula for a hydrogen atom ( $Z = 1$ ): $\frac{1}{λ} = R (\frac{1}{n_{f}^{2}} - \frac{1}{n_{i}^{2}})$

For the Lyman series (largest wavelength $λ_{L}$ ):

\frac{1}{λ_{L}} = R (\frac{1}{1^{2}} - \frac{1}{2^{2}}) = R (1 - \frac{1}{4}) = R (\frac{3}{4}) ⟹ λ_{L} = \frac{4}{3 R}

For the Balmer series (largest wavelength $λ_{B}$ ):

\frac{1}{λ_{B}} = R (\frac{1}{2^{2}} - \frac{1}{3^{2}}) = R (\frac{1}{4} - \frac{1}{9}) = R (\frac{9 - 4}{36}) = R (\frac{5}{36}) ⟹ λ_{B} = \frac{36}{5 R}

💡 Teacher's Secret Hint

Remember that the Rydberg constant $R$ is common for both calculations.

Step 3: Calculate the Ratio of Wavelengths○ Expand

Now, calculate the ratio of the largest wavelength of the Lyman series to that of the Balmer series:

\frac{λ_{L}}{λ_{B}} = \frac{\frac{4}{3 R}}{\frac{36}{5 R}} = \frac{4}{3 R} \times \frac{5 R}{36} = \frac{4 \times 5}{3 \times 36} = \frac{20}{108}

Simplify the ratio:

\frac{20}{108} = \frac{5 \times 4}{27 \times 4} = \frac{5}{27}

The ratio is $5 : 27$ .

💡 Teacher's Secret Hint

Ensure careful simplification of the fraction to arrive at the final ratio.

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Physics Question 43 – JEE-MAIN 2025

Clue 1 of 3: Understand Spectral Series

🧠 Full Solution Path