Question

The frequencies for series limit of Balmer and Paschen series are $v_{1}$ and $v_{3}$ respectively. If frequency of first line of Balmer series is $v_{2}$, then the relation is}

• A. $v_{1}-v_{3}=2v_{1}$
• B. $v_{1}+v_{2}=v_{3}$
• C. $v_{1}-v_{2}=v_{3}$
• D. $v_{1}+v_{3}=v_{2}$

Hint:

Energy differences between levels are additive. $E_{2 \to \infty} - E_{2 \to 3} = E_{3 \to \infty}$.

Answer 1

The Correct Option is C

Step 1: Formula
Energy $E = hv = Rc (\frac{1}{n_1^2} - \frac{1}{n_2^2})$.

Step 2: Analysis
- Balmer series limit ($n_1=2, n_2=\infty$): $v_1 \propto (\frac{1}{4} - 0) = 1/4$. - Balmer first line ($n_1=2, n_2=3$): $v_2 \propto (\frac{1}{4} - \frac{1}{9}) = 5/36$. - Paschen series limit ($n_1=3, n_2=\infty$): $v_3 \propto (\frac{1}{9} - 0) = 1/9$.

Step 3: Calculation
$v_1 - v_2 \propto \frac{1}{4} - \frac{5}{36} = \frac{9-5}{36} = \frac{4}{36} = \frac{1}{9}$.
This matches the value for $v_3$.

Step 4: Conclusion
Hence, $v_1 - v_2 = v_3$. Final Answer: (C)