Discussion
The ratio of frequencies of the first line of the Lyman series and the first line of the Balmer series is :
Answer: C
The ratio is 27/5.
By the Rydberg formula the frequency of a line is proportional to \(1/n_{1}^{2} - 1/n_{2}^{2}\). The first Lyman line is n = 2 to 1, and the first Balmer line n = 3 to 2:
\[ \begin{aligned} \nu_{L} &\propto 1 - \tfrac{1}{4} = \tfrac{3}{4} \\ \nu_{B} &\propto \tfrac{1}{4} - \tfrac{1}{9} = \tfrac{5}{36} \\ \frac{\nu_{L}}{\nu_{B}} &= \frac{3}{4}\times\frac{36}{5} = \frac{27}{5} \end{aligned} \]
Lyman lines are in the ultraviolet and Balmer lines in the visible, so the ratio must be greater than 1, which rules out 4/27 and 8/27. 27/8 comes from using the wrong second term for the Balmer line.
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