Hydrogen Spectrum - Balmer Series

Spectral wavelengths in hydrogen follow the Rydberg relation $\frac{1}{\lambda }=R\left(\frac{1}{n_1^2}-\frac{1}{n_2^2}\right)$, and a transition ending at $n_1=2$ belongs to the visible Balmer series. For the $n_2=3\to n_1=2$ jump (the H-alpha line), $\frac{1}{\lambda }=R\left(\frac{1}{4}-\frac{1}{9}\right)=R\left(\frac{9-4}{36}\right)=\frac{5R}{36}$. Hence $\lambda =\frac{36}{5R}=\frac{36}{5\times 1.097\times 10^7}\approx 6.56\times 10^{-7}$ m $=656$ nm. The value 486 nm is the H-beta line ($4\to 2$), not the requested transition. The value 121 nm is the Lyman-alpha line ($2\to 1$), which lies in the ultraviolet, not the visible. The value 365 nm is the Balmer series limit ($\infty \to 2$). The Balmer series, lying in the visible region, was historically the first hydrogen series discovered and gave early support to the Bohr model. Because the lower level is fixed at $n_1=2$, the longest-wavelength line of the series arises from the smallest energy gap, namely the $3\to 2$ transition. This matches the NCERT Balmer-series description. A plausibility check confirms 656 nm falls in the red part of the visible spectrum, exactly where H-alpha is observed experimentally.