The ratio of the shortest wavelength of two spectral series of hydrogen spectrum is found to be about 9. The spectral…

🧠 The Square-Law Search The "shortest wavelength" of a series is $\lambda = \frac{1}{R(1/n_1^2 - 0)} = \frac{n_1^2}{R}$. This means for the Hydrogen atom, the limit wavelength is directly proportional to the square of the shell it's falling into! If $\lambda_1 / \lambda_2 = 9$, then $n_1^2 / n_2^2 = 9$, which simplifies to $n_1 / n_2 = 3$.

🗺️ Trial & Error Map We need a pair of series where the shell numbers have a ratio of $3$:

• Pair 1: Lyman ($n=1$) and Balmer ($n=2$). Ratio $= 2 \neq 3$.
• Pair 2: Lyman ($n=1$) and Paschen ($n=3$). Ratio $= 3/1 = 3$. Bingo!
• Pair 3: Balmer ($n=2$) and Brackett ($n=4$). Ratio $= 2 \neq 3$.
• Pair 4: Paschen ($n=3$) and Pfund ($n=5$). Ratio $= 5/3 \neq 3$.

⚡ The "Integer Square" Hook Since we see the number $9$, our brain should immediately jump to $3^2$. Shortest wavelength scaling is $1^2 : 2^2 : 3^2 : 4^2$. Ratios: $1 : 4 : 9 : 16$. To get a ratio of $9$, we must be looking at the first ($1$) and the third ($9$) series.

⚠️ Common Traps If the question asked for the longest wavelength ratio, the math would be much messier. Always start with the "series limit" ($\infty \to n$) for these comparison questions because the math is $10 \times$ faster.

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