For any given series of spectral lines of atomic hydrogen, let v = vmax - vmin be the difference in maximum and minimum…

🧠 The "Gap of Gaps" We are comparing the range of wave numbers ($\Delta \bar{\nu}$) for two different series.

• $\bar{\nu}_{max}$ is the energy of the total drop from infinity (the Ionization limit of that series).
• $\bar{\nu}_{min}$ is the energy of the very first line (the smallest jump). The "width" of the series tells us how much energy is contained between the first excited state and the ionized state of that system.

🗺️ The Range Derivation For H ($Z=1$), the wave number is $\bar{\nu} = R(1/n_f^2 - 1/n_i^2)$.

1. Lyman Range: $n_f = 1$.
   • $\bar{\nu}_{max} = R(1 - 0) = R$
   • $\bar{\nu}_{min} = R(1 - 1/4) = 3R/4$
   • $\Delta \bar{\nu}_{L} = R - 3R/4 = R/4$
2. Balmer Range: $n_f = 2$.
   • $\bar{\nu}_{max} = R(1/4 - 0) = R/4$
   • $\bar{\nu}_{min} = R(1/4 - 1/9) = 5R/36$
   • $\Delta \bar{\nu}_{B} = R/4 - 5R/36 = (9R - 5R)/36 = 4R/36 = R/9$
3. The Ratio: $\frac{\Delta \bar{\nu}_{L}}{\Delta \bar{\nu}_{B}} = \frac{R/4}{R/9} = \frac{9}{4}$

⚡ The $n_1^3$ Scaling? No complex shortcut here, but notice that the "width" of a series gets much smaller as you move up. The Lyman range ($n=1$) is much wider than the Balmer range ($n=2$).

⚠️ Common Traps The question asks for ratios of "difference in wave numbers." Don't just divide the maximum wave numbers ($R$ vs $R/4$, which is $4:1$). You must subtract the first line's energy from each!

$\boxed{\text{Answer: (D)}}$