The ratio of the minimum frequency of the Paschen series in the spectrum to the maximum frequency of — Atomic Structure Chemistry Question
Question
The ratio of the minimum frequency of the Paschen series in the $He^+$ spectrum to the maximum frequency of the Balmer series in the $Li^{2+}$ spectrum is mathematically expressed as $\frac{x}{y}$, where $x$ and $y$ are coprime positive integers. Find the value of $x + y$.
💡 Solution & Explanation
Minimum frequency of Paschen series for $He^+$ ($Z=2$, $n_1=3, n_2=4$): $ u_{min} \propto 2^2 (\frac{1}{3^2} - \frac{1}{4^2}) = 4 (\frac{1}{9} - \frac{1}{16}) = 4 \times \frac{7}{144} = \frac{7}{36}$. Maximum frequency of Balmer series for $Li^{2+}$ ($Z=3$, $n_1=2, n_2=\infty$): $ u_{max} \propto 3^2 (\frac{1}{2^2} - \frac{1}{\infty}) = 9 (\frac{1}{4} - 0) = \frac{9}{4}$. Ratio $= \frac{7/36}{9/4} = \frac{7}{36} \times \frac{4}{9} = \frac{7}{81}$. So $x=7, y=81$, giving $x+y = 88$.