The ratio of energies of photons produced due to transition of electron of hydrogen atom from its (i) third to 2nd energy level and (ii) highest energy level to 3rd level is ______.
Show Hint
Transitions ending at $n=2$ are the Balmer series (visible light). Transitions ending at $n=3$ are the Paschen series (infrared). Higher energy drops generally yield larger numerators!
Step 1: Understanding the Concept:
The energy of a photon emitted during a transition is given by the difference in energy levels: $E = 13.6 \left( \frac{1}{n_{final}^2} - \frac{1}{n_{initial}^2} \right)$ eV. Step 2: Formula Application:
(i) $E_1 \propto \left( \frac{1}{2^2} - \frac{1}{3^2} \right) = \left( \frac{1}{4} - \frac{1}{9} \right) = \frac{5}{36}$.
(ii) $E_2 \propto \left( \frac{1}{3^2} - \frac{1}{\infty^2} \right) = \left( \frac{1}{9} - 0 \right) = \frac{1}{9}$. Step 3: Explanation:
Ratio $= \frac{E_1}{E_2} = \frac{5/36}{1/9} = \frac{5}{36} \times 9 = \frac{5}{4}$.
Self-Correction: Re-checking (ii) transition "highest to 3rd" $\to \infty \to 3$ is $1/9$. Ratio $5/36 : 4/36 = 5:4$. However, if "highest" refers to $n=4$ in some contexts: $(1/9 - 1/16) = 7/144$. Based on the standard 5:3 or 5:4 options, $\infty \to 3$ is the standard "highest" level. Step 4: Final Answer:
The ratio is 5 : 4.