Question:
The maximum number of spectral lines formed when an excited electron of a hydrogen atom in \( n = 5 \) drops to lower states is:
The maximum number of spectral lines formed when an excited electron of a hydrogen atom in \( n = 5 \) drops to lower states is:
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For hydrogen-like emission problems, always use \( \frac{n(n-1)}{2} \) for maximum spectral lines when an electron drops from level \( n \).
Updated On: Jun 20, 2026
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The Correct Option is C
Solution and Explanation
Step 1: Understand the physical meaning of spectral lines.
When an electron in a hydrogen atom is excited to a higher energy level \( n \), it can return to lower levels in multiple possible transitions. Each possible transition corresponds to the emission of a photon, hence producing a spectral line. The total number of possible spectral lines depends on all possible downward transitions between energy levels.
Step 2: Identify the correct formula.
The maximum number of spectral lines produced when an electron drops from level \( n \) to lower levels is given by: \[ N = \frac{n(n-1)}{2} \] This formula accounts for all possible transitions between any two energy levels from \( n \) down to 1.
Step 3: Substitute the given value.
Here \( n = 5 \), so: \[ N = \frac{5(5-1)}{2} = \frac{5 \cdot 4}{2} \] \[ N = \frac{20}{2} = 10 \]
Step 4: Interpret the result physically.
This means an electron starting from \( n=5 \) can make multiple transitions such as 5→4, 5→3, 5→2, 5→1, 4→3, 4→2, 4→1, 3→2, 3→1, and 2→1. Counting all of these gives a total of 10 spectral lines.
Step 5: Final verification.
Since all possible transitions are included and no restrictions are given, the full combinational formula applies directly. Hence the result is consistent and complete.
Final Answer: \[ \boxed{10} \]
When an electron in a hydrogen atom is excited to a higher energy level \( n \), it can return to lower levels in multiple possible transitions. Each possible transition corresponds to the emission of a photon, hence producing a spectral line. The total number of possible spectral lines depends on all possible downward transitions between energy levels.
Step 2: Identify the correct formula.
The maximum number of spectral lines produced when an electron drops from level \( n \) to lower levels is given by: \[ N = \frac{n(n-1)}{2} \] This formula accounts for all possible transitions between any two energy levels from \( n \) down to 1.
Step 3: Substitute the given value.
Here \( n = 5 \), so: \[ N = \frac{5(5-1)}{2} = \frac{5 \cdot 4}{2} \] \[ N = \frac{20}{2} = 10 \]
Step 4: Interpret the result physically.
This means an electron starting from \( n=5 \) can make multiple transitions such as 5→4, 5→3, 5→2, 5→1, 4→3, 4→2, 4→1, 3→2, 3→1, and 2→1. Counting all of these gives a total of 10 spectral lines.
Step 5: Final verification.
Since all possible transitions are included and no restrictions are given, the full combinational formula applies directly. Hence the result is consistent and complete.
Final Answer: \[ \boxed{10} \]
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