---
title: "A gas discharge tube contains a large sample of isolated hydrogen atoms whose electrons are all excited to the \\(n = 4\\) energy level. The energy levels of the hydrogen atom are given by \\(E_n = -\\dfrac{13.6 \\text{ eV}}{n^2}\\), and visible light corresponds to photon energies between \\(1.8 \\text{ eV}\\) and \\(3.1 \\text{ eV}\\). The electrons de-excite to the ground state (\\(n = 1\\)) through all possible direct and multi-step cascade pathways.  | Option | Photons in \\(n = 4 \\to 3 \\to 1\\) Cascade | Photons in \\(n = 4 \\to 2 \\to 1\\) Cascade | Distinct Visible Lines from Sample | |—|—|—|—| | (A) | One visible, one ultraviolet | Two ultraviolet | 1 | | (B) | Two ultraviolet | One visible, one ultraviolet | 3 | | (C) | One infrared, one visible | One visible, one ultraviolet | 2 | | (D) | One infrared, one ultraviolet | One visible, one ultraviolet | 2 |  Which row in the table correctly classifies the spectral regions of the emitted photons for the two cascade pathways and gives the total number of distinct visible-light wavelengths emitted by the sample?"
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url: "https://nerd-notes.com/ubq/123601/"
date_modified: "2026-09-28T12:00:22+00:00"
---

# A gas discharge tube contains a large sample of isolated hydrogen atoms whose electrons are all excited to the \(n = 4\) energy level. The energy levels of the hydrogen atom are given by \(E_n = -\dfrac{13.6 \text{ eV}}{n^2}\), and visible light corresponds to photon energies between \(1.8 \text{ eV}\) and \(3.1 \text{ eV}\). The electrons de-excite to the ground state (\(n = 1\)) through all possible direct and multi-step cascade pathways.

| Option | Photons in \(n = 4 \to 3 \to 1\) Cascade | Photons in \(n = 4 \to 2 \to 1\) Cascade | Distinct Visible Lines from Sample |
|—|—|—|—|
| (A) | One visible, one ultraviolet | Two ultraviolet | 1 |
| (B) | Two ultraviolet | One visible, one ultraviolet | 3 |
| (C) | One infrared, one visible | One visible, one ultraviolet | 2 |
| (D) | One infrared, one ultraviolet | One visible, one ultraviolet | 2 |

Which row in the table correctly classifies the spectral regions of the emitted photons for the two cascade pathways and gives the total number of distinct visible-light wavelengths emitted by the sample?

A gas discharge tube contains a large sample of isolated hydrogen atoms whose electrons are all excited to the \(n = 4\) energy level. The energy levels of the hydrogen atom are given by \(E_n = -\dfrac{13.6 \text{ eV}}{n^2}\), and visible light corresponds to photon energies between \(1.8 \text{ eV}\) and \(3.1 \text{ eV}\). The electrons de-excite to the ground state (\(n = 1\)) through all possible direct and multi-step cascade pathways.

| Option | Photons in \(n = 4 \to 3 \to 1\) Cascade | Photons in \(n = 4 \to 2 \to 1\) Cascade | Distinct Visible Lines from Sample |
|---|---|---|---|
| (A) | One visible, one ultraviolet | Two ultraviolet | 1 |
| (B) | Two ultraviolet | One visible, one ultraviolet | 3 |
| (C) | One infrared, one visible | One visible, one ultraviolet | 2 |
| (D) | One infrared, one ultraviolet | One visible, one ultraviolet | 2 |

Which row in the table correctly classifies the spectral regions of the emitted photons for the two cascade pathways and gives the total number of distinct visible-light wavelengths emitted by the sample?

![An energy level diagram showing four horizontal, parallel line segments representing atomic energy levels, vertically stacked. A vertical axis on the far left has an upward arrow labeled Energy (\(\text{eV}\)). The lowest horizontal line is labeled on the left with \(n = 1\) and on the right with \(-13.60 \text{ eV}\). The second line from the bottom is located above the first, labeled on the left with \(n = 2\) and on the right with \(-3.40 \text{ eV}\). The third line is positioned above the second line, closer to the second than the second is to the first, labeled on the left with \(n = 3\) and on the right with \(-1.51 \text{ eV}\). The fourth line is positioned above the third line, labeled on the left with \(n = 4\) and on the right with \(-0.85 \text{ eV}\). No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1790596822-vwDrmC.jpg)

- **A.** Photons in \(n = 4 \to 3 \to 1\): One visible, one ultraviolet | Photons in \(n = 4 \to 2 \to 1\): Two ultraviolet | Distinct visible lines: 1
- **B.** Photons in \(n = 4 \to 3 \to 1\): Two ultraviolet | Photons in \(n = 4 \to 2 \to 1\): One visible, one ultraviolet | Distinct visible lines: 3
- **C.** Photons in \(n = 4 \to 3 \to 1\): One infrared, one visible | Photons in \(n = 4 \to 2 \to 1\): One visible, one ultraviolet | Distinct visible lines: 2
- **D.** Photons in \(n = 4 \to 3 \to 1\): One infrared, one ultraviolet | Photons in \(n = 4 \to 2 \to 1\): One visible, one ultraviolet | Distinct visible lines: 2

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/123601/*