Hydrogen Line Spectrum
Structure of Atom | Rydberg Formula | Class 11
1. Introduction to Hydrogen Spectrum
When an electric discharge is passed through gaseous hydrogen, the $H_2$ molecules dissociate, and the energetically excited hydrogen atoms produced emit electromagnetic radiation of discrete frequencies. This resulting spectrum is known as the hydrogen emission spectrum.
Unlike a continuous spectrum (like a rainbow), the hydrogen spectrum consists of distinct isolated lines. This proved that energy levels in an atom are quantized.
2. Spectral Series
The spectral lines for atomic hydrogen are grouped into different series named after their discoverers. These transitions occur when an electron jumps from a higher energy level ($n_2$) to a lower energy level ($n_1$).
| Series Name | Lower State ($n_1$) | Higher State ($n_2$) | Spectral Region |
|---|---|---|---|
| Lyman | 1 | 2, 3, 4, ... | Ultraviolet (UV) |
| Balmer | 2 | 3, 4, 5, ... | Visible |
| Paschen | 3 | 4, 5, 6, ... | Infrared (IR) |
| Brackett | 4 | 5, 6, 7, ... | Infrared (IR) |
| Pfund | 5 | 6, 7, 8, ... | Infrared (IR) |
3. The Rydberg Formula
Johannes Rydberg presented a simple theoretical equation to calculate the wave number ($\bar{\nu}$) and wavelength ($\lambda$) of all the spectral lines in the hydrogen spectrum.
Where:
- $\bar{\nu}$ = Wave number ($cm^{-1}$ or $m^{-1}$)
- $\lambda$ = Wavelength
- $R_H$ = Rydberg constant = $109,677 \ cm^{-1}$ (or $1.097 \times 10^7 \ m^{-1}$)
- $Z$ = Atomic number ($Z=1$ for Hydrogen, $Z=2$ for $He^+$, $Z=3$ for $Li^{2+}$)
- $n_1$ = Lower energy level (integer)
- $n_2$ = Higher energy level (integer), where $n_2 > n_1$
4. Limiting Line & Wavelengths
Longest Wavelength ($\lambda_{max}$) / Lowest Energy: Occurs when the transition is from the immediately next energy level (e.g., $n_2 = n_1 + 1$). This is known as the first line of the series.
Shortest Wavelength ($\lambda_{min}$) / Highest Energy: Occurs when the electron falls from infinity ($n_2 = \infty$). This is called the Limiting line or series limit.
5. Number of Spectral Lines
If an electron in a hydrogen sample is in the $n^{th}$ excited state, the total maximum number of spectral lines emitted when it returns to the ground state ($n=1$) is given by:
If the transition is between any two levels $n_2$ and $n_1$ (where $n_2 > n_1$), the formula becomes:
$$ \text{Lines} = \frac{(n_2 - n_1)(n_2 - n_1 + 1)}{2} $$Practice Quiz
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