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The Lyman Series is an exciting group of spectral lines in the hydrogen spectrum. These lines result from electron transitions ending in the first energy level (n=1). This means electrons are falling from a higher level down to n=1. These transitions emit ultraviolet light, which is not visible to the naked eye.
For example:

• The first line results from an electron dropping from n=2 to n=1.
• The second line is from n=3 to n=1.
• The third line happens when falling from n=4 to n=1.

The Rydberg formula helps calculate the wavelengths of these transitions. For the Lyman series, the formula sets n鈧� = 1 and varies n鈧� for different lines. The result is the characteristic wavelengths of this series.

The Balmer Series is famous for its bright spectral lines. These occur when electrons transition to the second energy level (n=2). Unlike the Lyman Series, the Balmer Series includes visible light, which falls within the human eye's ability to see, making it special for historical spectroscopy.
Important transitions include:

• From n=3 to n=2, giving the first line in the series.
• From n=4 to n=2, resulting in the second line.
• From n=5 to n=2 for the third line.

Using the Rydberg formula, you can find the exact wavelengths by setting n鈧� = 2 and adjusting n鈧� accordingly. This calculation provides the key to understanding the beautiful colors in this series.

In the realm of the infrared spectrum, the Paschen Series stands out. It involves transitions where electrons land on the third energy level (n=3), emitting wavelengths primarily in the infrared region.
Key transitions are:

• The first line from n=4 to n=3.
• The second line when falling from n=5 to n=3.
• The third line due to a drop from n=6 to n=3.

To calculate these wavelengths using the Rydberg formula, you choose n鈧� = 3 and vary n鈧� for the different lines. This computation shows why the Paschen Series is not observable without special equipment, as it is outside the visible light range.

Energy Transitions are at the heart of spectral line series, linking them to electron movement between energy levels in atoms. When an electron changes its level, either gaining or losing energy, light is absorbed or emitted, forming spectral lines. These changes are discrete, meaning electrons jump from one energy level to another without stopping in-between.
The difference in energy between these two levels dictates the photon energy and thus the wavelength observed. This principle explains why specific wavelengths correspond to different series like Lyman, Balmer, and Paschen.
The Rydberg formula captures this concept by correlating the initial (n鈧�) and final (n鈧�) quantum numbers to calculate the spectral lines tied to these transitions.

Wavelength Calculation using the Rydberg formula is straightforward yet fascinating. It requires the known constant, the Rydberg constant (R鈧�), and a pair of quantum numbers: n鈧� and n鈧�.
The formula:\[ \frac{1}{\lambda} = R_H \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right)\]To find the wavelength (位), solve this equation for different series by plugging in specific values of n鈧� and n鈧�. Remember:

• Lyman Series sets n鈧� to 1.
• Balmer Series uses n鈧� of 2.
• Paschen Series implements n鈧� as 3.

As n鈧� increases from n鈧� + 1 upwards, each calculated wavelength corresponds to a spectral line in the series. The series limit is reached as n鈧� approaches infinity, resulting in the shortest possible wavelength.