Understanding the Wavelength of the Second Line in the Paschen Series

Introduction to the Paschen Series

The Paschen series is a part of the hydrogen spectral series, which occurs when electrons transition out of the n3 energy level to lower levels within the atom. This series of spectral lines is primarily observed in the infrared region of the electromagnetic spectrum. This article focuses on the wavelength of the second line in the Paschen series and how it can be calculated.

Fundamentals of Atomic Spectroscopy

Atomic spectroscopy involves the study of light emitted or absorbed by atoms in the form of spectral lines. The Paschen series is known for its transitions from the 3rd energy level (n3) to lower levels, including n2 and n1. The second line in the Paschen series refers to the transition from the n5 energy level to the n3 energy level, which is an important spectroscopic transition.

Calculating the Wavelength of the Second Line in the Paschen Series

The wavelength of the second line in the Paschen series can be calculated using the Rydberg formula, which is a fundamental equation in atomic physics. The general form of the Rydberg formula is:

(frac{1}{lambda} R_H left(frac{1}{n_f^2} - frac{1}{n_i^2}right))

Where (lambda) is the wavelength of the emitted or absorbed light, (R_H) is the Rydberg constant for hydrogen (approximately 1.0974 × 107 m-1), (n_f) is the final energy level, (n_i) is the initial energy level.

For the second line in the Paschen series, the transition from n5 to n3, we can substitute these values into the Rydberg formula:

(frac{1}{lambda} 1.0974 times 10^7 left(frac{1}{3^2} - frac{1}{5^2}right))

This simplifies to:

(frac{1}{lambda} 1.0974 times 10^7 left(frac{1}{9} - frac{1}{25}right))

(frac{1}{lambda} 1.0974 times 10^7 times frac{16}{225})

(frac{1}{lambda} 9.77 times 10^5 , m^{-1})

( lambda 1.0214 times 10^{-6} , m 1021.4 , nm 1.0214 , mu m)

Therefore, the wavelength of the second line in the Paschen series for the transition from n5 to n3 is approximately 1021.4 nm, or 1.0214 microns.

Applications of the Paschen Series

The Paschen series has significant applications in various fields of science and engineering, including astronomy, spectroscopy, and quantum physics. It is particularly useful in understanding the interactions between atoms and light, which can help in the development of new technologies such as spectroscopic measurements and precision measurements of fundamental constants.

Conclusion

Understanding the wavelength of the second line in the Paschen series is crucial for researchers and students in the field of atomic spectroscopy. With the accuracy provided by the Rydberg formula and modern computational tools like calculators, the wavelength can be calculated precisely. This knowledge not only enhances our understanding of atomic structure and behavior but also contributes to advancements in related technologies and applications.