Ch 39: Particles Behaving as Waves

Young & Freedman Calc - University Physics 14th Edition

Young & Freedman Calc14th EditionUniversity PhysicsISBN: 9780321973610Not the one you use?Change textbook

Young & Freedman Calc 14th Edition

Ch 39: Particles Behaving as Waves

Problem 28

Chapter 39, Problem 28

Find the longest and shortest wavelengths in the Lyman and Paschen series for hydrogen. In what region of the electromagnetic spectrum does each series lie?

Verified step by step guidance

Step 1: Understand the Lyman and Paschen series. The Lyman series corresponds to electronic transitions in a hydrogen atom where the electron falls to the n=1 energy level, while the Paschen series corresponds to transitions where the electron falls to the n=3 energy level.

Step 2: Use the Rydberg formula to calculate the wavelength of emitted light during these transitions:

\[ \frac{1}{\lambda} = R_H \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right) \]

, where $ \lambda $ is the wavelength, $ R_H $ is the Rydberg constant (approximately $ 1.097 \times 10^7 \ \text{m}^{-1} $), $ n_1 $ is the lower energy level, and $ n_2 $ is the higher energy level.

Step 3: For the longest wavelength in each series, set $ n_2 = n_1 + 1 $. For the Lyman series, $ n_1 = 1 $, so $ n_2 = 2 $. For the Paschen series, $ n_1 = 3 $, so $ n_2 = 4 $. Substitute these values into the Rydberg formula to find the longest wavelength.

Step 4: For the shortest wavelength in each series, set $ n_2 \to \infty $. This represents the limit where the electron transitions from a very high energy level to $ n_1 $. Substitute $ n_1 = 1 $ for the Lyman series and $ n_1 = 3 $ for the Paschen series into the Rydberg formula to find the shortest wavelength.

Step 5: Determine the region of the electromagnetic spectrum for each series. The Lyman series wavelengths typically fall in the ultraviolet region, while the Paschen series wavelengths fall in the infrared region. Compare the calculated wavelengths to the known ranges of the electromagnetic spectrum to confirm this classification.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Lyman Series

The Lyman series refers to the set of spectral lines resulting from electron transitions in a hydrogen atom where electrons fall to the n=1 energy level from higher levels (n=2, 3, 4, ...). The wavelengths of these transitions fall in the ultraviolet region of the electromagnetic spectrum, with the longest wavelength corresponding to the transition from n=2 to n=1, and the shortest from n=∞ to n=1.

Paschen Series

The Paschen series consists of spectral lines produced when electrons transition to the n=3 energy level in hydrogen from higher levels (n=4, 5, 6, ...). The wavelengths of these transitions are found in the infrared region of the electromagnetic spectrum, with the longest wavelength occurring from n=4 to n=3, and the shortest from n=∞ to n=3.

Electromagnetic Spectrum

The electromagnetic spectrum encompasses all types of electromagnetic radiation, arranged by wavelength or frequency. It includes various regions such as radio waves, microwaves, infrared, visible light, ultraviolet, X-rays, and gamma rays. Understanding where the Lyman and Paschen series fall within this spectrum is crucial for identifying their physical properties and applications in spectroscopy.

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