Ch 39: Particles Behaving as Waves

Young & Freedman Calc - University Physics 15th Edition

Young & Freedman Calc15th EditionUniversity PhysicsISBN: 9780135159552Not the one you use?Change textbook

Young & Freedman Calc 15th Edition

Ch 39: Particles Behaving as Waves

Problem 28

Chapter 38, 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.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.

Video duration:

Was this helpful?

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.

Recommended video:

Guided course

06:06

The Electromagnetic Spectrum

Related Practice

Textbook Question

How many photons per second are emitted by a $7.50$ -mW CO₂ laser that has a wavelength of $10.6$ mm?

views

Textbook Question

In a set of experiments on a hypothetical one-electron atom, you measure the wavelengths of the photons emitted from transitions ending in the ground level ( $n equals 1$ ), as shown in the energy-level diagram in Fig. E $39.27$ . You also observe that it takes $17.50$ eV to ionize this atom. What is the energy of the atom in each of the levels ( $n equals 1$ , $n equals 2$ , etc.) shown in the figure?

views

Textbook Question

The energy-level scheme for the hypothetical one-electron element Searsium is shown in Fig. $E 39.25$ . The potential energy is taken to be zero for an electron at an infinite distance from the nucleus. An $18$ -eV photon is absorbed by a Searsium atom in its ground level. As the atom returns to its ground level, what possible energies can the emitted photons have? Assume that there can be transitions between all pairs of levels.

Textbook Question

Use Balmer's formula to calculate (a) the wavelength, (b) the frequency, and (c) the photon energy for the Hg line of the Balmer series for hydrogen.

Textbook Question

A triply ionized beryllium ion, Be³⁺ (a beryllium atom with three electrons removed), behaves very much like a hydrogen atom except that the nuclear charge is four times as great. For the hydrogen atom, the wavelength of the photon emitted in the $n equals 2$ to $n equals 1$ transition is $122$ nm (see Example $39.6$ ). What is the wavelength of the photon emitted when a Be³⁺ ion undergoes this transition?

Textbook Question

Using a mixture of CO₂, N₂, and sometimes He, CO₂ lasers emit a wavelength of $10.6$ $μ$ m. At power of $0.100$ kW, such lasers are used for surgery. How many photons per second does a CO₂ laser deliver to the tissue during its use in an operation?