Problem 38 - Chemistry: The Central Science

Question

The Lyman series of emission lines of the hydrogen atom are those for which \(n_{f}=1\). (a) Determine the region of the electromagnetic spectrum in which the lines of the Lyman series are observed. (b) Calculate the wavelengths of the first three lines in the Lyman series - those for which \(n_{i}=2,3\), and 4 .

Step-by-Step Solution

Answer

In conclusion, the first three lines of the Lyman series have wavelengths of approximately: 1. \(\lambda_1 \approx 1.215 \times 10^{-7} m\) 2. \(\lambda_2 \approx 1.025 \times 10^{-7} m\) 3. \(\lambda_3 \approx 0.972 \times 10^{-7} m\) And the Lyman series is observed in the ultraviolet region of the electromagnetic spectrum.

Step 1: Understand the Rydberg formula

The Rydberg formula is used to find the wavelength of electromagnetic radiation emitted or absorbed during a transition between energy levels in the hydrogen atom. The formula is as follows: \[\frac{1}{\lambda} = R_H \left(\frac{1}{n_{f}^2} - \frac{1}{n_{i}^2}\right)\] Where \(\lambda\) is the wavelength of the emitted radiation, \(R_H\) is the Rydberg constant for hydrogen (\(R_H \approx 1.097 \times 10^7 m^{-1}\)), \(n_{f}\) is the final energy level, and \(n_{i}\) is the initial energy level.

Step 2: Determine the wavelengths of the Lyman series transitions

For the Lyman series, we have \(n_{f} = 1\). We need to calculate the wavelengths for the first three lines, for which \(n_{i} = 2, 3\), and 4. Let's calculate each of these wavelengths using the Rydberg formula: a) For \(n_{i} = 2\): \[\frac{1}{\lambda_1} = R_H \left(\frac{1}{1^2} - \frac{1}{2^2}\right)\] \[\lambda_1 = \frac{1}{R_H \left(\frac{3}{4}\right)}\] b) For \(n_{i} = 3\): \[\frac{1}{\lambda_2} = R_H \left(\frac{1}{1^2} - \frac{1}{3^2}\right)\] \[\lambda_2 = \frac{1}{R_H \left(\frac{8}{9}\right)}\] c) For \(n_{i} = 4\): \[\frac{1}{\lambda_3} = R_H \left(\frac{1}{1^2} - \frac{1}{4^2}\right)\] \[\lambda_3 = \frac{1}{R_H \left(\frac{15}{16}\right)}\] Now, let's calculate the numerical values of these wavelengths: \[\lambda_1 \approx \frac{1}{1.097 \times 10^7 m^{-1} \times \frac{3}{4}} \approx 1.215 \times 10^{-7} m\] \[\lambda_2 \approx \frac{1}{1.097 \times 10^7 m^{-1} \times \frac{8}{9}} \approx 1.025 \times 10^{-7} m\] \[\lambda_3 \approx \frac{1}{1.097 \times 10^7 m^{-1} \times \frac{15}{16}} \approx 0.972 \times 10^{-7} m\]

Step 3: Identify the region of the electromagnetic spectrum

Now we need to identify which part of the electromagnetic spectrum the Lyman series falls under. Here are the wavelength ranges for various regions of the electromagnetic spectrum: 1. Radio waves: \(\lambda > 10^{-1} m\) 2. Microwaves: \(10^{-1} m > \lambda > 10^{-3} m\) 3. Infrared: \(10^{-3} m > \lambda > 7 \times 10^{-7} m\) 4. Visible light: \(7 \times 10^{-7} m > \lambda > 4 \times 10^{-7} m\) 5. Ultraviolet: \(4 \times 10^{-7} m > \lambda > 10^{-8} m\) 6. X-rays: \(10^{-8} m > \lambda > 10^{-11} m\) 7. Gamma rays: \(\lambda < 10^{-11} m\) Looking at our calculated wavelengths for the Lyman series lines, we can see that they all fall within the ultraviolet range (between \(4 \times 10^{-7} m\) and \(10^{-8} m\)). Therefore, the Lyman series is observed in the ultraviolet region of the electromagnetic spectrum. In conclusion, the first three lines of the Lyman series have wavelengths of approximately: 1. \(\lambda_1 \approx 1.215 \times 10^{-7} m\) 2. \(\lambda_2 \approx 1.025 \times 10^{-7} m\) 3. \(\lambda_3 \approx 0.972 \times 10^{-7} m\) And the Lyman series is observed in the ultraviolet region of the electromagnetic spectrum.

Key Concepts

Rydberg Formula

Understanding the Rydberg formula is essential for students delving into the field of quantum mechanics and spectral analysis. Developed by Swedish physicist Johannes Rydberg, this formula establishes the relationship between the wavelength of the emitted or absorbed electromagnetic radiation and the electron transitions between different energy levels in an atom, particularly hydrogen.

The mathematical expression for the Rydberg formula is:
\[\[\begin{align*}\frac{1}{\lambda} = R_H \left(\frac{1}{n_f^2} - \frac{1}{n_i^2}\right)\end{align*}\]\]In this equation, \(\lambda\) represents the wavelength of the photon emitted or absorbed, while \(R_H\) stands for the Rydberg constant, which is roughly 1.097 x \(10^7 m^{-1}\) for hydrogen. The terms \(n_i\) and \(n_f\) denote the initial and final energy levels of the electron within the atom, respectively.

The beauty of the Rydberg formula lies in its predictive power, allowing us to calculate the exact position of spectral lines for given transitions. For students working on exercises involving any form of spectral analysis, comprehending how to apply this formula is a fundamental stepping stone.

Electromagnetic Spectrum

The electromagnetic spectrum is a term used to describe all possible frequencies of electromagnetic radiation, ranging from very long radio waves to very short gamma rays. The spectrum is normally divided into seven regions, in order of increasing frequency and decreasing wavelength: radio waves, microwaves, infrared light, visible light, ultraviolet light, X-rays, and gamma rays.

It's fascinating to observe that the visible light is just a tiny portion of the entire spectrum, which although invisible to the eye, is of immense importance to various technological and scientific applications like making phone calls, heating food in the microwave, or diagnosing medical conditions with X-rays.

When studying spectral lines such as the Lyman series, recognizing where these lines fall within the electromagnetic spectrum helps contextualize their energy levels and the types of instruments required for their detection. The Lyman series, for instance, is situated within the ultraviolet region of the spectrum.

Hydrogen Emission Lines

Hydrogen emission lines are a specific set of spectral lines that are visible when an electron in a hydrogen atom moves from a higher energy level to a lower energy level and emits a photon. These lines are categorized into several series, each correlating with transitions ending at different energy levels.

The Lyman series, specifically, covers electron transitions that conclude at the first energy level (n=1). Because the Lyman series is in the ultraviolet region, these lines are not visible to the naked eye, but they're fundamental to the study of astronomical objects and plasma physics.

Recognition of the Lyman series aids in the comprehension of quantum mechanics and the electronic structure of atoms. It serves as an important observational tool for astrophysicists, providing insights into the composition of stars and interstellar medium. By studying these emission lines, students can deepen their understanding of both atomic physics and the broader universe.