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 .

Short answer

(a) The Lyman series of hydrogen atom's emission lines are observed in the ultraviolet region of the electromagnetic spectrum, with wavelengths ranging from approximately \(9.724 \times 10^{-8} m\) to \(1.215 \times 10^{-7} m\). (b) The wavelengths of the first three lines in the Lyman series for \(n_i=2,3,\) and 4 are: \(\lambda_1 \approx 1.215 \times 10^{-7} m\) \(\lambda_2 \approx 1.025 \times 10^{-7} m\) \(\lambda_3 \approx 9.704 \times 10^{-8} m\)

Step-by-step solution

(a) Determine the region of the electromagnetic spectrum for the Lyman series

To determine the region of the electromagnetic spectrum, we'll need to find the range of wavelengths for the Lyman series. We can do this using the Rydberg formula for the hydrogen atom: \(\frac{1}{\lambda} = R_H (\frac{1}{n_{f}^2} - \frac{1}{n_{i}^2})\) where \(\lambda\) is the wavelength of the emission line, \(R_H\) is the Rydberg constant for hydrogen (approximately \(1.097 \times 10^7 m^{-1}\)), \(n_{f}\) is the final energy level, and \(n_{i}\) is the initial energy level. For the Lyman series, the final energy level is \(n_{f} = 1\). To find the range of wavelengths, we need to consider the shortest and longest wavelengths, which correspond to the largest and smallest possible values of \(n_{i}\) respectively. In the Lyman series, the smallest value of \(n_{i}\) is 2 and the largest value is infinity. First, let's find the shortest wavelength, setting \(n_{i}=2\) in the Rydberg formula: \(\frac{1}{\lambda_{min}} = R_H (1 - \frac{1}{4})\) \(\lambda_{min} = \frac{1}{R_H (\frac{3}{4})}\) Now, let's find the longest wavelength, setting \(n_{i}=\infty\) in the Rydberg formula: \(\frac{1}{\lambda_{max}} = R_H (1 - 0)\) \(\lambda_{max} = \frac{1}{R_H}\) Plugging in the value of the Rydberg constant, we find: \(\lambda_{min} \approx 9.724 \times 10^{-8} m\) \(\lambda_{max} \approx 1.215 \times 10^{-7} m\) This range of wavelengths corresponds to the ultraviolet region of the electromagnetic spectrum.

(b) Calculate the first three lines' wavelengths in the Lyman series

Using the Rydberg formula, we can now calculate the wavelengths of the first three lines in the Lyman series, corresponding to \(n_i = 2, 3,\) and 4. For \(n_i = 2\): \(\frac{1}{\lambda_1} = R_H (1 - \frac{1}{2^2})\) \(\lambda_1 = \frac{1}{R_H (\frac{3}{4})}\) For \(n_i = 3\): \(\frac{1}{\lambda_2} = R_H (1 - \frac{1}{3^2})\) \(\lambda_2 = \frac{1}{R_H (\frac{8}{9})}\) For \(n_i = 4\): \(\frac{1}{\lambda_3} = R_H (1 - \frac{1}{4^2})\) \(\lambda_3 = \frac{1}{R_H (\frac{15}{16})}\) Plugging in the value of the Rydberg constant, we find: \(\lambda_1 \approx 1.215 \times 10^{-7} m\) \(\lambda_2 \approx 1.025 \times 10^{-7} m\) \(\lambda_3 \approx 9.704 \times 10^{-8} m\)

Key concepts

Hydrogen Atom

The hydrogen atom is one of the simplest and most studied systems in quantum mechanics. It consists of a single proton at its nucleus and one electron orbiting this proton in various energy levels. This simplicity makes hydrogen an ideal system for understanding atomic structure.
In the hydrogen atom, the electron can jump between discrete energy levels. These transitions involve either absorbing or emitting a photon (a particle of light), which corresponds to the difference in energy between the levels. Different transitions lead to different series in the hydrogen atom's emission spectrum, including the famous Lyman series. This occurs when electrons fall from higher levels to the first energy level, which is the lowest and most stable energy configuration.

• Electrons occupy various energy levels or "shells." The ground state is when the electron is closest to the nucleus.
• Each energy level in hydrogen is associated with a principal quantum number, denoted as \( n \).
• Excitation or de-excitation of electrons between levels causes emission of specific wavelengths of light.

Understanding these basics of the hydrogen atom allows us to explore phenomena like the Lyman series, crucial for insights into atomic theory and spectral analysis.

Electromagnetic Spectrum

The electromagnetic spectrum includes all types of electromagnetic radiation, or energy that travels through space at the speed of light. This spectrum spans a wide range of wavelengths and frequencies, from radio waves to gamma rays.
Where a particular light wave falls on the electromagnetic spectrum depends on its wavelength or frequency. For example, visible light has wavelengths ranging from about 400 nm to 700 nm, whereas the ultraviolet region has shorter wavelengths.

• The electromagnetic spectrum is divided into regions based on wavelength, such as - Radio waves with long wavelengths - Microwaves - Infrared radiation - Visible light - Ultraviolet radiation - X-rays - Gamma rays with very short wavelengths
• Each region of the spectrum correlates to different physical processes and diagnostic applications.
• The Lyman series is specifically found in the ultraviolet region, where the wavelengths are shorter than those of visible light.

By understanding the electromagnetic spectrum, we can determine which series of emission lines belong to which part, like how the Lyman series is exclusively in the ultraviolet.

Rydberg Constant

The Rydberg constant is a fundamental physical constant used in atomic physics, particularly for calculating wavelengths of photon emissions in atomic spectra. Named after the Swedish physicist Johannes Rydberg, it is crucial in determining the spectral lines of chemical elements.

• The constant is denoted by \( R_H \) for hydrogen, the typical element when discussing the Rydberg constant.
• It has the approximate value of \( 1.097 \times 10^7 m^{-1} \).
• The Rydberg formula, \( \frac{1}{\lambda} = R_H \left( \frac{1}{n_f^2} - \frac{1}{n_i^2} \right) \), uses this constant to calculate the wavelength of light emitted when an electron transitions between energy levels \( n_i \) (initial) and \( n_f \) (final).

This constant helps predict and explain the wavelengths of light emitted or absorbed by an element, specifically in phenomena like the H series, including the Lyman series for hydrogen. By using the Rydberg constant, scientists can gain precise insights into atomic structures and spectral behaviors.

Ultraviolet Region

The ultraviolet (UV) region of the electromagnetic spectrum is situated just beyond the visible light region, spanning wavelengths from about 10 nm to 400 nm.
Ultraviolet light is not visible to the human eye but can have diverse applications and effects.

• UV radiation is divided into three main bands: - UVA (320-400 nm): These rays reach the earth’s surface and play a role in skin aging. - UVB (290-320 nm): These are partially absorbed by the ozone layer, and can cause sunburns. - UVC (100-290 nm): Almost entirely absorbed by the Earth's atmosphere.
• The Lyman series emissions, specific to the hydrogen atom, fall into the UV region, typically near the lower boundary at the higher energy or shorter wavelength end of the UV spectrum.
• Due to its higher energy levels, UV radiation can cause chemical reactions, and is commonly used in sterilization and fluorescence effects.

Understanding the UV region's role in the Lyman series underlines its importance in both scientific research and practical applications. The Lyman series emits light in the UV region, illustrating the jump of hydrogen's electron to its lowest energy state.