Problem 91 - Chemistry The Central Science

Question

The series of emission lines of the hydrogen atom for which \(n_{1}=3\) is called the Paschen series. (a) Determine the region of the electromagnetic spectrum in which the lines of the Paschen series are observed. (b) Calculate the wavelengths of the first three lines in the Paschen series - those for which \(n_{1}=4,5,\) and \(6 .\)

Step-by-Step Solution

Answer

(a) The Paschen series is observed in the infrared region of the electromagnetic spectrum. (b) The wavelengths of the first three lines in the Paschen series are approximately \(1.875 \times 10^{-6}\, m\), \(1.282 \times 10^{-6}\, m\), and \(1.093 \times 10^{-6}\, m\).

Step 1: Understand the Rydberg formula for hydrogen

The Rydberg formula for hydrogen is given by: \[ \frac{1}{\lambda} = R_H \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right) \] where \(\lambda\) is the wavelength of the emission line, \(R_H\) is the Rydberg constant for hydrogen (\(R_H \approx 1.097 \times 10^7 \, m^{-1}\)), \(n_1\) is the principal quantum number of the lower energy level, and \(n_2\) is the principal quantum number of the higher energy level. For the Paschen series, \(n_1 = 3\).

Step 2: Determine the wavelengths of the first three lines in the Paschen series

To find the wavelengths of the first three lines in the Paschen series, we need to use the Rydberg formula and set \(n_1 = 3\) and \(n_2 = 4, 5, 6\). 1. For the first line, \(n_2 = 4\): \[ \frac{1}{\lambda_1} = R_H \left( \frac{1}{3^2} - \frac{1}{4^2} \right) \] 2. For the second line, \(n_2 = 5\): \[ \frac{1}{\lambda_2} = R_H \left( \frac{1}{3^2} - \frac{1}{5^2} \right) \] 3. For the third line, \(n_2 = 6\): \[ \frac{1}{\lambda_3} = R_H \left( \frac{1}{3^2} - \frac{1}{6^2} \right) \]

Step 3: Calculate the wavelengths for the first three lines

Now we can plug in the values for \(R_H\) and calculate the wavelengths for the first three lines. 1. For the first line, \(\lambda_1\): \[ \frac{1}{\lambda_1} = 1.097 \times 10^7 \left( \frac{1}{9} - \frac{1}{16} \right) \] \[ \lambda_1 = \frac{1}{(1.097 \times 10^7)(\frac{7}{144})} \] \[ \lambda_1 \approx 1.875 \times 10^{-6}\, m \] 2. For the second line, \(\lambda_2\): \[ \frac{1}{\lambda_2} = 1.097 \times 10^7 \left( \frac{1}{9} - \frac{1}{25} \right) \] \[ \lambda_2 = \frac{1}{(1.097 \times 10^7)(\frac{16}{225})} \] \[ \lambda_2 \approx 1.282 \times 10^{-6}\, m \] 3. For the third line, \(\lambda_3\): \[ \frac{1}{\lambda_3} = 1.097 \times 10^7 \left( \frac{1}{9} - \frac{1}{36} \right) \] \[ \lambda_3 = \frac{1}{(1.097 \times 10^7)(\frac{27}{324})} \] \[ \lambda_3 \approx 1.093 \times 10^{-6}\, m \]

Step 4: Determine the region of the electromagnetic spectrum

The electromagnetic spectrum is divided into several regions based on the wavelength. Since all three of our wavelengths are in the range of \(10^{-6}\,m\), the region of the electromagnetic spectrum for the Paschen series is in the infrared region. So the answer is: (a) The Paschen series is observed in the infrared region of the electromagnetic spectrum. (b) The wavelengths of the first three lines in the Paschen series are approximately \(1.875 \times 10^{-6}\, m\), \(1.282 \times 10^{-6}\, m\), and \(1.093 \times 10^{-6}\, m\).

Key Concepts

Rydberg Formula

The Rydberg formula is a key tool in understanding the emission spectra of hydrogen atoms. It allows us to calculate the wavelengths of light emitted when an electron transitions between energy levels. The formula is expressed as follows:
\[ \frac{1}{\lambda} = R_H \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right) \]
Here,

• \( \lambda \) is the wavelength of the emitted light.
• \( R_H \approx 1.097 \times 10^7 \, m^{-1} \) is the Rydberg constant for hydrogen.
• \( n_1 \) and \( n_2 \) are the principal quantum numbers of the lower and higher energy levels, respectively.

This formula is crucial for calculating the wavelengths associated with various series of lines in hydrogen's emission spectrum, including the Paschen series, which we'll cover next.

Paschen Series

The Paschen series is one of the series of lines in the emission spectrum of hydrogen and is vital for understanding transitions in lower energy states. Specifically, this series involves transitions where the electron falls to the third energy level, denoted as \( n_1 = 3 \). This means it covers transitions like:

• From \( n_2 = 4 \) to \( n_1 = 3 \)
• From \( n_2 = 5 \) to \( n_1 = 3 \)
• From \( n_2 = 6 \) to \( n_1 = 3 \)

These transitions result in the emission of light at particular wavelengths, all of which fall within the infrared region of the electromagnetic spectrum.
Understanding the Paschen series helps us link the theoretical principles of quantum mechanics to observable physical phenomena.

Electromagnetic Spectrum

The electromagnetic spectrum encompasses all types of electromagnetic radiation, from radio waves to gamma rays. It is broadly divided into regions based on the wavelength or frequency of the waves:

• Radio Waves: Longest wavelengths
• Microwaves
• Infrared: Where the Paschen series lies
• Visible Light: The spectrum we can see
• Ultraviolet
• X-Rays
• Gamma Rays: Shortest wavelengths

The position of the Paschen series in the infrared region signifies that the emitted light from these transitions has a longer wavelength than visible light. This placement helps us understand the energy dynamics and interactions between light and matter.

Infrared Region

The infrared region of the electromagnetic spectrum is typically characterized by wavelengths that are longer than visible light but shorter than microwaves. This ranges from approximately 700 nm to 1 mm.
In the context of hydrogen's emission spectrum, particularly the Paschen series, emitted lines fall into this region.

• Infrared light is often associated with heat, as objects emit infrared radiation based on their temperature.
• This region is crucial in various applications, including thermal imaging and remote sensing.
• In astronomy, infrared observations enable the study of objects obscured by interstellar dust.

For students and scientists studying atomic and molecular transitions, grasping the implications of emissions in the infrared region offers invaluable insight into the underlying energetic processes.