Problem 84 - Chemistry: The Central Science

Question

The series of emission lines of the hydrogen atom for which \(n_{f}=3\) 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 \(n_{i}=4,5\), and 6 .

Step-by-Step Solution

Answer

(a) The lines of the Paschen series are observed in the infrared region of the electromagnetic spectrum. (b) The wavelengths of the first three lines in the Paschen series are approximately \(1.88 \times 10^{-6} m\), \(1.28 \times 10^{-6} m\), and \(1.0942 \times 10^{-6} m\).

Step 1: Recall the Rydberg formula for hydrogen

The Rydberg formula for hydrogen, which relates the wavelength of emitted light to the initial and final energy levels, is given by: \[ \frac{1}{\lambda} = R_H \left( \frac{1}{n_f^2} - \frac{1}{n_i^2} \right) \] Where: - \(\lambda\) is the wavelength of the emitted light, - \(R_H\) is the Rydberg constant for hydrogen, approximately equal to \(1.097 \times 10^7 m^{-1}\), - \(n_f\) is the final energy level and - \(n_i\) is the initial energy level.

Step 2: Find the wavelengths of the Paschen series emissions

We are given that \(n_f = 3\) for the Paschen series. To find the wavelengths of the first three lines, we will use the equation with \(n_i = 4, 5, 6\). For \(n_i = 4\): \[ \frac{1}{\lambda_1} = R_H \left( \frac{1}{3^2} - \frac{1}{4^2} \right) \] For \(n_i = 5\): \[ \frac{1}{\lambda_2} = R_H \left( \frac{1}{3^2} - \frac{1}{5^2} \right) \] For \(n_i = 6\): \[ \frac{1}{\lambda_3} = R_H \left( \frac{1}{3^2} - \frac{1}{6^2} \right) \] From these equations, we can calculate the wavelengths \(\lambda_1\), \(\lambda_2\), and \(\lambda_3\).

Step 3: Calculate the wavelengths

Calculating the wavelengths using the equations from step 2: For \(n_i = 4\): \[ \frac{1}{\lambda_1} = R_H \left( \frac{1}{9} - \frac{1}{16} \right) \] \[ \frac{1}{\lambda_1} = (1.097 \times 10^7 m^{-1}) \left( \frac{7}{144} \right) \] \[ \lambda_1 \approx 1.88 \times 10^{-6} m \] For \(n_i = 5\): \[ \frac{1}{\lambda_2} = R_H \left( \frac{1}{9} - \frac{1}{25} \right) \] \[ \frac{1}{\lambda_2} = (1.097 \times 10^7 m^{-1}) \left( \frac{16}{225} \right) \] \[ \lambda_2 \approx 1.28 \times 10^{-6} m \] For \(n_i = 6\): \[ \frac{1}{\lambda_3} = R_H \left( \frac{1}{9} - \frac{1}{36} \right) \] \[ \frac{1}{\lambda_3} = (1.097 \times 10^7 m^{-1}) \left( \frac{27}{324} \right) \] \[ \lambda_3 \approx 1.0942 \times 10^{-6} m \]

Step 4: Identify the region of the electromagnetic spectrum

The calculated wavelengths (\(\lambda_1\), \(\lambda_2\), and \(\lambda_3\)) are all within the infrared region of the electromagnetic spectrum, which is typically between \(700 \, nm\) and \(1 \, mm\). Conclusion: (a) The lines of the Paschen series are observed in the infrared region of the electromagnetic spectrum. (b) The wavelengths of the first three lines in the Paschen series are approximately \(1.88 \times 10^{-6} m\), \(1.28 \times 10^{-6} m\), and \(1.0942 \times 10^{-6} m\).

Key Concepts

Understanding the Rydberg Formula

The Rydberg formula is a fundamental equation in atomic physics that provides a mathematical description of the wavelengths of the spectral lines emitted by hydrogen atoms. It is crucial for calculating the precise wavelengths of the hydrogen emission lines in the various series, including the Paschen series.

Expressed mathematically, the Rydberg formula is given by: \[ \frac{1}{\lambda} = R_H \left( \frac{1}{n_f^2} - \frac{1}{n_i^2} \right) \]
where:\begin{itemize}\item \(\lambda\) is the wavelength of the emitted light,\item \(R_H\) is the Rydberg constant (\(1.097 \times 10^7 m^{-1}\)),\item \(n_f\) is the final energy level,\item \(n_i\) is the initial energy level.\end{itemize}
Using this formula, we can calculate the emitted light's wavelength as an electron transitions from a higher energy level (\(n_i\)) to a lower energy level (\(n_f\)). In the case of the Paschen series, electrons fall to the \(n_f = 3\) energy level, emitting photons in the infrared region.

Hydrogen Emission Lines and Series

When an electron within a hydrogen atom undergoes a transition between two different energy levels, it emits or absorbs a photon with a distinct wavelength, producing what is known as an emission or absorption line. These lines are characteristic of hydrogen and can be used to identify its presence in various astronomical and laboratory settings.

The hydrogen emission lines are grouped into series, named after their discoverers. They include the Lyman series (ultraviolet), the Balmer series (visible), and the Paschen series (infrared), among others. Each series corresponds to electrons falling to specific energy levels:

• Lyman series - transitions to \(n_f = 1\),
• Balmer series - transitions to \(n_f = 2\),
• Paschen series - transitions to \(n_f = 3\).

For example, the Paschen series occurs when electrons return to the third energy level (\(n_f = 3\)) from a higher level (\(n_i = 4, 5, 6, ...\)). This grouping helps us understand the organization of an atom's energy levels and the electromagnetic spectrum's different regions.

The Infrared Spectrum and Paschen Series

The infrared spectrum constitutes a part of the electromagnetic spectrum with wavelengths longer than visible light but shorter than microwaves. It ranges from approximately \(700 \, nm\) to \(1 \, mm\), which includes a wide array of applications from thermal imaging to communications.

The Paschen series, specifically, falls within this infrared region. When exploring the Paschen series, we are looking at a transition of an electron from a higher energy level to the third level. The emission wavelengths calculated for the Paschen series (\(\lambda_1\), \(\lambda_2\), \(\lambda_3\)) align with the infrared spectrum, being longer than the visible light wavelengths and, as such, invisible to the human eye. Infrared light's unique properties make it essential in many technological applications and in understanding various astronomical phenomena, such as the study of cool stars and interstellar dust clouds.