Question

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

Short 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: - \(n_{i} = 4 \rightarrow \lambda_1 = 1.87 \times 10^{-6} \, m\) - \(n_{i} = 5 \rightarrow \lambda_2 = 1.28 \times 10^{-6} \, m\) - \(n_{i} = 6 \rightarrow \lambda_3 = 1.09 \times 10^{-6} \, m\)

Step-by-step solution

Rydberg Formula

To find the wavelengths of the emission lines, we need to use the Rydberg formula: \[\frac{1}{\lambda} = R_H \left(\frac{1}{n_{f}^2} - \frac{1}{n_{i}^2}\right)\] Where: - \(\lambda\) is the wavelength of the emitted light - \(R_H\) is the Rydberg constant for hydrogen \((R_H = 1.097 \times 10^7 \, \mathrm{m^{-1}})\) - \(n_{f}\) is the quantum number of the final energy level - \(n_{i}\) is the quantum number of the initial energy level In this exercise, the final energy level is \(n_{f} = 3\). We need to find the first three lines of the Paschen series, corresponding to \(n_{i} = 4, 5, 6\).

Determine Wavelengths

Let's use the Rydberg formula to find the wavelengths for each line in the Paschen series: (a) For \(n_{i} = 4\): \[\frac{1}{\lambda_1} = R_H \left(\frac{1}{3^2} - \frac{1}{4^2}\right)\] \[\lambda_1 = \frac{1}{R_H \left(\frac{1}{9} - \frac{1}{16}\right)}\] (b) For \(n_{i} = 5\): \[\frac{1}{\lambda_2} = R_H \left(\frac{1}{3^2} - \frac{1}{5^2}\right)\] \[\lambda_2 = \frac{1}{R_H \left(\frac{1}{9} - \frac{1}{25}\right)}\] (c) For \(n_{i} = 6\): \[\frac{1}{\lambda_3} = R_H \left(\frac{1}{3^2} - \frac{1}{6^2}\right)\] \[\lambda_3 = \frac{1}{R_H \left(\frac{1}{9} - \frac{1}{36}\right)}\] Now, we can calculate the wavelengths for each line:

Calculate Wavelengths

(a) For \(n_{i} = 4\): \[\lambda_1 = \frac{1}{1.097 \times 10^7 \, \mathrm{m^{-1}} \left(\frac{1}{9} - \frac{1}{16}\right)} = 1.87 \times 10^{-6} \, m\] (b) For \(n_{i} = 5\): \[\lambda_2 = \frac{1}{1.097 \times 10^7 \, \mathrm{m^{-1}} \left(\frac{1}{9} - \frac{1}{25}\right)} = 1.28 \times 10^{-6} \, m\] (c) For \(n_{i} = 6\): \[\lambda_3 = \frac{1}{1.097 \times 10^7 \, \mathrm{m^{-1}} \left(\frac{1}{9} - \frac{1}{36}\right)} = 1.09 \times 10^{-6} \, m\] These wavelengths correspond to the Infrared region of the electromagnetic spectrum.

Final Results

(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: - \(n_{i} = 4 \rightarrow \lambda_1 = 1.87 \times 10^{-6} \, m\) - \(n_{i} = 5 \rightarrow \lambda_2 = 1.28 \times 10^{-6} \, m\) - \(n_{i} = 6 \rightarrow \lambda_3 = 1.09 \times 10^{-6} \, m\)

Key concepts

Rydberg Formula

The Rydberg formula is a fundamental equation used to predict the wavelength of light resulting from an electron moving between energy levels in a hydrogen atom. Named after the Swedish physicist Johannes Rydberg, this formula is vital for understanding atomic spectra. Let's break down the components:

The formula itself is: \[\begin{equation}\frac{1}{\lambda} = R_H \left(\frac{1}{n_f^2} - \frac{1}{n_i^2}\right)\end{equation}\]
Where \( \lambda \) represents the wavelength of the emitted light, \( R_H \) is the Rydberg constant, and \( n_f \) and \( n_i \) are the final and initial quantum numbers, respectively. Importantly, this formula indicates that light emission occurs in discrete wavelengths, reflecting the quantum nature of electrons within an atom. This foundational concept helps in predicting the color and energy of light that an electron emits as it transitions between different energy levels.

Hydrogen Emission Spectrum

When we explore the hydrogen emission spectrum, we dive into the different series of lines that are produced by the transitions of electrons. One such series is the Paschen series, which is associated with the transitions down to the third orbit, or energy level. The hydrogen atom has one electron orbiting the nucleus, and when this electron absorbs energy, it jumps to higher energy orbits. Upon returning to lower energy orbits, it emits light at certain wavelengths, creating the spectral lines unique to hydrogen.

These spectral lines are part of the electromagnetic spectrum and can be categorized into series depending on the energy transitions involved. For the Paschen series specifically, the emitted lines fall within the infrared spectrum, suggesting that they are beyond the visible range for human eyes. This behavior is governed by quantized energy levels, a principle central to quantum mechanics and an exquisite reflection of the discrete nature of atomic energy changes.

Quantum Numbers

To comprehend quantum numbers, imagine them as the 'address' of an electron within an atom. Each electron in an atom is described by a unique set of quantum numbers, which include the principal quantum number (\( n \)), angular momentum quantum number (\( l \)), magnetic quantum number (\( m \)), and spin quantum number (\( s \)). In the context of the Rydberg formula and the hydrogen emission spectrum, the principal quantum number (\( n \)) plays a starring role.

It's \( n \) that determines the energy level of the electron: the higher the value of \( n \), the further away from the nucleus and the higher the energy orbit. Quantum numbers are pivotal in predicting and explaining the electronic structure of atoms, which directly influences the emission spectrum of an atom. In the case of the Paschen series, for example, transitions to the \( n = 3 \) orbit from higher orbits (\( n_i = 4, 5, 6 \)) give us insight into the infrared emissions characteristic of this series.

Infrared Spectrum

The infrared spectrum refers to a portion of the electromagnetic spectrum that has longer wavelengths than visible light, typically ranging from 700 nm to 1 mm. This type of radiation is not detectable by the human eye, but we can feel it as heat. Speaking about the Paschen series, these transitions to the third energy level fall within the infrared range, confirming that the emitted photons have lower energy compared to visible light.

In practical applications, the infrared spectrum is used in numerous technologies, including remote controls, night-vision equipment, and astronomical observations. When studying the Paschen series, it's fascinating to discover that the seemingly invisible light these atoms emit plays such an integral role in both the natural world and our technological advancements. By learning about infrared emissions from hydrogen, students can begin to appreciate the broader implications of this kind of radiation.