Problem 42 - Chemistry The Central Science

Question

The Lyman series of emission lines of the hydrogen atom are those for which \(n_{1}=1 .\) (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 \(n_{1}=2,3,\) and \(4 .\)

Step-by-Step Solution

Answer

The Lyman series lines are observed in the ultraviolet region of the electromagnetic spectrum. The wavelengths of the first three lines in the Lyman series are approximately \(121.54\) nm, \(102.57\) nm, and \(97.243\) nm, respectively.

Step 1: (Hydrogen spectrum formula)

The formula we will use to determine the wavelength of hydrogen spectrum lines is known as the Rydberg formula: \[\frac{1}{\lambda} = R_H \left(\frac{1}{n_1^2} - \frac{1}{n_2^2}\right)\] Here, \(\lambda\) is the wavelength of the spectral line, \(R_H\) is the Rydberg constant for hydrogen (approximately \(1.09737\times10^7\) m\(^{-1}\)), \(n_1\) and \(n_2\) are positive integers representing the energy levels, with \(n_2 > n_1\). For the Lyman series, \(n_1=1\) so we will be plugging this value into the formula to get the required wavelengths.

Step 2: (Determine the region of the electromagnetic spectrum)

To find the region of the electromagnetic spectrum in which the Lyman series lines are observed, we need to find the wavelengths associated with \(n_2 \to \infty\), which gives the shortest wavelength of the series (since shorter wavelength means higher frequency and energy). Plugging \(n_1=1, n_2=\infty\) into the Rydberg formula, we have: \[\frac{1}{\lambda_\text{min}} = R_H \left(\frac{1}{1^2} - \frac{1}{\infty}\right) = R_H\] Thus, the minimum wavelength (\(\lambda_\text{min}\)) of the Lyman series is: \[\lambda_\text{min} = \frac{1}{R_H} \approx \frac{1}{1.09737\times10^7\,\text{m}^{-1}} \approx 91.175\times10^{-9}\,\text{m} \] Since this series lies in the ultraviolet region of the electromagnetic spectrum, we can say that the lines of the Lyman series are observed in the ultraviolet region.

Step 3: (Calculate the wavelengths of the first three lines in the Lyman series)

To find the wavelengths of the first three lines in the Lyman series, we need to calculate the wavelengths for \(n_2=2\), \(3\), and \(4\). 1. For \(n_2=2\): \[\frac{1}{\lambda_1} = R_H \left(\frac{1}{1^2} - \frac{1}{2^2}\right)\] \[\lambda_1 =\frac{1}{R_H\left(\frac{3}{4}\right)}\approx \frac{1}{(1.09737\times10^7\,\text{m}^{-1})\left(\frac{3}{4}\right)} \approx 121.54\times10^{-9}\,\text{m} \] 2. For \(n_2=3\): \[\frac{1}{\lambda_2} = R_H \left(\frac{1}{1^2} - \frac{1}{3^2}\right)\] \[\lambda_2 =\frac{1}{R_H\left(\frac{8}{9}\right)}\approx \frac{1}{(1.09737\times10^7\,\text{m}^{-1})\left(\frac{8}{9}\right)} \approx 102.57\times10^{-9}\,\text{m} \] 3. For \(n_2=4\): \[\frac{1}{\lambda_3} = R_H \left(\frac{1}{1^2} - \frac{1}{4^2}\right)\] \[\lambda_3 =\frac{1}{R_H\left(\frac{15}{16}\right)}\approx \frac{1}{(1.09737\times10^7\,\text{m}^{-1})\left(\frac{15}{16}\right)} \approx 97.243\times10^{-9}\,\text{m} \] Therefore, the wavelengths of the first three lines in the Lyman series are approximately \(121.54\) nm, \(102.57\) nm, and \(97.243\) nm, respectively.

Key Concepts

Rydberg Formula

The Rydberg formula is a fundamental equation in atomic physics, used to predict the wavelengths of spectral lines of hydrogen. It is expressed as follows:\[ \frac{1}{\lambda} = R_H \left(\frac{1}{n_1^2} - \frac{1}{n_2^2}\right) \]where:

• \(\lambda\) is the wavelength of the light emission.
• \(R_H\) is the Rydberg constant, approximately \(1.09737\times10^7\, \text{m}^{-1}\).
• \(n_1\) and \(n_2\) are integers, with \(n_2 > n_1\).

The formula is particularly useful for identifying transitions between different energy levels in a hydrogen atom. For the Lyman series, \(n_1 = 1\) and \(n_2\) can be any integer greater than \(1\), which allows us to calculate the wavelengths of the emitted light during these transitions.

Electromagnetic Spectrum

The electromagnetic spectrum encompasses all types of electromagnetic radiation, categorized by their wavelengths or frequencies.

• Waves on the spectrum range from extremely long radio waves to very short gamma rays.
• Humans can only see light within the visible spectrum, which is a small part of the range.

Each type of radiation on the electromagnetic spectrum has unique properties and applications. For example, radio waves are used for communication, and X-rays are used in medical imaging. In the case of hydrogen emission lines, their specific placement within the spectrum helps in understanding atomic structure and energy transitions.

Ultraviolet Region

The ultraviolet (UV) region is a part of the electromagnetic spectrum with wavelengths shorter than visible light but longer than X-rays.

• UV wavelengths range from about 10 nm to 400 nm.
• The Lyman series of hydrogen emission lines falls within this range, specifically in the ultraviolet region.

This part of the spectrum is beyond human vision but can affect materials and organisms (such as causing suntans and sunburns). Therefore, the ultraviolet region is essential in both nature and technology, influencing processes like fluorescence and disinfection.

Hydrogen Emission Lines

Hydrogen emission lines are the specific wavelengths of light that hydrogen atoms emit when electrons fall to lower energy levels.

• When an electron transitions from a higher energy level to a lower one, it releases energy in the form of light.
• Each series of emission lines corresponds to electrons falling to a particular final energy level.

The Lyman series, for example, involves transitions where electrons fall to the first energy level (\(n_1=1\)). These emission lines are significant because they help scientists understand quantum mechanics and atomic structures. By studying these lines, researchers learn about electron energy levels and can even determine properties of distant stars.