Question

In 1885 , Johann Balmer, a mathematician, derived the following relation for the wavelength of lines in the visible spectrum of hydrogen $$ \lambda=\frac{364.5 \mathrm{n}^{2}}{\left(\mathrm{n}^{2}-4\right)} $$ where \(\lambda\) is in nanometers and \(n\) is an integer that can be 3,4 , \(5, \ldots\) Show that this relation follows from the Bohr equation and the equation using the Rydberg constant. Note that in the Balmer series, the electron is returning to the \(\mathbf{n}=2\) level.

Short answer

Question: Show that Balmer's relation follows from the Bohr equation and the Rydberg constant. Answer: We have shown that the Balmer's relation of the hydrogen atom can be derived from the Bohr equation and the Rydberg constant by applying the conditions of the Balmer series (returning to the n1=2 energy level in a hydrogen atom) and substituting the Rydberg constant value. Thus, Balmer's relation does follow from the Bohr equation and the Rydberg constant.

Step-by-step solution

Recalling Balmer's relation

We know that Balmer's relation for the wavelength of lines in the visible spectrum of hydrogen atom is $$ \lambda=\frac{364.5 n^{2}}{\left(n^{2}-4\right)} $$ #Step 2: Write down the Rydberg's formula for hydrogen-like species#

Recalling the Rydberg's formula

The Rydberg's formula for hydrogen-like species' spectral lines wavelength is $$ \frac{1}{\lambda}=RZ^2\left(\frac{1}{n_1^2}-\frac{1}{n_2^2}\right) $$ #Step 3: Apply the conditions for the Balmer series#

Applying Balmer series conditions

In the Balmer series, the electron returns to the \(n_1=2\) energy level in a hydrogen atom (\(Z=1\)). We have to plug these values into the Rydberg's formula. $$ \frac{1}{\lambda}=R\left(\frac{1}{2^2}-\frac{1}{n_2^2}\right) $$ #Step 4: Rearrange the equation to get the wavelength expression#

Rearrange the equation

Rearranging the Rydberg's formula to solve for \(\lambda\): $$ \lambda=\frac{1}{R\left(\frac{1}{4}-\frac{1}{n_2^2}\right)} $$ #Step 5: Substitute the Rydberg constant value to get the final expression#

Substitute the Rydberg constant

Now, substituting the value of Rydberg constant \(R \approx 1.097 \times 10^7 m^{-1}\) in the equation and converting from meters to nanometers, $$ \lambda=\frac{10^9}{1.097\times10^7\left(\frac{1}{4}-\frac{1}{n_2^2}\right)} $$ $$ \lambda=\frac{364.5n_2^{2}}{n_2^{2}-4} $$ Since this equation is the same as the given Balmer's relation, it has been shown that the Balmer series follows from the Bohr equation and the Rydberg constant.

Key concepts

Bohr model

The Bohr model, introduced by Niels Bohr in 1913, revolutionized the understanding of atomic structure by incorporating quantum theory. This model positions electrons in discrete energy levels, or shells, around the nucleus, which can only hold a certain number of electrons. An electron can transition between these levels by absorbing or emitting a photon of a specific wavelength, which corresponds to the energy difference between these levels.

In the context of the Balmer series, the Bohr model explains the appearance of specific spectral lines as electrons fall from higher energy levels to the second energy level (n=2). These transitions result in the emission of light in the visible spectrum, which is a central aspect of the hydrogen spectrum. Ensuring students grasp this concept is crucial as it establishes a foundation for understanding more complex atomic behavior.

Rydberg formula

The Rydberg formula describes the wavelengths of the spectral lines of many chemical elements. For hydrogen, the formula is particularly simple and has the form:
\[\begin{equation}\frac{1}{\lambda} = RZ^2 \left(\frac{1}{n_1^2} - \frac{1}{n_2^2}\right)\end{equation}\]
where \(\lambda\) is the wavelength of the emitted or absorbed light, \(R\) is the Rydberg constant (approximately 1.097 x 10^7 m^-1 for hydrogen), \(Z\) is the atomic number, \(n_1\) is the lower energy level, and \(n_2\) is the higher energy level.

The formula is used to predict the wavelength of light resulting from an electron moving between energy levels of a hydrogen atom. This demonstrates a direct relationship between the observed spectral lines and the structure of an atom, which is integral when studying atomic spectra.

Hydrogen spectrum

The hydrogen spectrum is a series of lines that represent the wavelengths of light emitted or absorbed by electrons transitioning between energy levels in a hydrogen atom. These spectral lines are not continuous but discrete, concurring with the quantum nature of energy levels proposed by the Bohr model.

The Balmer series, only a part of the hydrogen spectrum, involves transitions to the second energy level and produces visible light. When introducing students to the hydrogen spectrum, emphasize the importance of the Balmer series, as it was one of the first sets of spectral lines to be understood and described in the framework of early quantum theory. Understanding the hydrogen spectrum is foundational to concepts such as energy quantization and photon absorption/emission in atomic physics.

Wavelength calculation

Wavelength calculation in the context of atomic spectra involves applying the appropriate formula, such as the Rydberg formula, to determine the wavelengths of light associated with electron transitions. The exercise provided deals with determining the wavelengths of the Balmer series, where the final formula for the wavelength is:\[\begin{equation}\lambda = \frac{364.5n_2^{2}}{n_2^{2}-4}\end{equation}\]
In these equations, it's essential to understand the role of terms such as \(n_2\), which indicates the energy level to which the electron is transitioning. Also, converting units from meters to nanometers can be necessary since the wavelengths of visible light are typically expressed in nanometers. To aid in students' comprehension, real-world examples of wavelength calculation can demonstrate the practical application of this concept in fields like astrophysics and telecommunications.