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 \(\mathbf{n}\) 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: Balmer's relation, \(\lambda = \frac{364.5 n_i^2}{n_i^2 -4}\), is derived from the Bohr energy equation, \(E_n = -\frac{13.6 \mathrm{eV}}{n^2}\), and the Rydberg formula, \(\frac{1}{\lambda} = R \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right)\). The derivation involves combining these equations through energy conservation and Planck's equation, resulting in a formula for the wavelength of emitted photons in the Balmer series.

Step-by-step solution

Write the Bohr's energy equation

The energy of an electron in any energy level \((n)\) in the hydrogen atom according to Bohr's theory is: $$ E_n = -\frac{13.6 \mathrm{eV}}{n^2} $$

Write the equation involving the Rydberg constant

In the case of hydrogen, the Rydberg formula for the wavelength \((\lambda)\) of emitted photons is given by: $$ \frac{1}{\lambda} = R \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right) $$ where \(R \approx 1.097 \times 10^7 \mathrm{m}^{-1}\) is the Rydberg constant, \(n_1\) and \(n_2\) are the initial and final energy levels, respectively (with \(n_2 > n_1\)).

Balmer series - return to n=2 level

In Balmer series, the electron returns to the \(n=2\) level. Let's denote the initial energy level as \(n_i\). Then \(n_1 = n_i\) and \(n_2=2\). Substitute these values in the equation from step 2: $$ \frac{1}{\lambda} = R \left( \frac{1}{n_i^2} - \frac{1}{2^2} \right) $$

Calculate the energy difference using the Bohr's energy equation

The energy difference between the two energy levels is the difference between the energies of the levels (\(\Delta E\)): $$ \Delta E = E_2 - E_{n_i} $$ Using Bohr's energy equation, we get: $$ \Delta E = -\frac{13.6 \mathrm{eV}}{2^2} - \left( -\frac{13.6 \mathrm{eV}}{n_i^2} \right) $$

Relate the energy difference to the wavelength using the Planck's equation

The energy difference between the two levels is equal to the energy of the emitted photon: $$ \Delta E = h \cdot \nu $$ where \(h\) is the Planck's constant and \(\nu\) is the frequency of the photon. We can rewrite the frequency as \(\nu = \frac{c}{\lambda}\), where \(c\) is the speed of light. $$ \Delta E = h\frac{c}{\lambda} $$

Combine the equations to find Balmer's relation

First, rewrite the equation from Step 5 in terms of \(\lambda\): $$ \lambda = \frac{hc}{\Delta E} $$ We have found \(\Delta E\) in Step 4. Substitute this expression into the above equation: $$ \lambda = \frac{hc}{ -\frac{13.6 \mathrm{eV}}{2^2} - \left( -\frac{13.6 \mathrm{eV}}{n_i^2} \right)} $$ Simplify this equation, and convert the Planck's constant times the speed of light from Joule meters to electron volt nanometers (1 eV = 1240 nm): $$ \lambda = \frac{364.5 n_i^2}{n_i^2 -4} $$ This is the Balmer's relation. We can see that it is derived from the Bohr theory and the equation involving the Rydberg constant.

Key concepts

Bohr Model

The Bohr Model, introduced by Niels Bohr in 1913, was a significant advancement in understanding atomic structure. It describes the hydrogen atom with the following key points:

• Electrons orbit the nucleus in specific circular paths, or energy levels, much like planets orbit the sun.
• Each of these orbits corresponds to a distinct energy level, characterized by a quantum number, n.
• Electrons in these orbits do not lose energy, hence explaining the stability of atoms.

When an electron jumps from a higher energy orbit to a lower one, it emits energy in the form of a photon. This release of energy is responsible for the spectral lines observed in the hydrogen spectrum.
Understanding this model helps us see why specific wavelengths appear in the hydrogen spectrum during electron transitions.

Rydberg Formula

The Rydberg Formula is crucial for calculating the wavelengths of light emitted when electrons move between energy levels in atoms. Named after the Swedish physicist Johannes Rydberg, this formula is particularly useful for hydrogen.
Mathematically, it is expressed as:
\[\frac{1}{\lambda} = R \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right)\]
where:

• \(\lambda\) is the wavelength of emitted light.
• \(R\) is the Rydberg constant, approximately \(1.097 \times 10^7 \text{ m}^{-1}\).
• \(n_1\) and \(n_2\) are integers representing the electron's initial and final energy levels, respectively.

In the Balmer series context, this formula helps calculate visible wavelengths emitted when an electron transitions to the \(n=2\) energy level from a higher one.

Hydrogen Spectrum

The Hydrogen Spectrum is the range of electromagnetic radiation emitted by hydrogen atoms. When an electron moves from a higher energy level to a lower energy state, it releases energy as light.
This spectrum includes several series, each named after the scientist who discovered them, such as:

• Lyman Series: Transitions ending at \(n=1\), observed in the ultraviolet region.
• Balmer Series: Transitions ending at \(n=2\), seen in visible light.
• Paschen Series: Transitions ending at \(n=3\), found in the infrared region.

Each line in the spectrum corresponds to a specific transition. By analyzing these spectral lines, scientists gain insights into the energy levels and behavior of electrons within an atom.

Energy Levels

Energy levels in an atom are fixed distances away from the nucleus where electrons can reside. In the Bohr model, these are considered circular orbits.
Key details include:

• Each energy level is identified by a principal quantum number, \(n\).
• The energy of each level can be calculated using Bohr's formula: \(E_n = -\frac{13.6 \text{ eV}}{n^2}\).
• Lower \(n\) values correspond to more stable, lower energy levels.

Transitioning between these energy levels involves either absorbing or emitting photons.
This change results in energy shifts that produce observable effects, like the spectral lines seen in the hydrogen spectrum. Understanding these levels allows us to predict electron behavior and explain spectral phenomena.