Atomic spectra#

What You Need to Know

  • Atomic Spectroscopy measures the frequencies or wavelengths of radiation absorbed or emitted by atoms.

  • Spectra of Atoms show discrete lines, indicating that atoms absorb or emit only specific, few frequencies. In other words, the energies of atoms are quantized.

  • Bohr’s Theory: Bohr attempted to explain atomic spectra by combining classical mechanics with the concept of quantization. For the simplest hydrogen atom, Bohr’s theory worked perfectly, providing a closed formula that explains all the spectral lines of the hydrogen atom.

  • Limitation of Bohr’s Theory: However, for atoms with more than one electron, Bohr’s theory fails to generalize, leading scientists to develop a more rigorous theory—Quantum Mechanics.

Spectroscopy of atoms#

  • When heated or subjected to electrical discharge, atoms emit radiation of characteristic frequencies. The spectrum from each atom is unique.

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Fig. 19 Atomic spectorscopy of hydrogen atom. hydrogen in gas discharge tube radiates discrete wavelengts which can be detected as discrete lines by passing the radiation through prism.#

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Fig. 20 Using spectorscopy one could detect presence of different elements in the sun.#

Spectral lines and Rydberg’s formula#

  • The existance of discrete spectral lines are impossible to describe with classical mechanics. In 1885, Johann Blamer demonstrated that a subset of the hydrogen atom spectrum (the Balmer series) could be described by the equation

\[v = 8.2202\times10^{14}\left(1-\frac{4}{n^2}\right)\]

where \(n=3,4,5,...\). Later, Johannes Rydberg generalized this formula to account for the entire hydrogen atom spectrum yielding the Rydberg formula

Rydberg formula

\[\tilde{v} = R_H\left(\frac{1}{n_1^2}-\frac{1}{n_2^2}\right)\]

where

  • \(R_H = 1.097 \times 10^7 \ \text{m}^{-1}\) is the Rydberg constant.

  • \(n_1 = 1,2,3,...\), and \(n_2 = n_1+1,n_1+2,...\).

  • While these equations fit the hydrogen atom spectrum nicely, they do not prescribe any physics to the system. They do not present a model of the hydrogen atom but rather a heuristic equation that fits the data. Nonetheless, scientists were perplexed by the presence of the integers \(n_1\) and \(n_2\).

atomic series

Fig. 21 Atomic spectral lines are named after their disocerers. Each series contains all transitions to a distinct ground or excited state level \(n=1,2,3\).#

Bohr’s model of H atom#

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Fig. 22 Evolution of atomic model from pre-quantum to present day complete quantum theory of atoms.#

  • In 1911, Niels Bohr proposed a model for the hydrogen atom that was able to recapitulate the hydrogen atom spectrum.

  • The model consists of an electron orbiting a proton in circular orbits. The proton is considered to be fixed in space because it is so much more massive than the electron.

  • Most importantly Bohr had to introduce new ad-hoc requirements to keep the electron stable. Namely the electron demonstrates wavelike characteristics which has an integer number \(n=1,2,3,...\) of modes around the circular orbit.

  • Thanks to this quantization rule expression for the H atom energy is obtained which is a function of an integer number \(n=1,2,3,...\)

FrigginBohr

Fig. 23 A man goes to visit Niels Bohr, and sees a horseshoe hanging over Bohr’s door (a scandinavian superstition). The man says, “But Niels, you are a scientist! Surely you do not believe in this superstition?” Niels replies, “Of course I don’t believe in it!” The man is confused. “Why do you have it if you don’t believe in it?” Bohr replies, “It is supposed to work, even if you don’t believe in it!”#

Quantizes states of electron#

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Fig. 24 One could rationalize discrete Bohr’s orbits by the integer number of wavelengths one could fit into the orbit.#

Restricitng waves with wavelength \(\lambda\) to sit around orbit separated by integer number yields

\[\begin{equation} 2\pi r = n\lambda_e \quad n=1,2,3,... \end{equation}\]

where \(\lambda_e\) is the deBroglie wavelength of an electron and can be written as

\[\begin{equation} \lambda_e = \frac{h}{m_ev}. \end{equation}\]

Plugging the deBroglie wavelength equation we get first equation that electron in H atom should satisfy.

\[\begin{equation} m_evr = \frac{nh}{2\pi} = n\hbar, \end{equation}\]

We have introduce \(\hbar = \frac{h}{2\pi}\) as a short-hand because it comes up frequently in quantum mechanics. The term on the left-hand side of the last equation, \(m_evr\), is the angular momentum of the electron. Thus Bohr ‘s model demonstrates a quantization of the angular momentum of the electron.

Force balance#

After introducing ad-hoc quantization rule, Bohr’s model then resorts to classical mechanics to obtain energy function. Bohr posited that for stationary states of the electron the electrostatic force between the proton and electron must match centrifugal force

Electrostatic force

\[\begin{equation} f_{el} = \frac{e^2}{4\pi\varepsilon_0r^2} \end{equation}\]
  • where \(4\pi\varepsilon_0\) is present to achieve SI units.

Fentrifugal force

\[\begin{equation} f_{cf} = \frac{m_ev^2}{r} \end{equation}\]
  • where \(m_e\) is the mass and \(v\) is the velocity of the electron. Equating these two forces yields

\[\begin{equation} \frac{e^2}{4\pi\varepsilon_0r^2} = \frac{m_ev^2}{r}. \end{equation}\]
  • The combination of the force balance equation and the quantized angular momentum equation quantizes the values of \(r\), the radius of the electron’s circular orbit, that can be taken. To demonstrate this we solve the quantized angular momentum equation for \(v\) and plug the result into the force balance equation and solve for \(r\):

\[\begin{split}\begin{align} \frac{e^2}{4\pi\varepsilon_0r^2} &= \frac{m_e\left( \frac{n\hbar}{m_er}\right)^2}{r} \\ \Rightarrow \frac{e^2}{4\pi\varepsilon_0} &= \frac{(n\hbar)^2}{m_er} \\ \Rightarrow e^2m_er &= 4\pi\varepsilon_0(n\hbar)^2 \\ \Rightarrow r &= \frac{4\pi\varepsilon_0(n\hbar)^2}{e^2m_e} = n^2 \cdot a_0 \quad n=1,2,3,... \end{align}\end{split}\]
  • The radius of the first Bohr orbit is denoted \(a_0 = \frac{4\pi\varepsilon_0\hbar^2}{e^2m_e}\) or units of Bohr.

Energy of H atom#

The energy of the system can is a sum of the Coulomb attraction between the electron and the proton and the kinetic energy of the electron:

\[\begin{equation} E(r) = \frac{1}{2}m_ev^2 - \frac{e^2}{4\pi\varepsilon_0r} \end{equation}\]

To determine the energy of an electron that is limited to be in the circular wavelike orbits described above, we must use the force balance relationship. We do that by substituting \(m_ev^2 = \frac{e^2}{4\pi\varepsilon_0r}\) into the energy equation to yield

\[\begin{split}\begin{align} E(r) &= \frac{1}{2}\frac{e^2}{4\pi\varepsilon_0r} - \frac{e^2}{4\pi\varepsilon_0r} \\ &= -\frac{1}{2}\frac{e^2}{4\pi\varepsilon_0r} \\ &= -\frac{1}{2}\frac{e^2}{4\pi\varepsilon_0}\frac{e^2m_e}{4\pi\varepsilon_0(n\hbar)^2} \\ &= -\frac{m_ee^4}{32\pi^2\varepsilon_0\hbar^2}\frac{1}{n^2} \\ &= -\frac{m_ee^4}{8\varepsilon_0^2h^2}\frac{1}{n^2} \quad n=1,2,3,... \end{align}\end{split}\]

where we plugged in the quantized values for \(r\) derived from the for balance relationship. Taking differences in energy between two energy levels, \(n_1\) and \(n_2>n_1\), yields

\[\begin{equation} \Delta E = \frac{m_ee^4}{8\varepsilon_0^2h^2}\left(\frac{1}{n_1^2} - \frac{1}{n_2^2}\right) \end{equation} \]

Equating this to the frequency (use \(E = h\nu\) and \(\tilde{\nu} = \frac{\nu}{c}\)) of emitted light yields:

\[\begin{equation} \tilde{v} = \frac{m_ee^4}{8\varepsilon_0^2ch^3}\left(\frac{1}{n_1^2} - \frac{1}{n_2^2}\right) = R_H \left(\frac{1}{n_1^2} - \frac{1}{n_2^2}\right) \end{equation}\]

We see that this yields an expression for the Rydberg constant in terms of fundamental constants \(R_H = \frac{m_ee^4}{8\varepsilon_0^2ch^3}\)

Problems#

Problem 1#

The so-called Lyman series of lines in the emission spectrum of hydrogen corresponds to transitions from various excited states to the n = 1 orbit. Calculate the wavelength of the lowest-energy line in the Lyman series to three significant figures. In what region of the electromagnetic spectrum does it occur?

Problem 2#

  • A. Calculate the energy of a photon that is produced when an electron in a hydrogen atom goes from an orbit with n=4 to and orbit with \(n=1\)

  • B. What happens to the energy of the photon as the initial value of \(n\) approaches infinity?

Problem 3#

Use Rydberg’s formula to calculate firs few lines of Lymann series (\(n_1=1\))

Problem 4#

A line in the Lymann series of hydrogen has a wavelength of \(1.03 \cdot 10^{-7} m\) Find the original level of the electron.

Problem 5#

Using Bohr theory calculate ionization energy of singly ionized helium \(He^{+}\)

Problem 6#

  • Calculate radii of Bohr orbit for first few levels.

  • (Optional) Using python plot \(r_n\) vs \(n\)